Big Ideas Math

Big Ideas Math Answers 2nd Grade 1st Chapter Numbers and Arrays PDF is provided here. We have given the solutions to all the questions in Big Ideas Math Answers Grade 2 Chapter 1 Numbers and Arrays in pdf format. BIM Grade 2 Textbook Chapter 1 Numbers and Arrays Answer Key helps the students to complete their homework in time and also to enhance their knowledge. It will also useful to learn the concepts in depth.

Download Big Ideas Math Book 2nd Grade Answer Key Chapter 1 Numbers and Arrays PDF

Apply the real-time math examples by learning the tricks using BIM Grade 2 Chapter 1 Numbers and Arrays. Have a look at the topics before you start the preparation. The quick method of solving math questions si helpful to the students to save their time. We have provided the answers for each and every section of chapter 1 numbers and arrays in the following sections.

The topics covered in Bid Ideas Math Book Grade 2 Answer Key Chapter 1 Numbers and Arrays are Even and Odd Numbers, Model Even and Odd Numbers, Equal Groups, Use Arrays, and Make Arrays. The performance task given at the end helps you to test your skills. Make use of the below links and learn the basic topics covered here.

Vocabulary

• Numbers and Arrays Vocabulary

Lesson 1: Even and Odd Numbers

• Lesson 1.1 Even and Odd Numbers
• Even and Odd Numbers Home & Practice 1.1

Lesson 2: Model Even and Odd Numbers

• Lesson 1.2 Model Even and Odd Numbers
• Model Even and Odd Numbers Homework & Practice 1.2

Lesson 3: Equal Groups

• Lesson 1.3 Equal Groups
• Equal Groups Homework & Practice 1.3

Lesson 4: Use Arrays

• Lesson 1.4 Use Arrays
• Use Arrays Homework & Practice 1.4

Lesson 5: Make Arrays

• Lesson 1.5 Make Arrays
• Make Arrays Homework & Practice 1.5

Performance Task

• Numbers and Arrays Performance Task
• Numbers and Arrays Activity
• Numbers and Arrays Chapter Practice

Numbers and Arrays Vocabulary

Organize It

Use the review words to complete the graphic organizer.

Answer: Plus sign
‘+’ is the mathematical symbol used to represent the notion of positive as well as the operation of addition.

Define It

Use your vocabulary cards to identify the word.

Question 1.

Answer: Even

Question 2.

Answer: Even

Question 3.

Answer: Odd

Chapter 1 Vocabulary Cards

Lesson 1.1 Even and Odd Numbers

Explore and Grow

Use linking cubes to model each story.
There are 6 students in the gym. Does each student have a partner?
There are 5 students in the library. Does each student have a partner?

Think and Grow

Show and Grow

Question 1.

Answer: Even

Explanation: There are 6 parts, An even number can be shown as 2 equal parts.

Question 2.

Answer: Odd

Explanation: There are 9 parts, An odd number cannot be shown as 2 equal parts.

Color cubes to show the number. Circle even or odd.

Question 3.

Answer: Odd

Explanation: 11 is odd number and cannot be shown as 2 equal parts.

Question 4.

Answer: Even

Explanation: 16 is even number and can be shown as 2 equal parts.

Apply and Grow: Practice

Color cubes to show the number. Circle even or odd.

Question 5.

Answer: Odd

Explanation: 13 is odd number and cannot be shown as 2 equal parts.

Question 6.

Answer: Even

Explanation: 10 is even number and can be shown as 2 equal parts.

Is the number even or odd?

Question 7.

Answer: Odd

Explanation: 1 is odd number and cannot be shown as 2 equal parts.

Question 8.

Answer: Even

Explanation: 4 is even number and can be shown as 2 equal parts.

Question 9.

Answer: Even

Explanation: 18 is even number and can be shown as 2 equal parts.

Question 10.

Answer: Odd

Explanation: 17 is odd number and cannot be shown as 2 equal parts.

Question 11.

Answer: Odd

Explanation: 19 is odd number and cannot be shown as 2 equal parts.

Question 12.

Answer: Even

Explanation: 20 is even number and can be shown as 2 equal parts.

Question 13.
Number Sense
Circle even or odd to describe each group. Then write each number in the correct group.

Answer:

Think and Grow: Modeling Real Life

There is an even number of students in your class. There are more than 16 but fewer than 20 students. How many students are in your class?

Answer: 18 Students
Explanation: Between 16 and 20, Even number is 18.

Show how you know:

Show and Grow

Question 14.
There is an odd number of cows in a field. There are more than 13 but fewer than 17 cows. How many cows are in the field?

Answer: 15 cows
Explanation: Between 13 and 17, Odd number is 15.

Question 15.
There are 14 geese on a farm. There are 2 more chickens than geese. Is there an even or odd number of chickens?

Answer: Even
Explanation: There are total 16 Chickens on a farm, which is Even number.

Even and Odd Numbers Home & Practice 1.1

Question 1.

Answer: Odd

Explanation: Total 11 parts, which cannot be shown as 2 equal parts.

Question 2.

Answer: Even

Explanation: Total 10 parts, which can be shown as 2 equal parts.

Color cubes to show the number. Circle even or Odd

Question 3.

Answer: Odd

Explanation: 9 is Odd number which cannot be shown in 2 equal parts.

Question 4.

Answer: Even

Explanation: 14 is Even number which can be shown in 2 equal parts.

Question 5.

Answer: Even

Explanation: 18 is Even number which can be shown in 2 equal parts.

Question 6.

Answer: Odd

Explanation: 15 is Odd number which cannot be shown in 2 equal parts.

Is the number even or odd?

Question 7.

Answer: Even

Explanation: 2 is Even number which can be shown in 2 equal parts.

Question 8.

Answer: Odd

Explanation: 5 is Odd number which cannot be shown in 2 equal parts.

Review & Refresh

Question 9.
DIG DEEPER!
You break apart a linking cube train to make two equal parts. There is 1 cube left over. Is the number of cubes even or odd? Explain.

Answer: Odd
Explanation: There are 5 parts in cube train, when it is broken in 2 equal parts then there is 1 cube left (2+2+1) which is Odd number.

Question 10.
Modeling Real Life
There is an even number of eggs in a nest. There are more than 10 but fewer than 14 eggs. How many eggs are in the nest?

Answer: 12 eggs
Explanation: 12 is even number and also between 10 and 14, 12 is the only even number.

Question 11.
Modeling Real Life
You have 6 green crayons. You have 1 more blue crayon than green crayons. Do you have an even or an odd number of blue crayons?

Answer: Odd

Explanation:  Total 7 blue crayon which is Odd number so it cannot be shown in 2 equal parts.

Review & Refresh

Question 12.
6 + 6 = ___
Answer: 12
Explanation: 12 is Even number which can be shown in 2 equal parts.

Question 13.
9 + 9 = ___
Answer: 18
Explanation: 18 is Even number which can be shown in 2 equal parts.

Question 14.
7 + 7 = ___
Answer: 14
Explanation: 14 is Even number which can be shown in 2 equal parts.

Question 15.
8 + 8 = ___
Answer: 16
Explanation: 16 is Even number which can be shown in 2 equal parts.

Lesson 1.2 Model Even and Odd Numbers

Explore and Grow

Use linking cubes to model each sum. Is the sum even or odd?

Answer: Even

Explanation: 4+4=8, 8 is Even number which can be shown in 2 equal parts.

Answer: Odd

Explanation: 5+4=9, 9 is Odd number which cannot be shown in 2 equal parts.

Show and Grow

Question 1.

Answer: Odd

Explanation: 7=4+3, Odd number

Question 2.

Answer: Even

Explanation: 10=5+5, Even number

Question 3.

Answer: Even

Explanation: 14=7+7, Even number

Question 4.

Answer: Odd

Explanation: 17=9+8, Odd number

Question 5.

Answer: Even

Explanation: 18=9+9, Even number

Question 6.

Answer: Odd

Explanation: 13=7+6, Odd number

Question 7.

Answer: Odd

Explanation: 15=8+7, Odd number

Question 8.

Answer: Even

Explanation: 20=10+10, Even number

Question 9.
YOU BE THE TEACHER
Descartes uses doubles plus 1 to model an odd number. Is he correct? Explain.

Answer: Yes
Explanation: 3+4=7, Odd number which cannot be shown as 2 equal parts.

Question 10.
You do 6 sit-ups on Saturday and 7 on Sunday. Do you do an even or odd number of sit-ups in all?

Answer:
Explanation: 6+7=13, It is an Odd number

Think and Grow: Modeling Real Life

There is an even number of marbles in one bag and an odd number of marbles in another bag. Is there an even or an odd number of marbles in all?

Which equation could match the story?

There is an ___ number of marbles in all.
Answer: Odd
Explanation: 8+7=15, 8 is even and 7 is odd number, in total 15 is Odd number of marbles.

Show and Grow

Question 11.
Two buckets each have an odd number of seashells. Is there an even or an odd number of seashells in all?

Which equation could match the story?

There is an __ number of seashells in all.
Answer: Odd
Explanation: 9+5=14, 9 and 5 are odd numbers which in total 14 is Even number.

Question 12.
DIG DEEPER!
You have an odd number of flowers. You and your friend have an even number of flowers in all. Does your friend have an even or an odd number of flowers?

Your friend has an ___ number of flowers.
Answer: Even
Explanation: Friend has only even number of flowers.

Model Even and Odd Numbers Homework & Practice 1.2

Question 1.

Answer: Even

Explanation: 8=4+4, Even number

Question 2.

Answer: Odd

Explanation: 17=9+8, Odd number

Question 3.

Answer: Odd

Explanation: 9=5+4, Odd number

Question 4.

Answer: Even

Explanation: 16=8+8, Even number

Question 5.
Reasoning
Fill in the blanks using even or odd.
The sum of two even numbers is ___.
Answer: Even
The sum of two odd numbers is ___.
Answer: Even
The sum of an even number and an odd number is __.
Answer: Odd

Question 6.
You do 10 jumping jacks on Saturday and 10 on Sunday. Do you do an even or odd number of jumping jacks in all?

Answer: Even
Explanation: 10+10=20, which is Even number.

Question 7.
Modeling Real Life
You and your friend each have an even number of googly eyes. Do you and your friend have an even or an odd number of googly eyes in all?

Which equation could match the story?

You have an __ number of googly eyes in all.
Answer: 4+6=10
Explanation: 4 and 6 are Even numbers. So totally 10 googly eyes which is even number.

Question 8.
DIG DEEPER!
You hop an even number of times. You and your friend hop an odd number of times in all. Does your friend hop an even or an odd number of times?
Your friend hops an ___ number of times.
Answer: Your Friend hops an Odd number of times.
Explanation: you hop an even and overall odd number, so your friend has odd number of times.

Review & Refresh

Circle the shape that shows equal shares.

Question 9.

Answer:

Question 10.

Answer:

Lesson 1.3 Equal Groups

Explore and Grow

Circle groups of two oranges. Complete the sentence.

Answer: 4 groups of 2 is 8.

Show and Grow

Question 1.

Answer: 5 groups of 2
5+5=10

Question 2.

Answer: 2 groups of 3
2+2+2=6

Question 3.
Circle groups of 3. Write a repeated addition equation.

Answer: 5 groups of 3
5+5+5+5+5=25

Apply and Grow: Practice

Question 4.
Circle groups of 5. Write a repeated addition equation.

Answer: 3 groups of 5
5+5=15

Question 5.
Circle groups of 4. Write a repeated addition equation.

Answer: 4 groups of 4
Answer: 4+4+4+4=16

Question 6.
YOU BE THE TEACHER
Newton says he can circle 5 equal groups. Is he correct? Explain.

Answer:

Yes he is correct. Newton can circle 3 groups of 5 parts. So, 3+3+3+3+3=15

Think and Grow: Modeling Real Life

You have 3 boxes. There are 5 pencils in each box. How many pencils are there in all?

Model:
Repeated addition equation:

Answer: 15 pencils in total. 5+5+5=15.

Show and Grow

Question 7.
You have 5 bags. There are 4 notebooks in each bag. How many notebooks are there in all?

Answer: 20 notebooks in total. 5+5+5+5=20.

Question 8.
DIG DEEPER!
There are 4 boxes. Each box has the same number of glue sticks. There are 16 in all. How many glue sticks are in each box?

Answer: There are 4 glue sticks in each box.

Question 9.
Explain how you solved Exercises 7 and 8. What did you do differently?
_______________________________________
_______________________________________

Answer: Adding the number with the given number of times.

Equal Groups Homework & Practice 1.3

Question 1.

Answer: 3 groups of 2
3+3=6

Question 2.

Answer: 4 groups of 3
4+4+4=12

Question 3.
Circle groups of 2. Write a repeated addition equation.

Answer: 4 groups of 2

Answer: 2+2+2+2=8

Question 4.
Structure
Show two different ways to put the buttons in equal groups.
One Way:

Another Way:

Answer:

Question 5.
Modeling Real Life
You have 3 jars of paint brushes.There are 6 paint brushes in each jar. How many paint brushes are there in all?

Answer: 18 paintbrushes
Explanation: 6+6+6=18
Review & Refresh

Question 6.

How many students chose baseball? ___

Answer: 5 students chose baseball
Which sport is the most favorite?

Answer: Favorite sport is Soccer

Lesson 1.4 Use Arrays

Explore and Grow

How many equal groups are there? Write an addition equation to tell how many cars there are in all.

Number of equal groups: ___
Answer: 2
Addition equation:
3+3=6

Show and Grow

Question 1.

Answer: 2 rows of 4
4+4=8

Question 2.

Answer: 3 rows of 4
4+4+4=12

Question 3.

Answer: 4 rows of 4
4+4+4+4=16

Apply and Grow: Practice

Question 4.

Answer: 2 rows of 5
5+5=10

Question 5.

Answer: 3 rows of 3
3+3+3=9

Question 6.

Answer: 3 rows of 5
5+5+5=15

Question 7.

Answer: 5 rows of 4
5+5+5+5=20

Question 8.
Logic
Which arrays show the same number of circles?

Answer: Red and Blue circles show the same number.

Think and Grow: Modeling Real Life

The arrays show the desks in two classrooms. Which classroom has more desks?

Repeated addition equations:

Classroom A: 5+5+5+5+5=25
Classroom B: 7+7+7+7=28
Answer: Classroom B has more number of desks.

Show and Grow

Question 9.
The arrays show gardens of green and yellow pepper plants. Are there more green pepper plants or yellow pepper plants?

Answer: Green pepper plants
Explanation: There are 15 Green pepper plants and 12 Yellow pepper plants.

Use Arrays Homework & Practice 1.4

Question 1.

Answer: 3 rows of 2
2+2+2=6

Question 2.

Answer: 2 rows of 2
2+2=4

Question 3.

Answer: 1 rows of 1
1+1+1+1=4

Question 4.

Answer: 3 rows of 6
6+6+6= 18

Question 5.
Number Sense
Use the array to complete the equation.

Answer: 6+6=12

Question 6.
Modeling Real Life
The arrays show toy cars. Are there more orange cars or blue cars?

Answer: Orange cars
Explanation: There are 16 orange cars and 15 blue cars.

Question 7.
DIG DEEPER!
The arrays show a sheet of stickers separated into two pieces. How many rows and columns of stickers did the sheet have before it was separated?

Answer: 5 rows and 5 columns

Review & Refresh

Question 8.

Answer: 6 flat surfaces
8 vertices
12 edges

Question 9.

Answer: 0 flat surfaces
0  vertices
0 edges

Lesson 1.5 Make Arrays

Explore and Grow

Use counters to model the story. Write an addition equation to match.

There are 4 rows of students. There are 3 students in each row. How many students are there in all?

Addition equation:

Answer: 12 students
Explanation: 3+3+3+3=12 students.

Show and Grow

Question 1.
A photo album has 3 rows of photos. There are 2 photos in each row. How many photos are there in all?

Answer: 6 photos
Explanation: 2+2+2=6

Question 2.
You have 4 rows of stickers. There are 5 stickers in each row. How many stickers do you have in all?

Answer: 20 stickers
Explanation: 5+5+5+5=20

Apply and Grow: Practice

Question 3.
An ice cube tray has 5 rows. There are 2 ice cubes in each row. How many ice cubes are there in all?

Answer: 10 ice cubes
Explanation: 2+2+2+2+2=10

Question 4.
A bookcase has 3 shelves. There are 5 stuffed animals on each shelf. How many stuffed animals are there in all?

Answer: 15 stuffed animals
Explanation: 5+5+5=15 stuffed animals

Question 5.
A closet has 4 shelves. There are 2 games on each shelf. How many games are there in all?

Answer: 8 games
Explanation: 2+2+2+2=8

Question 6.
Structure
Make an array to match the equation.
5 + 5 + 5 + 5 = 20

Answer:

Think and Grow: Modeling Real Life

A marching band has 3 equal rows of drummers. There are 15 drummers in all. How many drummers are in each row?

Model:
Repeated addition equation:

Answer: 5 Drummers
Explanation: 5+5+5=15

Show and Grow

Question 7.
A quilt has 4 equal rows of patches. There are 24 patches in all. How many patches are in each row?

Answer: 6 patches
Explanation: 6+6+6+6=24

Question 8.
A building has 3 equal rows of windows. There are 18 windows in all. How many columns are there?

Answer: 6 columns
Explanation: 3+3+3+3+3+3=18

Make Arrays Homework & Practice 1.5

Question 1.
A parking lot has 3 rows. There are 5 parking spots in each row. How many parking spots are there in all?

Answer: 15 parking spots
Explanation: 5+5+5=15

Question 2.
A bookcase has 4 shelves. There are 3 books on each shelf. How many books are there in all?

Answer: 12 books
Explanation: 3+3+3+3=12

Question 3.
Reasoning
Newton has 10 tokens. Which equations can Newton use to make an array with his tokens?

Answer: 2+2+2+2+2=10  and 5+5=10

Question 4.
Modeling Real Life
A theater has 4 equal rows of seats. There are 16 seats in all. How many seats are in each row?

Answer: 4 seats
Explanation: 4+4+4+4=16

Question 5.
Modeling Real Life
A chorus has 5 equal rows of singers. There are 30 singers in all. How many singers are in each row?

Answer: 6 singers
Explanation: 6+6+6+6+6=30

Review & Refresh

Question 6.
There are 9 lions in all. How many lions are in the den?

Answer: 0 lions are in den.
Explanation: All 9 lions are outside the den. Therefore, 0 lions are in the den.

Numbers and Arrays Performance Task

Question 1.
Your art supplies are packaged in boxes as described below.

a. What do you have the most of ?

Answer: 16 Crayons
Explanation: Colored Pencils= 5+5+5=15
Markers= 7+7=14
Crayons= 4+4+4+4=16
Crayons are more than Colored Pencils and Markers.
b. Do you have an even or an odd number of markers?

Answer: Even number of markers

Question 2.
a. You have 5 equal rows of paint bottles. You have 20 paint bottles in all. How many paint bottles are in each row?

Answer: 4 paint bottles are in each row.
b. You add another row of paint bottles. How many paint bottles do you have now?
Answer: 24 paint bottles
c. Describe another way to arrange the paint bottles you have now.

Answer: 6+6+6+6=24 paint bottles

Numbers and Arrays Activity

Array Flip and Find

To Play: Place the Array Flip and Find Cards face down in the boxes. Take turns flipping two cards. If your cards show the same total, keep the cards. If your cards show different totals, flip the cards back over. Play until all matches are made.

Numbers and Arrays Chapter Practice

1.1 Even and Odd Numbers

Question 1.

Answer:  Odd number

Explanation: 5 cube part

Question 2.

Answer: Even number

Explanation: 14 cube part

Color cubes to show the number. Circle even or odd.

Question 3.

Answer:

Question 4.

Answer:

Question 5.
Modeling Real Life
You see an odd number of boats. There are more than 15 but fewer than 19 boats. How many boats do you see?

Show how you know:

Answer: 17 boats
Explanation: Between 15 and 19 there is Odd number 17.

1.2 Model Even and Odd Numbers

Question 6.

Answer: 9=5+4

Question 7.

Answer: 18=9+9

Question 8.

Answer: 10=5+5

Question 9.

Answer: 7+6=13

1.3 Equal Groups

Question 10.

Answer: 4 groups of 2
Answer: 4+4=8

Question 11.
Circle groups of 3. Write a repeated addition equation.

Answer: 4 groups of 3
Answer: 3+3+3+3=12

Question 12.
Structure
Show two different ways to put the balls in equal groups.
One Way:

Another Way:

Answer:

1.4 Use Arrays

Question 13.

Answer: 3 rows of 2
Answer: 2+2+2=6

Question 14.

Answer: 2 rows of 4
Answer: 4+4=8

Question 15.

Answer: 5 rows of 3
Answer: 3+3+3+3+3=15

Question 16.

Answer: 4 rows of 5
Answer: 5+5+5+5=20

1.5 Make Arrays

Question 17.
A cupboard has 4 shelves. There are 3 glasses on each shelf. How many glasses are there in all?

Answer: 12 glasses
Answer: 3+3+3+3=12

Question 18.
A bingo card has 5 rows. There are 5 squares in each row. How many squares are there in all?

Answer: 25 squares
Answer: 5+5+5+5+5=25

Question 19.
Modeling Real Life
A pet store has 5 equal rows of fish tanks. There are 20 fish tanks in all. How many fish tanks are in each row?

Answer: 4 fish tanks
Explanation: 4+4+4+4+4=20

Conclusion:

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Are you searching everywhere regarding the Big Ideas Math 8th Grade Answer Key Chapter 4 Graphing and Writing Linear Equations? If so, halt your search as this is the one-stop destination for all your needs. Practice using the Graphing and Writing Linear Equations Big Ideas Math Grade 8 Answers and understand the concepts easily. Begin your preparation right away and seek the homework help needed right after class in a matter of seconds.

Big Ideas Math Book 8th Grade Answer Key Chapter 4 Graphing and Writing Linear Equations

Make the most out of the handy resources available for Big Ideas Math Ch 4 Graphing and Writing Linear Equations and stand out from the rest of the crowd. BIM Book 8th Grade Chapter 4 Solutions include questions belonging to Lessons 4.1 to 4.7, Cumulative Practice, Assessment Tests, Review Tests, etc. Big Ideas Math 8th Grade Chapter 4 Solution Key is given by subject experts after extensive research. Access the quick links over here during your preparation and get the assistance needed at the comfort of your home.

Performance Task

• Graphing and Writing Linear Equations STEAM Video/Performance Task
• Graphing and Writing Linear Equations Getting Ready for Chapter 4

Lesson: 1 Graphing Linear Equations

• Lesson 4.1 Graphing Linear Equations
• Graphing Linear Equations Homework & Practice 4.1

Lesson: 2 Slope of a Line

• Lesson 4.2 Slope of a Line
• Slope of a Line Homework & Practice 4.2

Lesson: 3 Graphing Proportional Relationships

• Lesson 4.3 Graphing Proportional Relationships
• Graphing Proportional Relationships Homework & Practice 4.3

Lesson: 4 Graphing Linear Equations in Slope-Intercept Form

• Lesson 4.4 Graphing Linear Equations in Slope-Intercept Form
• Graphing Linear Equations in Slope-Intercept Form Homework & Practice 4.4

Lesson: 5 Graphing Linear Equations in Standard Form

• Lesson 4.5 Graphing Linear Equations in Standard Form
• Graphing Linear Equations in Standard Form Homework & Practice 4.5

Lesson: 6 Writing Equations in Slope-Intercept Form

• Lesson 4.6 Writing Equations in Slope-Intercept Form
• Writing Equations in Slope-Intercept Form Homework & Practice 4.6

Lesson: 7 Writing Equations in Point-Slope Form

• Lesson 4.7 Writing Equations in Point-Slope Form
• Writing Equations in Point-Slope Form Homework & Practice 4.7

Chapter: 4 – Graphing and Writing Linear Equations

• Graphing and Writing Linear Equations Connecting Concepts
• Graphing and Writing Linear Equations Chapter Review
• Graphing and Writing Linear Equations Practice Test
• Graphing and Writing Linear Equations Cumulative Practice

Graphing and Writing Linear Equations STEAM Video/Performance Task

STEAM Video

“Hurricane
A hurricane is a storm with violent winds. How can you prepare your home for a hurricane?
Watch the STEAM Video “Hurricane!” Then answer the following questions.

1. Robert says that the closer you are to the eye of a hurricane, the stronger the winds become. The wind speed on an island is 50 miles per hour when the eye of a hurricane is 140 miles away.
a. Describe the wind speed on the island when the eye of the hurricane is 100 miles away.
b. Describe the distance of the island from the eye of the hurricane when the wind speed on the island is 25 miles per hour.
c. Sketch a line that could represent the wind speed y (in miles per hour) on the island when the eye of x the hurricane is miles away from the island. Wind speed
2. A storm dissipates as it travels over land. What does this mean?

Performance Task

Anatomy of a Hurricane
After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given information about the atmospheric pressure inside a hurricane.

You will be asked to use a model to find the strength of a hurricane after x hours of monitoring. Why is it helpful to predict how strong the winds of a hurricane will become?

Graphing and Writing Linear Equations Getting Ready for Chapter 4

Chapter Exploration
1. Work with a partner.
a. Use the equation y = \(\frac{1}{2}\)x + 1 to complete the table. (Choose any two x-values and find the y-values.)
b. Write the two ordered pairs given by the table. These are called solutions of the equation.

c. PRECISION Plot the two solutions. Draw a line exactly through the points.
d. Find a different point on the line. Check that this point is a solution of the equation y = \(\frac{1}{2}\)x + 1.
e. LOGIC Do you think it is true that any point on the line is a solution of the equation y = \(\frac{1}{2}\)x + 1? Explain.
f. Choose five additional x-values for the table below. (Choose both positive and negative x-values.) Plot the five corresponding solutions. Does each point lie on the line?

g. LOGIC Do you think it is true that any solution of the equation y = \(\frac{1}{2}\)x + 1 is a point on the line? Explain.
h. Why do you think y = ax + b is called a linear equation?

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
linear equation
slope
y-intercept
solution of a linear equation
x-intercept

Lesson 4.1 Graphing Linear Equations

EXPLORATION 1

Creating Graphs
Work with a partner. It starts snowing at midnight in Town A and Town B. The snow falls at a rate of 1.5 inche sper hour.
a. In Town A, there is no snow on the ground at midnight. How deep is the snow at each hour between midnight and 6 A.M.? Make a graph that represents this situation.

b. Repeat part(a) for TownB, which has 4 inches of snow on the ground at midnight.
c. The equations below represent the depth y(in inches) of snow x hours after midnight in Town C and Town D. Graph each equation.
Town C y = 2x + 3
Town D y = 8

d. Use your graphs to compare the snowfall in each town.
Answer:

Try It

Graph the linear equation.
Question 1.
y = 3x
Answer:
Make to table of values
Replace x with a number and find the value of y

Plot the values of x and y obtained above, on the graph

Draw the line through the points

Question 2.
y = – 2x – 1
Answer:

Plot the values of x and y

Now the line through the points

Question 3.
y = –\(\frac{1}{2}\)x + 2
Answer:

Plot the ordered pairs

Graph the linear equation.
Question 4.
y = 3
Answer:
The graph of y = 3 is a horizontal like passing through (0, 3)
Draw a horizontal line through this point.

Question 5.
y = – 1.5
Answer:
The graph of y = -1.5 is a horizontal line passing through (0, -1.5)
Draw a horizontal line through this point.

Question 6.
x = – 4
Answer:
The graph of x = – 4 is a vertical line passing through (-4, 0)
Draw a vertical line through this point.

Question 7.
x = \(\frac{1}{2}\)
Answer:
The graph of x = \(\frac{1}{2}\) is a vertical line passing through (\(\frac{1}{2}\), 0)
Draw a vertical line through this point.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

GRAPHING A LINEAR EQUATION Graph the linear equation.
Question 8.
y = – x + 1
Answer:
Make of a table of values
Replace x with a number and find the value of y

Plot the values of x and y obtained, on the graph,

Question 9.
y = 0.8x – 2
Answer:
Replace x with a number and find the value of y

Question 10.
x = 2.5
Answer:
The graph of x = 2.5 is a vertical line passing through (2.5, 0)
Draw a vertical line through this point.

Question 11.
y = \(\frac{2}{3}\)
Answer:
The graph of y = \(\frac{2}{3}\) is a horizontal line passing through (0, \(\frac{2}{3}\))
Draw a horizontal line through this point.

Question 12.
WHICH ONE DOESN’T BELONG?
Which equation does not belong with the other three? Explain your reasoning.

Answer:
y = x – 2
4x + 3 = y
y = x² + 6
x + 5 = y

Self-Assessment for Problem Solving
Solve each exercise. The rate your understanding of the success criteria in your journal.

Question 13.
A game show contestant earns y dollars for completing a puzzle in x minutes. This situation is represented by the equation y = – 250x + 5000. How long did a contestant who earned $500 take to complete the puzzle? Justify your answer.
Answer:
Given,
A game show contestant earns y dollars for completing a puzzle in x minutes.
This situation is represented by the equation y = – 250x + 5000.
y = -250x + 5000
500 = -250x + 5000
500 – 5000 = -250x + 5000 – 5000
-4500 = -250x
x = 18

Question 14.
The total cost y (in dollars) to join a cheerleading team and attend x competitions is represented by the equation y = 10x + 50.

a. Graph the linear equation.

Answer:

b. You have $75 to spend. How many competitions can you attend?
Answer:
75 ≤ 10x + 50
75 – 50 ≤ 10x
25 ≤ 10x
2.5 ≥ x
By this I can say that I can attend 2 competitions if I have $75 to spend.

Question 15.
The seating capacity for a banquet hall is represented by y = 8x + 56, where x is the number of extra tables you need. How many extra tables do you need to double the original seating capacity?
Answer:
Given,
The seating capacity for a banquet hall is represented by y = 8x + 56, where x is the number of extra tables you need.
y = 8x + 56
2 × 56 = 8x + 56
112 = 8x + 56
8x = 112 – 56
8x = 56
x = 7 tables

Graphing Linear Equations Homework & Practice 4.1

Review & Refresh

Tell whether the triangles are similar. Explain.
Question 1.

Answer:
x° + 46° + 95° = 180°
x° + 141° = 180°
x° = 180° – 141°
x° = 39°
Thus the angles of the triangle are 39°, 46°, 95°
y° + 39° + 46° = 180°
y° + 75° = 180°
y° = 180° – 75°
y° = 95°
Three angles of the triangle are 39°, 46°, 95°
The triangles have two pairs of congruent angles.

Question 2.

Answer:
x° + 40° + 51° = 180°
x° + 91° = 180°
x° = 180° – 91°
x° = 89°
Three angles of the triangle are 40°, 51°, 89°
y° + 40° + 79° = 180°
y° + 119° = 180°
y° = 180° – 119°
y° = 61°

Describe the translation of the point to its image.
Question 3.
(1, – 4) → (3, 0)
Answer:
A(1, -4) = A'(1 + 2, -4) = (3, -4)
A'(3, 4) = B(3, -4 + 4) = (3, 0)
Translate 2 units right and 4 units up.

Question 4.
(6, 4) → (- 4, – 6)
Answer:
We are given the points
(6, 4) → (- 4, – 6)
A(6, 4) = A'(6 – 10, 4) = (-4, 4)
A'(-4, -4) = B(-4, 4 – 10) = (-4, -6)

Question 5.
(4, – 2) → (- 9, 3)
Answer:
We are given the points
A(4, -2)
B(-9, 3)
A(4, -2) = A'(4 – 13, -2) = (-9, -2)
A'(-9, -2) = B(-9, -2 + 4) = (-9, 3)

Concepts, Skills, & Problem Solving

CREATING GRAPHS Make a graph of the situation. (See Exploration 1, p. 141.)
Question 6.
The equation y = – 2x + 8 represents the amount (in fluid ounces) of dish detergent in a bottle after x days of use.
Answer:

Question 7.
The equation y = 15x + 20 represents the cost (in dollars) of a gym membership after x months.
Answer:

PRECISION Copy and complete the table with two solutions. Plot the ordered pairs and draw the graph of the linear equation. Use the graph to find a third solution of the equation.
Question 8.

Answer:

(x, y) = (2, 5)

Question 9.

Answer:

(x, y) = (3, 3)

GRAPHING A LINEAR EQUATION Graph the linear equation.
Question 10.
y = – 5x
Answer:

Question 11.
y = 9x
Answer:

Question 12.
y = 5
Answer:
The graph of y = 5 is a horizontal line passing through (0, 5)
Draw a horizontal line through this point.

Question 13.
x = – 6
Answer:

Question 14.
y = x – 3
Answer:

Question 15.
y = – 7x – 1
Answer:

Question 16.
y = – \(\frac{x}{8}\) + 4
Answer:

Question 17.
y = 0.75x – 0.5
Answer:

Question 18.
y = – \(\frac{2}{3}\)
Answer:

Question 19.
y = 6.75
Answer:

Question 20.
x = – 0.5
Answer:
The graph of x = -0.5 is a vertical line passing through (-0.5, 0)
Draw a vertical line through this point.

Question 21.
x = \(\frac{1}{4}\)
Answer:
The graph of x = \(\frac{1}{4}\) is a vertical line passing through (\(\frac{1}{4}\), 0)
Draw a vertical line through this point.

Question 22.
YOU BE THE TEACHER
Your friend graphs the equation y = 4. Is your friend correct? Explain your reasoning.
Answer:

No my friend is not correct because the graph for the equation y = 4 is a  horizontal line not a vertical line, and it passes through the point (0, 4) not (4, 0)

Question 23.
MODELING REAL LIFE
The equation y = 20 represents the cost y (in dollars) for sending x text messages in a month. Graph the linear equation. What does the graph tell you about your texting plan?

Answer:

Question 24.
MODELING REAL LIFE
The equation y = 2x + 3 represents the cost y (in dollars) of mailing a package that weighs x pounds.

a. Use a graph to estimate how much it costs to mail the package.
b. Use the equation to find exactly how much it costs to mail the package.
Answer:
Given the equation y = 2x + 3
The ordered pairs will be (0, 3), (2,7), (4, 11)
Now plot the ordered pairs

y = 2(1.126) + 3
= 5.252 ≈ 5.25

SOLVING A LINEAR EQUATION Solve for y. Then graph the linear equation.
Question 25.
y – 3x = 1
Answer:
y – 3x = 1
y = 3x + 1

Draw a line through the points

Question 26.
5x + 2y = 4
Answer:
5x + 2y = 4
2y = 4 – 5x
y = – \(\frac{5}{2}\)x + 2

Question 27.
– \(\frac{1}{3}\)y + 4x = 3
Answer:
– \(\frac{1}{3}\)y + 4x = 3
– \(\frac{1}{3}\)y = 3 – 4x
y = 12x – 9

Question 28.
x + 0.5y = 1.5
Answer:
x + 0.5y = 1.5
0.5y = -x + 1.5
y = -2x + 3

Question 29.
MODELING REAL LIFE
The depth y (in inches) of a lake after x years is represented by the equation y = 0.2x + 42. How much does the depth of the lake increase in four years? Use a graph to justify your answer.

Answer:
y = 0.2x + 42
Depth of the lake now: y = 0.2(0) + 42 = 42
Depth of the lake after 4 years: y = 0.2(4) + 42 = 42.8

42.8 – 42 = 0.8 inches

Question 30.
MODELING REAL LIFE
The amount y (in dollars) of money in your savings account after x months is represented by the equation y = 12.5x + 100.
a. Graph the linear equation.

Answer:

b. How many months will it take you to save a total of $237.50?
Answer:
y = 12.5x + 100
237.5 = x + 100
237.5 – 100 = 12.5x + 100 – 100
12.5x = 137.5
x = 11

Question 31.
PROBLEM SOLVING
The radius y (in millimeters) of a chemical spill after x days is represented by the equation y = 6x + 50.

a. Graph the linear equation.

Answer:

b. The leak is noticed after two weeks. What is the area of the leak when it is noticed? Justify your answer.
Answer:
y = 6(14) + 50
y = 84 + 50
y = 134 mm
2πr = 2π = 841.95 sq. mm

Question 32.
GEOMETRY
The sum S of the interior angle measures of a polygon with n sides is S = (n – 2) • 180°.
a. Plot four points (n, S) that satisfy the equation. Is the equation a linear equation? Explain your reasoning.

Answer:

b. Does the value n = 3.5 make sense in the context of the problem? Explain your reasoning.
Answer:
The value n = 3.5 does not make sense because the number of angles cannot be other than integer greater or equal to 2.

Question 33.
DIG DEEPER!
One second of video on your cell phone uses the same amount of memory as two pictures. Your cell phone can store 2500 pictures.
a. Create a graph that represents the number y of pictures your cell phone can store when you take x seconds of video.

Answer:

b. How many pictures can your cell phone store in addition to a video that is one minute and thirty seconds long?
Answer:
Determine the number of pictures you can store in addition to a video of 1 min 30 seconds.
1 min 30 seconds = (60 + 90) 3 seconds = 90 seconds
2500 – (2 . 90)
2500 – 180 = 2320 pictures

Lesson 4.2 Slope of a Line

EXPLORATION 1

Measuring the Steepness of a Line
Work with a partner. Draw any nonvertical line in a coordinate plane.
a. Develop a way to measure the steepness of the line. Compare your method with other pairs.
b. Draw a line that is parallel to your line. What can you determine about the steepness of each line? Explain your reasoning.
Answer:

EXPLORATION 2

Using Right Triangles
Work with a partner. Use the figure shown.

a. △ABC is a right triangle formed by drawing a horizontal line segment from point A and a vertical line segment from point B. Use this method to draw another right triangle, △DEF, with its longest side on the line
b. What can you conclude about the two triangles in part(a)? Justify your conclusion. Compare your results with other pairs.
c. Based on your conclusions in part(b), what is true about \(\frac{BC}{AC}\) and the corresponding measure in △DEF? Explain your reasoning. What do these values tell you about the line?
Answer:

Try It

Find the slope of the line.
Question 1.

Answer:
(x1, y1) = (-2, 3)
(x2, y2) = (3, 2)
m = (y2 – y1)/(x2 – x1)
m = (2 -3)/(3 – (-2))
m = -1/5
Thus slope = -1/5

Question 2.

Answer:
(x1, y1) = (-4, -1)
(x2, y2) = (2, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – (-1))/(2 – (-4))
m = 2/6
Thus slope = 1/3

Find the slope of the line through the given points.
Question 3.
(1, -2), (7, -2)
Answer:
(x1, y1) = (1, -2)
(x2, y2) = (7, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – (-2))/(7 – 1)
m = 0/6
Thus slope = 0

Question 4.
(-3, -3), (-3, -5)
Answer:
(x1, y1) = (-3, -3)
(x2, y2) = (-3, -5)
m = (y2 – y1)/(x2 – x1)
m = (-5 + 3)/(-3 + 3)
m = -2/0
Thus slope = undefined

Question 5.
WHAT IF
The blue line passes through (-4, -3) and (-3, 2). Are any of the lines parallel? Explain.
Answer:
(x1, y1) = (-4, -3)
(x2, y2) = (-3, 2)
m = (y2 – y1)/(x2 – x1)
m = (2 + 3)/(-3 + 4)
m = 5/1
m = 5
The slpe of the blue line is 5 and the slope of the red line is also 5.
The blue lines and red lines have same slopes so they are parallel.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
VOCABULARY
What does it mean for a line to have a slope of 4?
Answer:
If a line have a slope of 4 it means that the line rises 4 units for every 1 units it runs.

FINDING THE SLOPE OF A LINE Find the slope of the line through the given points.
Question 7.
(1, -1), (6, 2)
Answer:
(x1, y1) = (1, -1)
(x2, y2) = (6, 2)
m = (y2 – y1)/(x2 – x1)
m = (2 – (-1))/(6 – 1)
m = 3/5

Question 8.
(2, -3), (5, -3)
Answer:
(x1, y1) = (2, -3)
(x2, y2) = (5, -3)
m = (y2 – y1)/(x2 – x1)
m = (5 – 2)/(-3 + 3)
m = 3/0
m = undefined

Question 9.
FINDING SLOPE
Are the lines parallel? Explain your reasoning.

Answer:
Red line:
(x1, y1) = (-1, 0)
(x2, y2) = (1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 0)/1 – (-1))
m = -2/2
m = -1
Blue Line:
(x1, y1) = (-1, 3)
(x2, y2) = (1, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – 3)/(1 – (-1))
m = -4/2
m = -2
The slope of the blue line and red line are not the same. So they are not parallel.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
The table shows the lengths (in inches) of your hair months after your last haircut. The points in the table lie on a line. Find and interpret the slope of the line. After how many months is your hair 4 inches long?

Answer:
Determine the slope of the line using two points from the table:
(2, 1), (4, 2)
m = (2 – 1)/4 – 2
m = 1/2
m = 0.5
This means that each month the hair grows 0.5 inches
As the hair grows 0.5 inches/ month, it will be 4 inches long after 4/0.5 = 8 months.

Question 11.
A customer pays an initial fee and a daily fee to rent a snowmobile. The total payment for 3 days is 92 dollars. The total payment for 5 days is 120 dollars. What is the daily fee? Justify your answer.
Answer:
Given,
A customer pays an initial fee and a daily fee to rent a snowmobile.
The total payment for 3 days is 92 dollars. The total payment for 5 days is 120 dollars.
m = (120 – 92)/5 – 3
m = 28/2
m = 14

Question 12.
You in-line skate from an elevation of 720 feet to an elevation of 750 feet in 30 minutes. Your friend in-line skates from an elevation of 600 feet to an elevation of 690 feet in one hour. Compare your rates of change in elevation.

Answer:
Given,
You in-line skate from an elevation of 720 feet to an elevation of 750 feet in 30 minutes.
Your friend in-line skates from an elevation of 600 feet to an elevation of 690 feet in one hour.
(750 – 720)/30 = 30/30 = 1 ft/min
(690 – 600)/60 = 90/60 = 1.5 ft/min

Slope of a Line Homework & Practice 4.2

Review & Refresh

Graph the linear equation.
Question 1.
y = 4x – 3
Answer:

Question 2.
x = -3
Answer:

Question 3.
y = 2
Answer:

Question 4.
y = \(\frac{3}{2}\)x – \(\frac{1}{2}\)
Answer:

Find the missing values in the ratio table.
Question 5.

Answer:

x/10 = 1/3
x = 10/3
x = 3.33
1/3 = 5/y
y = 5 × 3
y = 15
1/3 = 7/z
z = 3 × 7
z = 21

Question 6.

Answer:

Concepts, Skills, &Problem Solving

USING RIGHT TRIANGLES Use the figure shown (See Exploration 2, p. 147.)

Question 7.
Find the slope of the line.
Answer:
(x1, y1) = B(-4, 2)
(x2, y2) = A(-2, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – 2)/(-2 – (-4))
m = -1/2
Thus the slope m = -1/2

Question 8.
Let point D be at (-4, 1). Use the sides of △BDA to find the slope of the line.
Answer:

m = -BD/DA = -1/2

FINDING THE SLOPE OF A LINE Find the slope of the line.
Question 9.

Answer:
(x1, y1) = (-2, 0)
(x2, y2) = (2, 3)
m = (y2 – y1)/(x2 – x1)
m = (3 – 0)/(2 – (-2))
m = 3/4

Question 10.

Answer:
(x1, y1) = (-2, 5)
(x2, y2) = (2, 0)
m = (y2 – y1)/(x2 – x1)
m = (0 – 5)/(2 – (-2))
m = -5/4

Question 11.

Answer:
(x1, y1) = (-4, 1)
(x2, y2) = (1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 1)/(1 + 2)
m = -3/5

Question 12.

Answer:
(x1, y1) = (-5, -4)
(x2, y2) = (1, -3)
m = (y2 – y1)/(x2 – x1)
m = (-3 – (-4))/(1 – (-5))
m = 1/6

Question 13.

Answer:
(x1, y1) = (-1, 3)
(x2, y2) = (3, 3)
m = (y2 – y1)/(x2 – x1)
m = (3 – 3)/(3 – (-1))
m = 0/4
m = 0

Question 14.

Answer:
(x1, y1) = (1, 3)
(x2, y2) = (1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 3)/(1 – 1)
m = -5/0
m = undefined

FINDING THE SLOPE OF A LINE Find the slope of the line through the given points.
Question 15.
(4, -1), (-2, -1)
Answer:
(x1, y1) = (4, -1)
(x2, y2) = (-2, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – (-1))/(-2 – 4)
m = 0/-6
m = 0

Question 16.
(5, -3), (5, 8)
Answer:
(x1, y1) = (5, -3)
(x2, y2) = (5, 8)
m = (y2 – y1)/(x2 – x1)
m = (8 – 3)/(5 – 5)
m = 5/0
m = undefined

Question 17.
(-7, 0), (-7, -6)
Answer:
(x1, y1) = (-7, 0)
(x2, y2) = (-7, -6)
m = (y2 – y1)/(x2 – x1)
m = (-6 – 0)/(-7 – (-7))
m = -6/0
m = undefined

Question 18.
(-3, 1), (-1, 5)
Answer:
(x1, y1) = (-3, 1)
(x2, y2) = (-1, 5)
m = (y2 – y1)/(x2 – x1)
m = (5 – 1)/(-1 + 3)
m = 4/2
m = 2

Question 19.
(10, 4), (4, 15)
Answer:
(x1, y1) = (10, 4)
(x2, y2) = (4, 15)
m = (y2 – y1)/(x2 – x1)
m = (15 – 4)/(4 – 10)
m = 11/-6
m = -11/6

Question 20.
(-3, 6), (2, 6)
Answer:
(x1, y1) = (-3, 6)
(x2, y2) = (2, 6)
m = (y2 – y1)/(x2 – x1)
m = (6 – 6)/(2 – (-3))
m = 0/5
m = 0

Question 21.
REASONING
Draw a line through each point using slope of m = \(\frac{1}{4}\). Do the lines intersect? Explain.

Answer:

The 2 lines are parallel because they have the same slope and they do not intersect.

Question 22.
YOU BE THE TEACHER
Your friend finds the slope of the line shown. Is your friend correct? Explain your reasoning.

Answer:

No my friend is not correct because the denominator should be 2 – 4
(x1, y1) = (2, 3)
(x2, y2) = (4, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – 3)/(4 – 2)
m = -2/2
m = -1

IDENTIFYING PARALLEL LINES Which lines are parallel? How do you know?
Question 23.

Answer:
Blue line:
(x1, y1) = (-5, 2)
(x2, y2) = (-4, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – 2)/(-4 – (-5))
m = -3/1
m = -3
Red line:
(x1, y1) = (-2, 1)
(x2, y2) = (-1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 1)/(-1 – (-2))
m = -3/1
m = -3
Green Line:
(x1, y1) = (1, 3)
(x2, y2) = (2, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – 3)/(2 – 1)
m = -4/1
m = -4
Blue line and red line have slope of -3, so they are parallel.

Question 24.

Answer:
Blue line:
(x1, y1) = (-2, 3)
(x2, y2) = (-5, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 3)/(-5 – (-2))
m = -5/-3
m = 5/3
Red line:
(x1, y1) = (1, 2)
(x2, y2) = (-2, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 2)/(-2 – 1)
m = -4/-3
m = 4/3
Green Line:
(x1, y1) = (4, 1)
(x2, y2) = (1, -3)
m = (y2 – y1)/(x2 – x1)
m = (-3 – 1)/(1 – 4)
m = -4/-3
m = 4/3
Red line and green line have slope of 4/3 by this we can say that they are parallel.

IDENTIFYING PARALLEL LINES Are the given lines parallel? Explain your reasoning.
Question 25.
y = -5, y = 3
Answer:

Both lines are horizontal and have slope = 0

Question 26.
y = 0, x = 0
Answer:

The line y = 0 have slope = 0 and are horizontal lines.
The line x = 0 have slope = undefined and are vertical lines.
So, they are not parallel.

Question 27.
x = -4, x = 1
Answer:

Both lines are vertical and have an undefined slope.

FINDING SLOPE The points in the table lie on a line. Find the slope of the line.
Question 28.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (10 – 2)/(3 – 1) = (18 – 10)/(5 – 3) = (26 – 18)/(7 – 5)
m = 8/2 = 8/2 = 8/2
m = 4 = 4 = 4
Slope = 4

Question 9.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (2 – 0)/(2 – (-3)) = (4 – 2)/(7 – 2) = (6 – 4)/(12 – 7)
m = 2/5 = 2/5 = 2/5
m = 2/5

Question 30.
MODELING REAL LIFE
Carpenters refer to the slope of a roof as the pitch of the roof. Find the pitch of the roof.

Answer:
Pitch of the roof = rise/run
= 4/12 = 1/3

Question 31.
PROJECT
The guidelines for a wheelchair ramp suggest that the ratio of the rise to the run be no greater than 1 : 12.

a. CHOOSE TOOLS Find a wheelchair ramp in your school or neighborhood. Measure its slope. Does the ramp follow the guidelines?

Answer:
rise/run < 1/12
m = 0.06
1/12 = 0.0833
0.06 < 0.0833
As m < 1/12 the wheelchair ramp follows the guides.

b. Design a wheelchair ramp that provides access to a building with a front door that is 2.5 feet above the sidewalk. Illustrate your design.
Answer:
AC/AB = 1/12
2.5/AB = 1/12
AB = 2.5 × 12
AB = 30
So the end of the ramp should be placed at least 30 feet from the front door.

USING AN EQUATION Use an equation to find the value of k so that the line that passes through the given points has the given slope.
Question 32.
(1, 3), (5, k); m = 2
Answer:
A(1, 3)
B(5, k)
m = 2
2 = (k – 3)/(5 – 1)
2 × 4 = k – 3
8 = k – 3
k = 8 + 3
k = 11

Question 33.
(-2, k), (2, 0); m = -1
Answer:
Given,
A(-2, k)
B(2, 0)
m = -1
-1 = (0 – k)/2 – (-2)
-1 = -k/4
-4 = -k
k = 4

Question 34.
(-4, k), (6, -7); m = –\(\frac{1}{5}\)
Answer:
Given,
A(-4, k)
B(6, -7)
m = –\(\frac{1}{5}\)
–\(\frac{1}{5}\) = (-7 – k)/6 – (-4)
-2 = -7 – k
-2 + 7 = -k
5 = -k
k = -5

Question 35.
(4, -4), (k, -1); m = \(\frac{3}{4}\)
Answer:
\(\frac{3}{4}\) = (-1 – (-4))/(k – 4)
4 = k – 4
k = 4 + 4
k = 8

Question 36.
MODELING REAL LIFE
The graph shows the numbers of prescriptions filled over time by a pharmacy.

a. Find the slope of the line.
Answer:
(0, 0), (20, 5)
m = (5 – 0)/(20 – 0)
m = 5/20
m = 1/4
b. Explain the meaning of the slope as a rate of change.
Answer:
This means that every 4 minutes a prescription is filled.

Question 37.
CRITICAL THINKING
Which is steeper: the boatramp, or a road with a 12% grade? Note: Explain. (Road grade is the vertical increase divided by the horizontal distance.)

Answer:
Mramp = rise/run = 6/36 = 1/6
Mroad = 12% = 12/100 = 0.12
0.16 = 0.166… > 0.12
Mramp > Mroad
Therefore the slope of the ramp is steeper than the slope of the road.

Question 38.
REASONING
Do the points A(-2, -1), B(1, 5), and C(4, 11) lie on the same line? Without using a graph, how do you know?
Answer:
Given,
A(-2, -1), B(1, 5), and C(4, 11)
mAB = (5 – (-1))/(1 – (-2)) = 6/3 = 2
mBC = (11 – 5)/(4 – 1) = 6/3 = 2
By seeing the slopes we can say that the points A, B, C lie on the same line.

Question 39.
PROBLEM SOLVING
A small business earns a profit of $6500 in January and $17,500 in May. What is the rate of change in profit for this time period? Justify your answer.
Answer:
Pjan = 6500
Pmay = 17,500
Pmay – Pjan/5 – 1
= (17,500 – 6500)/4
= 11,000/4 = 2750

Question 40.
STRUCTURE
Choose two points in the coordinate plane. Use the slope formula to find the slope of the line that passes through the two points. Then find the slope using the formula \(\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\). Compare your results.
Answer:
P1(2, 5)
P2(3, 10)
m1 = (10 – 5)/(3 – 2) = 5/1 = 5
m2 = (5 – 10)/(1 – 3) = -5/-1 = 5
m1 = m2

Question 41.
DIG DEEPER!
The top and the bottom of the slide are level with the ground, which has a slope of 0.

a. What is the slope of the main portion of the slide?
b. Describe the change in the slope when the bottom of the slide is only 12 inches above the ground. Explain your reasoning.
Answer:
18 inches = 1.5 feet
mMC = rise/run = (8 – 1.5)/(12 – 1 – 1) = 6.5/10 = 0.65
AD = 1
mMC = CR/MR
= (8 – 1)/(12 – 1 – 1) = 7/10 = 0.7
The slope increases from 0.65 to 0.70 because the rise increasses, while the run stays the same.

Lesson 4.3 Graphing Proportional Relationships

EXPLORATION 1

Using a Ratio Table to Find Slope
Work with a partner. The graph shows amounts of vinegar and water that can be used to make a cleaning product.
a. Use the graph to make a ratio table relating the quantities. Explain how the slope of the line is represented in the table.

b. Make a ratio table that represents a different ratio of vinegar to water. Use the table to describe the slope of the graph of the new relationship.
Answer:

EXPLORATION 2

Deriving an Equation
Work with a partner. Let (x, y) represent any point on the graph of a proportional relationship.

a. Describe the relationship between the corresponding side lengths of the triangles shown in the graph. Explain your reasoning.
b. Use the relationship in part(a) to write an equation relating y, m, and x. Then solve the equation for y. How can you find the side lengths of the triangles in the graph?
c. What does your equation in part(b) describe? What does represent? Explain your reasoning.
Answer:

Try It

Question 1.
WHAT IF
The cost of frozen yogurt is represented by y = 0.75x. Graph the equation and interpret the slope.
Answer:
The equation shows that the slope m is 0.75. So the graph passes through the points (0, 0) and (1, 0.75).
Plot the ordered pairs and draw the graph.

The slope indicates that the unit cost is $0.75 per ounce.

Question 2.
How much would a spacecraft that weighs 3500 kilograms on Earth weigh on Titan?
Answer:
y = 1/7 x
y = 1/7 × 3500
y = 500 kg
So a spacecraft would weigh 500 kg on Titan.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

GRAPHING A PROPORTIONAL RELATIONSHIP Graph the equation.
Question 3.
y = 4x
Answer:

Question 4.
y = -3x
Answer:

Question 5.
y = 8x
Answer:

Question 6.
WRITING AND USING AN EQUATION
The number of objects a x machine produces is proportional to the time (in minutes) that the machine runs. The machine produces five objects in four minutes.
a. Write an equation that represents the situation.

Answer:
As 5 objects are produced in 4 minutes, the slope of the line is m = 5/4.
The equation that represents the situation is
y = 5/4 x
y = 1.25 x

b. Graph the equation in part (a) and interpret the slope.

Answer:
Use the slope. The equation shows that the slope m is 1.25. So the graph passes through the points (0, 0) and (1, 1.25)

c. How many objects does the machine produce in one hour?
Answer:

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.
The amount y (in liters) of water that flows over a natural waterfall in x seconds is represented by the equation y = 500x. The graph shows the number of liters of water that flow over an artificial waterfall. Which waterfall has a greater flow? Justify your answer.

Answer:
Given the equation y = 500x
15000 – 3000 = 12000
12000/4 = 3000
Mnatural = 500
3000 > 500
Therefore the artificial waterfall has greater flow.

Question 8.
The speed of sound in air is 343 meters per second. You see lightning and hear thunder 12 seconds later.
a. Is there a proportional relationship between the amount of time that passes and your distance from a lightning strike? Explain.

Answer:
y = kx
where k is the speed of sound, x the time and y the distance.
Yes, there is a proportional relationship between the amount of time that passes and your distance from the lightning strike as the further you are, the more time will pass until the sound reaches you.

b. Estimate your distance from the lightning strike.
Answer:
y = 343 × 12
= 4116 meters

Graphing Proportional Relationships Homework & Practice 4.3

Review & Refresh

Find the slope of the line.
Question 1.

Answer:
(x1, y1) = (0, -3)
(x2, y2) = (3, 0)
m = (y2 – y1)/(x2 – x1)
m = (0 – (-3))/(3 – 0)
m = (0 + 3)/(3 – 0)
m = 3/3
m = 1

Question 2.

Answer:
(x1, y1) = (0, 1)
(x2, y2) = (3, -5)
m = (y2 – y1)/(x2 – x1)
m = (-5 – 1)/(3 – 0)
m = -6/3
m = -2

Question 3.

Answer:
(x1, y1) = (0, 0)
(x2, y2) = (2, 8)
m = (y2 – y1)/(x2 – x1)
m = (8 – 0)/(2 – 0)
m = 8/2
m = 4

Solve the equation. Check your solution.
Question 4.
2x + 3x = 10
Answer:
Given the equation
2x + 3x = 10
5x = 10
x = 10/5
x = 2

Question 5.
x + \(\frac{1}{6}\) = 4 – 2x
Answer:
Given the equation
x + \(\frac{1}{6}\) = 4 – 2x
x + 2x = 4 – \(\frac{1}{6}\)
3x = 4 – \(\frac{1}{6}\)
3x = \(\frac{23}{6}\)
x = \(\frac{23}{18}\)

Question 6.
2(1 – x) = 11
Answer:
2(1 – x) = 11
2 – 2x = 11
2 – 11 = 2x
2x = -9
x = -9/2

Concepts, Skills, & Problem Solving

USING EQUIVALENT RATIOS The graph shows amounts of water and flour that can be used to make dough. (See Exploration 1, p. 155.)

Question 7.
Use the graph to make a ratio table relating the quantities. Explain how the slope of the line is represented in the table.
Answer:

m = rise/run
= (10 – 5)/(6 – 3)
= 5/3
That means to every 5 cups of flour there is an increase of 3 cups of water.
The slope m is 5/3.

Question 8.
Make a ratio table that represents a different ratio of flour to water. Use the table to describe the slope of the graph of the new relationship.
Answer:

From the table we find that for every increase of 7 cups of flour there is an increase of 4 cups of water.
The slope is 7/4.

Question 9.
GRAPHING AN EQUATION
The amount y(in dollars) that you raise by selling fundraiser tickets is represented by the equation y = 5x. Graph the equation and interpret the slope.
Answer:

The slope indicates that the unit cost is $5 per ticket.

IDENTIFYING PROPORTIONAL RELATIONSHIPS Tell whether and are in a proportional relationship. Explain your reasoning. If so, write an equation that represents the relationship.
Question 10.

Answer:
The graph doesn’t represent a proportional relationship because it doesn’t pass through the point (0, 0).

Question 11.

Answer:
The graph represents a proportional relationship because it is linear and passes through the point (0, 0)
(0, 0), (2, 8)
m = (8 – 0)/(2 – 0)
m = 8/2
m = 4
The equation is y = 4x

Question 12.

Answer:
(2 – 1)/(6 – 3) = 1/3
(3 – 2)/(9 – 6) = 1/3
(4 – 3)/(12 – 9) = 1/3
As the rate of change is constant, it means that the graph is a line.
(1 – y)/(3 – 0) = 1/3
(1 – y)/3 = 1/3
1 – y = 1
y = 1 – 1
y = 0
Therefore the point (0, 0) belomgs to the graph.
So the table represents a proportional relationship
y = 1/3 x

Question 13.

Answer:
(8 – 4)/(5 – 2) = 4/3
(13 – 8)/(8 – 5) = 5/3
(23 – 13)/10 – 8 = 10/2 = 5

Question 14.
MODELING REAL LIFE
The cost y (in dollars) to rent a kayak is proportional to the number x of hours that you rent the kayak. It costs $27 to rent the kayak for 3 hours.

a. Write an equation that represents the situation.
b. Interpret the slope of the graph of the equation.
c. How much does it cost to rent the kayak for 5 hours? Justify your answer.
Answer:
y = kx
27 = k × 3
k = 27/3
k = 9
The equation is k = 9x
b. The slope k = 3 shows that the cost of renting the kayak per hour is $9.
c. y = 9 × 5
y = 45

Question 15.
MODELING REAL LIFE
The distance y (in miles) that a truck travels on x gallons of gasoline is represented by the equation y = 18x. The graph shows the distance that a car travels.

a. Which vehicle gets better gas mileage? Explain how you found your answer.

Answer:
y = 18x
(0, 0), (2, 50)
m = (50 – 0)/(2 – 0)
m = 50/2
m = 25
25 > 18
Therefore the car has better mileage.

b. How much farther can the vehicle you chose in part(a) travel on 8 gallons of gasoline?
Answer:
y = 25 × 8 – 18 × 8
= 200 – 144
= 56 miles

Question 16.
PROBLEM SOLVING
Toenails grow about 13 millimeters per year. The table shows fingernail growth.

a. Do fingernails or toenails grow faster? Explain.

Answer:
y = 0.25x
m = (1.4 – 0.7)/(2 – 1)
m = 0.7
y = 0.7x
Because 0.7 > 0.25, the fingernails grow faster.

b. In the same coordinate plane, graph equations that represent the growth rates of toenails and fingernails. Compare and interpret the steepness of each graph.
Answer:

Question 17.
REASONING
The quantities and are in a proportional relationship. What do you know about the ratio of y to x for any point (x, y) on the graph of x and y?
Answer:
y = kx
where k is constant
y/x = k
This means the ratio of y to x is constant.

Question 18.
DIG DEEPER!
The graph relates the temperature change y (in degrees Fahrenheit) to the altitude change x (in thousands of feet).

a. Is the relationship proportional? Explain.

Answer: The relationship is proportional because the graph is linear and passes through the origin.

b. Write an equation of the line. Interpret the slope.

Answer:
(0,0), (10, -35)
m = (-35 – 0)/(10 – 0)
= -35/10
= -3.5
y = -3.5x

c. You are at the bottom of a mountain where the temperature is 74°F. The top of the mountain is 5500 feet above you. What is the temperature at the top of the mountain? Justify your answer.
Answer:
x = 5.5 – 0 = 5.5 thousand feet
y = -3.5x = -3.5(5.5) = -19.25
74 – 19.25 = 54.75°F

Question 19.
CRITICAL THINKING
Consider the distance equation d = rt, where d is the distance (in feet), r is the rate (in feet per second), and t is the time (in seconds). You run for 50 seconds. Are the distance you run and the rate you run at proportional? Use a graph to justify your answer.
Answer:
d = rt
d = 50r
Having the form y = kx the equation represents a proportional relationship.

Lesson 4.4 Graphing Linear Equations in Slope-Intercept Form

EXPLORATION 1

Deriving an Equation
Work with a partner. In the previous section, you learned that the graph of a proportional relationship can be represented by the equation y = mx, where m is the constant of proportionality.

a. You translate the graph of a proportional relationship 3 units up as shown below. Let (x, y) represent any point on the graph. Make a conjecture about the equation of the line. Explain your reasoning.

b. Describe the relationship between the corresponding side lengths of the triangles. Explain your reasoning.
c. Use the relationship in part(b) to write an equation relating y, m, and x. Does your equation support your conjecture in part(a)? Explain.
d. You translate the graph of a proportional relationship b units up. Write an equation relating y, m, x, and b. Justify your answer.
Answer:

Try It

Find the slope and the y-intercept of the graph of the linear equation.
Question 1.
y = 3x – 7
Answer:
Given the equation
y = 3x – 7
Write the equation in slope – intercept form: y = mx + b
The slope of the line is m and the y – intercept of the line is b.
y = 3x – 7
Slope = 3 and y – intercept = -7

Question 2.
y – 1 = –\(\frac{2}{3}\)x
Answer:
Write the equation in slope – intercept form: y = mx + b
The slope of the line is m and the y – intercept of the line is b.
y – 1 = –\(\frac{2}{3}\)x
y = –\(\frac{2}{3}\)x + 1
Slope = –\(\frac{2}{3}\) and y – intercept = 1

Graph the linear equation. Identify the x-intercept.
Question 3.
y = x – 4
Answer:
y = x – 4
Comparing the above equation with slope – intercept equation.
slope = 1, y-intercept = -4
Ploy y – intercept and slope
slope = rise/run = 1/1
Plot the point that is 1 unit right and 1 unit up from (0, -4) = (1, -3)

Thus the intercept is 4.

Question 4.
y = –\(\frac{1}{2}\)x + 1
Answer:
y = –\(\frac{1}{2}\)x + 1
Comparing the above equation with slope – intercept equation.
Slope = –\(\frac{1}{2}\), y-intercept = 1
y-intercept = 1. So plot (0, 1)
Slope = rise/run = -1/2
Plot the point that is 2 units right and 1 unit down from (0, -4) = (2, 0)

So, the x-intercept is 2.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
IN YOUR OWN WORDS
Consider the graph of the equation y = mx + b.
a. How does changing the value of m affect the graph of the equation?

Answer:
The value of m is the slope of the graph. If the value of m changes it means the slope of the graph is changing, whether it will rise or fall from left or right is dependent on the value of m.

b. How does changing the value of b affect the graph of the equation?
Answer:
The value of b is the y-intercept of the graph. If the value of b changes it means it affects where the graph crosses the y – axis.

IDENTIFYING SLOPE AND y-INTERCEPT Find the slope and the y-intercept of the graph of the linear equation.
Question 6.
y = -x + 0.25
Answer:
y = mx + c
slope = -1 and y – intercept = 0.25

Question 7.
y – 2 = –\(\frac{3}{4}\)x
Answer:
Given the equation
y – 2 = –\(\frac{3}{4}\)x
y = –\(\frac{3}{4}\)x + 2
slope = –\(\frac{3}{4}\) and y – intercept = 2

GRAPHING A LINEAR EQUATION Graph the linear equation. Identify the x-intercept.
Question 8.
y = x – 7
Answer:

The line crosses the x-axis at (7, 0)
So, the x – intercept is 7.

Question 9.
y = 2x + 8
Answer:

The line crosses the x – axis at (-4, 0)
So, the x – intercept is -4.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
The height y (in feet) of a movable bridge after rising for seconds is represented by the equation y = 3x + 6. Graph the equation. Interpret the y-intercept and slope. How many seconds does it take the bridge to reach a height of 76 feet? Justify your answer.

Answer:
Given,
y = 3x + 6.
slope = 3, y – intercept = 16

The y – intercept is 16. So, the initial height of the bridge is 16 feet.
The slope is 3. So, the bridge rises 3 feet per second.
The bridge will reach a height of 76 feet in 20 seconds.

Question 11.
The number of perfume bottles in storage after x months is represented by the equation y = -20x + 460. Graph the equation. Interpret the y-intercept and the slope. In how many months will there be no perfume bottles left in storage? Justify your answer.

Answer:
Given the equation
y = -20x + 460
Slope = -20, y-intercept = 460

The y-intercept is 460. So, the initial number of perfume in the storage is 460.
The slope is -20. So, the number of perfume bottle decrease with 20 bottles per months.
There will be no perfume bottle left in the storage in 23 months.

Graphing Linear Equations in Slope-Intercept Form Homework & Practice 4.4

Review & Refresh

Tell whether x and y are in a proportional relationship. Explain your reasoning. If so, write an equation that represents the relationship.
Question 1.

Answer:
(8 – 6)/(2 – 1) = 2/1 = 2
(10 – 8)/(3 – 2) = 2/1 = 2
(12 – 10)/(4 – 3) = 2/1 = 2
The rate of change in the table is constant.
(6 – y)/(1 – 0) = 2
6 – y = 2
y = 6 – 2
y = 4
Therefore the graph does not pass through the origin.
So x and y are not proportional.

Question 2.

Answer:
(4 – 0)/(-8 – 0) = 4/-8 = -1/2 = -0.5
(2 – 4)/(-4 – (-8)) = -2/4 = -1/2 = -0.5
(-2 – 2)/(4 – (-4)) = -4/8 = -1/2 = -0.5
(-4 – (-2))/(8 – 4) = -2/4 = -1/2 = -0.5
As the rate of change is constant, x and y are in a proportional relationship.
y = -0.5x

Solve the equation for y.
Question 3.
x = 4y – 2
Answer:
Given the equation
x = 4y – 2
x – 2 = 4y
y = x/4 + 1/2

Question 4.
3y = -6x + 1
Answer:
Given the equation
3y = -6x + 1
y = -2x + 1/3

Question 5.
1 + y = –\(\frac{4}{5}\)x – 2
Answer:
Given the equation
1 + y = –\(\frac{4}{5}\)x – 2
y = –\(\frac{4}{5}\)x – 3

Question 6.
2.5y = 5x – 5
Answer:
Given the equation
2.5y = 5x – 5
y = 2x – 2

Question 7.
1.3y + 5.2 = -3.9x
Answer:
Given the equation
1.3y + 5.2 = -3.9x
1.3y = -3.9x – 5.2
y = -3x – 4

Question 8.
y – \(\frac{2}{3}\)x = -6
Answer:
Given the equation
y – \(\frac{2}{3}\)x = -6
y = \(\frac{2}{3}\)x -6

Concepts, Skills, &Problem Solving

GRAPHING A LINEAR EQUATION Graph the equation. (See Exploration 1, p. 161.)
Question 9.
The graph of y = 3.5x is translated up 2 units.
Answer:
Given the equation
y = 3.5x
The line obtained by translating the graph of the line y = 3.5x up 2 units has the same slope (3.5) and y – intercept 2 units greater, which means b = 0 + 2 = 2

Question 10.
The graph of y = -5x is translated down 3 units.
Answer:
y = -5x
The line obtained by translating the graph of the line y = -5x down 3 units has the same slope and the y – intercept 3 units smaller, which means b = 0 – 3 = -3

MATCHING EQUATIONS AND GRAPHS Match the equation with its graph. Identify the slope and the y-intercept.
Question 11.
y = 2x + 1
Answer:
Given the eqation
y = 2x + 1
slope = 2 and y – intercept = 1

Question 12.
y = \(\frac{1}{3}\)x – 2
Answer:
slope = 1/3 and y – intercept = -2

Question 13.
y = –\(\frac{2}{3}\)x + 1
Answer:

Answer:
Slope = -2/3 and y – intercept = 1
The graph which passes through the point (0, 1) and has a negative slope is the matching graph of the given equation.

IDENTIFYING SLOPES AND y-INTERCEPTS Find the slope and the y-intercept of the graph of the linear equation.
Question 14.
y = 4x – 5
Answer:
y = mx + b
slope = 4 and y — intercept = -5

Question 15.
y = -7x + 12
Answer:
y = -7x + 12
y = mx + b
slpoe = -7 and y – intercept = 12

Question 16.
y = –\(\frac{4}{5}\)x – 2
Answer:
y = mx + b
slope = -4/5
y – intercept = -2

Question 17.
y = 2.25x + 3
Answer:
y = mx + b
slope = 2.25 and y – intercept = 3

Question 18.
y + 1 = \(\frac{4}{3}\)x
Answer:
y = mx + b
y + 1 = \(\frac{4}{3}\)x
y = \(\frac{4}{3}\)x – 1
slope = \(\frac{4}{3}\), y – intercept = -1

Question 19.
y – 6 = \(\frac{3}{5}\)x
Answer:
y = mx + b
y – 6 = \(\frac{3}{5}\)x
y = \(\frac{3}{5}\)x + 6
slope = 3/8 and y – intercept = 6

Question 20.
y – 3.5 = -2x
Answer:
y = mx + b
y – 3.5 = -2x
y = -2x + 3.5
slope = -2 and y – intercept = 3.5

Question 21.
y = -5 – \(\frac{1}{2}\)x
Answer:
y = mx + b
y = -5 – \(\frac{1}{2}\)x
y =- \(\frac{1}{2}\)x – 5
slope = – \(\frac{1}{2}\) and y – intercept = -5

Question 22.
y = 11 + 1.5x
Answer:
y = mx + b
y = 1.5x + 11
slope = 1.5 and y – intercept = 11

Question 23.
YOU BE THE TEACHER
Your friend finds the slope and y-intercept of the graph of the equation y = 4x – 3. Is your friend correct? Explain your reasoning.

Answer:
y = 4x – 3
No my friend is not correct because the y – intercept is -3.

Question 24.
MODELING REAL LIFE
The number y of seasonal allergy shots available at a facility x days after receiving a shipment is represented by y = -15x + 375.
a. Graph the linear equation.
b. Interpret the slope and the y-intercept.
Answer:
y = -15x + 375
x = 0
y = -15(0) + 375 = 375
y = 0
0 = -15x + 375
15x = 375
x = 375/15
x = 25

The slope shows that the number of seasonal allergy shots decrease by 15 shots each day.
The y – intercept shows that the number of shots immediately after receiving a shipment is 375.

GRAPHING AN EQUATION Graph the linear equation. Identify the x-intercept.
Question 25.
y = x + 3
Answer:
Given the equation
y = x + 3
slope = 1 and y – intercept = 3
Slope = rise/run = 1/1
Plot the point that is 1 unit right and 1 unit up from (0, 3) = (1, 4)

So, the x – intercept is -3.

Question 26.
y = 4x – 8
Answer:
y = 4x – 8
Comparing the above equation with slope – intercept equation.
slope = 4 and y – intercept = -8
Slope = rise/run = 4/1 = 4
Plot the point that is 1 unit right and 4 unit up from (0, -8) = (1, -4)

Question 27.
y = -3x + 9
Answer:
y = -3x + 9
slope = -3 and y – intercept = 9
slope rise/run = -3/1 = -3

So, the intercept is 3.

Question 28.
y = -5x – 5
Answer:
y = -5x – 5
slope = -5 and y – intercept = -5
slope = rise/run = -5/1
Plot the point that is 1 unit right and 5 unit up from (0, -5) = (1, -10)

So, the x – intercept is -1.

Question 29.
y + 14 = -7x
Answer:
y + 14 = -7x
y = -7x – 14
slope = -7 and y – intercept = -14
Slope = rise/run = -7/1
Plot the point that is 1 unit right and 7 unit down from (0, -14) = (1, -21)

So, the x – intercept is -2.

Question 30.
y = 8 – 2x
Answer:
Given the equation
y = 8 – 2x
y = -2x + 8
slope = -2 and y – intercept = 8
slope = rise/run = -2/1
Plot the point 1 unit right and 2 units down from (0, 8) = (1, 6)

So, the x – intercept is 4.

Question 31.
PRECISION
You go to a harvest festival and pick apples.
a. Which equation represents the cost (in dollars) of going to the festival and picking x pounds of apples? Explain.

b. Graph the equation you chose in part(a).
Answer:
Picking a pound of apples costs $0.75, therefore x pounds cost 0.75 × x = 0.75x
y = 0.75x + 5

Question 32.
REASONING
Without graphing, identify the equations of the lines that are parallel. Explain your reasoning.

Answer:
The lines which area parallel are those having the same slope.
y = 2x + 4
y = 2x – 3
y = 2x + 1
y = 1/2x + 1
y = 1/2x + 2

Question 33.
PROBLEM SOLVING
A skydiver parachutes to the ground. The height y (in feet) of the skydiver after x seconds is y = -10x + 3000.

a. Graph the linear equation.
b. Interpret the slope, y-intercept, and x-intercept.
Answer:
y = -10x + 3000
x = 0
y = -10(0) + 3000 = 3000
y = 0
0 = -10 + 3000
10x = 3000
x = 3000/10 = 300

b. The slope shows that each second the skydiver descends 10 feet.
The y – intercept shows that the skydiver begins its dive from 3000 feet.
The x – intercept shows that he reaches the ground after 300 seconds.

Question 34.
DIG DEEPER!
Six friends create a website. The website earns money by selling banner ads. It costs $120 a month to operate the website.
a. A banner ad earns $0.005 per click. Write a linear equation that represents the monthly profit after paying operating costs.
b. Graph the equation in part(a). On the graph, label the number of clicks needed for the friends to start making a profit. Explain.
Answer:
y = 0.005x – 120
x = 0
y = 0.005(0) – 120
y = -120
y = 0
0 = 0.005x – 120
0.005x = 120
x = 24000

x > 24,000

Lesson 4.5 Graphing Linear Equations in Standard Form

EXPLORATION 1

Using Intercepts
Work with a partner. You spend $150 on fruit trays and vegetable trays for a party.

a. You buy x fruit trays and y vegetable trays. Complete the verbal model. Then use the verbal model to write an equation that relates x and y.

b. What is the greatest number of fruit trays that you can buy? vegetable trays? Can you use these numbers to graph your equation from part (a) in the coordinate plane? Explain.

c.Use a graph to determine the different combinations of fruit trays and vegetable trays that you can buy. Justify your answers algebraically.
d. You are given an extra $50 to spend. How does this affect the intercepts of your graph in part(c)? Explain your reasoning.
Answer:

Try It

Graph the linear equation.
Question 1.
x + y = -2
Answer:
Given the equation
y = mx + b
x + y = -2
y = -x – 2
Comparing the value of b and m from y = mx + b
m = -1 and b = -2
Plot y – intercept = (0, b) = (0, -2)
Slope = -1
run/rise = -1/1
Plot the point 1 unit down and 1 unit to the right = (1, -3)
Now plot the points and draw the graph

Question 2.
–\(\frac{1}{2}\)x + 2y = 6
Answer:
–\(\frac{1}{2}\)x + 2y = 6
2y = 6 + \(\frac{1}{2}\)x
y = 0.25x + 3
Comparing the value of b and m from y = mx + b
m = 0.25 and b = 3
Plot y – intercept = (0, b) = (0, 3)
Slope = 0.25
run/rise = 0.25/1
Plot the point 0.25 unit up and 1 unit to the right = (1, 3.25)
Now plot the points and draw the graph

Question 3.
–\(\frac{2}{3}\)x + y = 0
Answer:
–\(\frac{2}{3}\)x + y = 0
y = \(\frac{2}{3}\)x
Comparing the value of b and m from y = mx + b
m = \(\frac{2}{3}\) and b = 0
Plot y – intercept = (0, b) = (0, 0)
Slope =\(\frac{2}{3}\)
run/rise = \(\frac{2}{3}\)
Plot the point 0.25 unit up and 1 unit to the right = (3, 2)
Now plot the points and draw the graph

Question 4.
2x + y = 5
Answer:
2x + y = 5
y = -2x + 5
Comparing the value of b and m from y = mx + b
m = -2 and b = 5
Plot y – intercept = (0, b) = (0, 5)
Slope = -2
run/rise = \(\frac{-2}{1}\)
Plot the point 0.25 unit up and 1 unit to the right = (1, 3)
Now plot the points and draw the graph

Graph the linear equation using intercepts.
Question 5.
2x – y = 8
Answer:
y = 0
2x – y = 8
2x – 0 = 8
2x = 8
x = 4
The x – intercept is (4, 0)
Y – intercept :
x = 0
2x – y = 8
2(0) – y = 8
y = -8

Question 6.
x + 3y = 6
Answer:
X-intercept:
y = 0
x + 3y = 6
x + 3(0) = 6
x + 0 = 6
x = 6
The x – intercept is (6, 0)
Y – intercept:
x = 0
x + 3y = 6
0 + 3y = 6
y = 2
The y – intercept is (0, 2)

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

STRUCTURE Determine whether the equation is in standard form. If not, rewrite the equation in standard form.
Question 7.
y = x – 6
Answer:
y = x – 6
The standard form of equation is: Ax + By = C
The given equation is not in the standard form.
y = x – 6
x – y = 6

Question 8.
y – \(\frac{1}{6}\)x + 5 = 0
Answer:
The standard form of equation is: Ax + By = C
The given equation is not in the standard form.
y – \(\frac{1}{6}\)x + 5 = 0
\(\frac{1}{6}\)x – y = 5

Question 9.
4x + y = 5
Answer:
The standard form of equation is: Ax + By = C
The given equation is in the form of the standard form.

Question 10.
WRITING
Describe two ways to graph the equation 4x + 2y = 6.
Answer:
The two ways to graph the equation:
1. Graph the equation using standard form
2. Graph the equation using intercept.

GRAPHING A LINEAR EQUATION Graph the linear equation.
Question 11.
4x + y = 5
Answer:
Given the equation
4x + y = 5
y = -4x + 5
Comparing the value of b and m from y = mx + b
m = -4 and b = 5
Plot y – intercept = (0, b) = (0, 5)
Slope = -4
run/rise = \(\frac{-4}{1}\)
Plot the point 4 unit down and 1 unit to the right = (1, 1)
Now plot the points and draw the graph

Question 12.
\(\frac{1}{3}\)x + 2y = 8
Answer:
X – intercept:
y = 0
\(\frac{1}{3}\)x + 2y = 8
\(\frac{1}{3}\)x + 2(0) = 8
\(\frac{1}{3}\)x = 8
x = 24
The x – intercept is (24, 0)
Y – intercept:
x = 0
\(\frac{1}{3}\)x + 2y = 8
\(\frac{1}{3}\)(0) + 2y = 8
2y = 8
y = 4
The y – intercept is (0, 4)

Question 13.
5x – y = 10
Answer:
X – intercept:
y = 0
5x – 0 = 10
5x = 10
x = 2
The x-intercept is (2, 0)
Y – intercept:
x = 0
5x – y = 10
5(0) – y = 10
-y = 10
y = -10
The y – intercept is (0, -10)

Question 14.
x – 3y = 9
Answer:
X – intercept:
y = 0
x – 3(0) = 9
x = 9
The x – intercept is (9, 0)
Y – intercept:
x = 0
0 – 3y = 9
-3y = 9
y = -3
The y – intercept is (0, -3)

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 15.
You have $30 to spend on paint and clay. The equation 2x + 6y = 30 represents this situation, where x is the number of paint bottles and y is the number of tubs of clay. Graph the equation. Interpret the intercepts. How many bottles of paint can you buy if you buy 3 tubs of clay? Justify your answer.
Answer:
Given,
You have $30 to spend on paint and clay.
The equation 2x + 6y = 30 represents this situation,
where x is the number of paint bottles and y is the number of tubs of clay.
X – intercept:
y = 0
2x + 6y = 30
2x + 6(0) = 30
2x = 30
x = 15
The x – intercept is (15, 0)
Y – intercept:
x = 0
2x + 6y = 30
2(0) + 6y = 30
6y = 30
y = 5
The y – intercept is (0, 5)

From the graph, I can buy 6 bottles of point if I buy 3 tubs of clay.

Question 16.
You complete two projects for a class in 60 minutes. The equation x + y = 60 represents this situation, where x is the time (in minutes) you spend assembling a birdhouse and y is the time (in minutes) you spend writing a paper.

a. Graph the equation. Interpret the intercepts.

Answer:
x + y = 60
y = -x + 60

b. You spend twice as much time assembling the birdhouse as you do writing the paper. How much time do you spend writing the paper? Justify your answer.
Answer:
We are given,
y = 2x
2x = -x + 60
2x + x = 60
3x = 60
x = 20
y = 2 (20)
y = 40

Graphing Linear Equations in Standard Form Homework & Practice 4.5

Review & Refresh

Find the slope and the y-intercept of the graph of the linear equation.
Question 1.
y = x – 1
Answer:
y = mx + b
Slope = -1 and y – intercept = -1

Question 2.
y = -2x + 1
Answer:
y = -2x + 1
y = mx + b
Slope = -2 and y – intercept = 1

Question 3.
y = \(\frac{8}{9}\)x – 8
Answer:
y = \(\frac{8}{9}\)x – 8
y = mx + b
Slope = \(\frac{8}{9}\) and y – intercept = -8

Tell whether the blue figure is a reflection of the red figure.
Question 4.

Answer:
The blue figure is not a reflection of the red figure because, for example the reflection of the upper leg of the upper leg of the red triangle across the y-axis is the top vertex of the blue triangle, not a point.

Question 5.

Answer:
The blue figure is a reflection of the red figure because to each point in the red figure corresponds a symmetrical point in the blue figure.

Question 6.

Answer:
The blue figure is a reflection of the red figure because to each point in the red figure corresponds a symmetrical point in the blue figure.

Concepts, Skills, &Problem Solving

USING INTERCEPTS Define two variables for the verbal model. Write an equation in slope-intercept form that relates the variables. Graph the equation using intercepts. (See Exploration 1, p. 167.)
Question 7.

Answer:
x = amount of peaches (in pounds)
y = the amount of apples (in pounds)
2x + 1.5y = 15
y = 0 = 2x + 1.5(0) = 15
2x = 15
x = 7.5
x = 0
2(0) + 1.5y = 15
1.5y =15
y = 10

Question 8.

Answer:
x = the biked distance (in miles)
y = the walked distance (in miles)
y = 0
16x + 2(0) = 32
16x = 32
x = 2
x = 0
16(0) + 2y = 32
2y = 32
y = 16

REWRITING AN EQUATION Write the linear equation in slope-intercept form.
Question 9.
2x + y = 17
Answer:
Given the equation
2x + y = 17
y = 17 – 2x
y = -2x + 17

Question 10.
5x – y = \(\frac{1}{4}\)
Answer:
Given the equation
5x – y = \(\frac{1}{4}\)
-y = \(\frac{1}{4}\) – 5x
y = 5x – \(\frac{1}{4}\)

Question 11.
–\(\frac{1}{2}\)x + y = 10
Answer:
Given the equation
–\(\frac{1}{2}\)x + y = 10
y = \(\frac{1}{2}\)x + 10

GRAPHING AN EQUATION Graph the linear equation.
Question 12.
-18x + 9y = 72
Answer:
Given the equation
-18x + 9y = 72
X – intercept:
y = 0
-18x + 9(0) = 72
-18x = 72
x = -4
The x – intercept is (-4, 0)
Y – intercept:
x = 0
-18x + 9y = 72
-18(0) + 9y = 72
9y = 72
y = 8

Question 13.
16x – 4y = 2
Answer:
Given the equation
16x – 4y = 2
X – intercept:
y = 0
16x – 4y = 2
16x – 4(0) = 2
16x = 2
x = 0.125
The X – intercept is (0.125, 0)
Y – intercept:
x = 0
16(0) – 4y = 2
-4y = 2
y = -2

Question 14.
\(\frac{1}{4}\)x + \(\frac{3}{4}\)y = 1
Answer:
Given the equation
\(\frac{1}{4}\)x + \(\frac{3}{4}\)y = 1
x + 3y = 4
y = 0
x + 3(0) = 4
x = 4
x = 0
0 + 3y = 4
3y = 4
y = 4/3

MATCHING Match the equation with its graph.
Question 15.
15x – 12y = 60
Answer:
y = 0
15x – 12(0) = 60
15x = 60
x = 60/15
x = 4
x = 0
15(0) – 12y = 60
-12y = 60
y = -5
The graph having the x – intercept 4 and y – intercept -5

Question 16.
5x + 4y = 20
Answer:
Given the linear equation
5x + 4y = 20
y = 0
5x + 4(0) = 20
5x = 20
x = 4
x = 0
5(0) + 4y = 20
4y = 20
y = 5

Question 17.
10x + 8y = -40
Answer:

10x + 8y = -40
y = 0
10x + 8(0) = -40
10x = -40
x = -4
x = 0
10(0) + 8y = -40
8y = -40
y = -5

Question 18
YOU BE THE TEACHER
Your friend finds the x-intercept of -2x + 3y = 12. Is your friend correct? Explain your reasoning.

Answer:
-2x + 3y = 12
y = 0
-2x + 3(0) = 12
-2x = 12
x = -6
Your friend is not correct because the x – intercept is the value of x corresponding to y = 0.
Your friend computed the y – intercept.

Question 19.
MODELING REAL LIFE
A charm bracelet costs $65, plus $25 for each charm. The equation -25x + y = 65 represents the cost y (in dollars) of the bracelet, where x is the number of charms.
a. Graph the equation.
b. How much does a bracelet with three charms cost?
Answer:

y = 25x + 65
Substitute the value of x in the equation
y = 25(3) + 65
y = 75 + 65
y = 140

USING INTERCEPTS TO GRAPH Graph the linear equation using intercepts.
Question 20.
3x – 4y = -12
Answer:
Given the equation
3x – 4y = -12
3x – 4(0) = -12
3x = -12
x = -4
The x – intercept is (-4, 0)
Y – intercept:
x = 0
3(0) – 4y = -12
-4y = -12
y = 3
The y – intercept is (0, 3)

Question 21.
2x + y = 8
Answer:
X – intercept:
y = 0
2x + y = 8
2x + 0 = 8
2x = 8
x = 4
The x – intercept is (4, 0)
Y – intercept:
x = 0
2x + y = 8
2(0) + y = 8
y = 8
The y – intercept is (0, 8)

Question 22.
\(\frac{1}{3}\)x – \(\frac{1}{6}\)y = –\(\frac{2}{3}\)
Answer:
X – intercept:
y = 0
\(\frac{1}{3}\)x – \(\frac{1}{6}\)(0) = –\(\frac{2}{3}\)
\(\frac{1}{3}\)x = –\(\frac{2}{3}\)
x = -2
The x – intercept is (-2, 0)
Y – intercept:
x = 0
\(\frac{1}{3}\)(0) – \(\frac{1}{6}\)y = –\(\frac{2}{3}\)
y = 4
The y – intercept is (0, 4)

Question 23.
MODELING REAL LIFE
Your cousin has $90 to spend on video games and movies. The equation 30x + 15y = 90 represents this situation, where x is the number of video games purchased and y is the number of movies purchased. Graph the equation. Interpret the intercepts.
Answer:
30x + 15y = 90
x = 0
30(0) + 15y = 90
15y = 90
y = 6
y = 0
30x + 15(0) = 90
30x = 90
x = 3

The x – intercept shows that 3 video games are purchased when no movies are purchased.
The y – intercept shows that 6 movies are purchased when no video games are purchased.

Question 24.
PROBLEM SOLVING
A group of friends go scuba diving. They rent a boat for x days and scuba gear for y people, represented by the equation 250x + 50y = 1000.

a. Graph the equation and interpret the intercepts.
b. How many friends can go scuba diving if they rent the boat for 1 day? 2 days?
c. How much money is spent in total?
Answer:
250x + 50y = 1000
x = 0
250(0) + 50y = 1000
50y = 1000
y = 20
when y = 0
250x + 50(0) = 1000
250x = 1000
x = 4

b.
250(1) + 50y = 1000
250 + 50y = 1000
50y = 1000 – 250
50y = 750
y = 15
when x = 2
250(2) + 50y = 1000
500 + 50y = 1000
50y = 1000 – 500
50y = 500
y = 500/50
y = 10

Question 25.
DIG DEEPER!
You work at a restaurant as a host and a server. You earn $9.45 for each hour you work as a host and $3.78 for each hour you work as a server.

a. Write an equation in standard form that models your earnings.
b. Graph the equation.
Answer:
You earn $9.45 for each hour you work as a host and $3.78 for each hour you work as a server.
Number of hours worked as host + $3.78.
Number of hours worked as server = $113.40
9.45x + 3.78y = 113.40
x = 0
9.45(0) + 3.78y = 113.40
3.78y = 113.40
y = 30
when y = 0
9.45x + 3.78(0) = 113.40
9.45x = 113.40
x = 12

Question 26.
LOGIC
Does the graph of every linear equation have an x-intercept? Justify your reasoning.
Answer:
y = mx + b
y = 0
0 = mx + b
mx = -b
x = -b/m for m ≠ 0
If m = 0 the equation has no solution. Therefore the equation y = b has no x – intercept.

Question 27.
CRITICAL THINKING
For a house call, a veterinarian charges $70, plus $40 per hour.

a. Write an equation that represents the total fee y (in dollars) the veterinarian charges for a visit lasting x hours.

b. Find the x-intercept. Does this value make sense in this context? Explain your reasoning.
c. Graph the equation.
Answer:
Total fee = fixed charge + number of hours . cost per hour
y = 70 + 40x
y = 0
0 = 70 + 40x
-70 = 40x
x = -1.75
x = 0
y = 70 + 40(0)
y = 70

Lesson 4.6 Writing Equations in Slope-Intercept Form

EXPLORATION 1

Writing Equations of Lines
Work with a partner.For each part, answer the following questions.

• What are the slopes and the y-intercepts of the lines?
• What are equations that represent the lines?
• What do the lines have in common?

Answer:

EXPLORATION 2

Interpreting the Slope and the y-Intercept
Work with a partner. The graph represents the distance y (in miles) of a car from Phoenix after t hours of a trip.

a. Find the slope and the y-intercept of the line. What do they represent in this situation?
b. Write an equation that represents the graph.
c. How can you determine the distance of the car from Phoenix after 11 hours?
Answer:

Try It

Write an equation in slope-intercept form of the line that passes through the given points.
Question 1.

Answer:
m = (y2 – y1)/(x2 – x1)
= (4 – 2)/(1 – 0)
= 2/1
= 2
Because the line crosses the y – axis at (0, 2)
y = mx + b
y = 2x + 2

Question 2.

Answer:
m = (y2 – y1)/(x2 – x1)
= (-1 – 3)/(0 – (-3))
= -4/3
Because y = -1 when x = 0, the y – intercept is -1
y = mx + b
y = -4/3 x – 1

Write an equation of the line that passes through the given points.
Question 3.

Answer:
m = (y2 – y1)/(x2 – x1)
= (5 – 5)/(0 – (-4))
= 0/4
Because y = 5 when x = 0, the y – intercept is 5
y = mx + b
y = (0)x + 5
y = 5

Question 4.

Answer:
m = (y2 – y1)/(x2 – x1)
= (1 – 1)/(3 – 0)
= 0/3
= 0
Because the line crosses the y – axis at (0, 1) the y – intercept is 1
y = mx + b
y = (0)x + 1
y = 1

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING EQUATIONS IN SLOPE-INTERCEPT FORM Write an equation in slope-intercept form of the line that passes through the given points.
Question 5.

Answer:
m = (y2 – y1)/(x2 – x1)
= (5 – 2)/(1 – 0)
= 3/1
= 3
Because y = 2 when x = 0, the y – intercept is 2
y = mx + b
y = (3)x + 2
y = 3x + 2

Question 6.

Answer:
m = (y2 – y1)/(x2 – x1)
= (-1 – 5)/(1 – (-1))
= -6/2
= -3
Because the line crosses the y – axis at (0, 2) the y – intercept is 2
y = mx + b
y = -3x + 2

Question 7.
WRITING AN EQUATION
Write an equation of the line that passes through (0, -5) and (2, -5).
Answer:
m = (y2 – y1)/(x2 – x1)
= (-5 – (-5))/(2 – 0)
= 0/2
= 0
Because y = -5 when x = 0, the y – intercept is -5
y = mx + b
y = (0)x + -5
y = -5

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 8.
You load boxes onto an empty truck at a constant rate. After 3 hours, there are 100 boxes on the truck. How much longer do you work if you load a total of 120 boxes? Justify your answer.
Answer:
Let x be the number of hours you work if you load a total of 120 boxes.
100/3 = 120/x
100x = 3 × 120
x = 360/100
x = 3.6 hours
3.6 – 3 = 0.6 hours

Question 9.
The table shows the amounts (in tons) of waste left in a landfill after x months of waste relocation. Interpret the slope and the y-intercept of the line that passes through the given points. How many months does it take to empty the landfill? Justify your answer.

Answer:
m = (12 – 15)/ (6 – 0)
m = -3/6
m = -0.5
b = 15
The y – intercept shows that there are 150 tons of waste in the beginning.
y = -0.5x + 15
y = 0
0 = -0.5x + 15
x = 30
So the ladfill will be emptied after 30 months.

Question 10.
DIG DEEPER!
A lifetime subscription to a website costs $250. A monthly subscription to the website costs $10 to join and $15 per month. Write equations to represent the costs of each plan. If you want to be a member for one year, which plan is less expensive? Explain.
Answer:
Given,
A lifetime subscription to a website costs $250. A monthly subscription to the website costs $10 to join and $15 per month.
Total cost for plan 1 = the lifetime subscription
y = 250
Total cost for Plan 2 = Fixed tax + Number of months . monthly cost
y = 10 + 15x
Plan 1: y = 250
Plan 2: y = 10 + 15(12) = 190
As 190 < 250, plan 1 is less expensive.

Writing Equations in Slope-Intercept Form Homework & Practice 4.6

Review & Refresh

Write the linear equation in slope-intercept form.
Question 1.
4x + y = 1
Answer:
Given the equation
4x + y = 1
y = -4x + 1

Question 2.
x – y = \(\frac{1}{5}\)
Answer:
Given the equation
x – y = \(\frac{1}{5}\)
x – \(\frac{1}{5}\) = y

Question 3.
–\(\frac{2}{3}\)x + 2y = -7
Answer:
Given the equation
–\(\frac{2}{3}\)x + 2y = -7
2y = -7 + \(\frac{2}{3}\)x
y = \(\frac{1}{3}\)x – \(\frac{7}{2}\)

Plot the ordered pair in a coordinate plane.
Question 4.
(1, 4)
Answer:

Question 5.
(-1, -2)
Answer:

Question 6.
(0, 1)
Answer:

Question 7.
(2, 7)
Answer:

Concepts, Skills, & Problem Solving

INTERPRETING THE SLOPE AND THE y-INTERCEPT The graph y represents the cost (in dollars) to open an online gaming account and buy x games. (See Exploration 2, p. 173.)

Question 8.
Find the slope and the y-intercept of the line. What do they represent in this situation?
Answer:
(0, 15), (3, 45)
m = (45 – 15)/(3 – 0)
m = 30/3 10
Thus the slope of the line is m – 3.
b = 15
The slope represents the cost of one game, while the y – intercept is the cost of opening the gaming account.

Question 9.
Write an equation that represents the graph.
Answer:
m = 10
b = 15
y = mx + b
y = 10x + 15

Question 10.
How can you determine the total cost of opening an account and buying 6 games?
Answer:
y = 10x + 15
y = 10(6) + 15
y = 60 + 15
y = 75

WRITING EQUATIONS IN SLOPE-INTERCEPT FORM Write an equation in slope-intercept form of the line that passes through the given points.
Question 11.

Answer:
m = (y2 – y1)/(x2 – x1)
= (4 – 3)/(0 – (-1))
= 1/1
= 1
Because the line crosses the y – axis at (0, 4) the y – intercept is 4
y = mx + b
y = (1)x + 4
y = x + 4

Question 12.

Answer:
m = (y2 – y1)/(x2 – x1)
= (6 – 0)/(-3 – 0)
= 6/-3
= -2
Because the line crosses the y – axis at (0, 2) the y – intercept is 2
y = mx + b
y = -2x + 0
y = -2x

Question 13.

Answer:
m = (y2 – y1)/(x2 – x1)
= (2 – 1)/(4 – 0)
= 1/4
Because the line crosses the y – axis at (0, 1) the y – intercept is 1
y = mx + b
y = 1/4 x + 1

Question 14.

Answer:
m = (y2 – y1)/(x2 – x1)
= (1 – 2)/(0 – (-2))
= -1/2
Because y = 1 when x = 0, the y – intercept is 1
y = mx + b
y = -1/2 x + 2

Question 15.

Answer:
m = (y2 – y1)/(x2 – x1)
= (-3 – (-4))/(0 – (-3))
= 1/3
Because y = -3 when x = 0, the y – intercept is -3
y = mx + b
y = 1/3 x – 3

Question 16.

Answer:
m = (y2 – y1)/(x2 – x1)
= (-1 -4)/(0 – (-2))
= -5/2
Because y = -1 when x = 0, the y – intercept is -1
y = mx + b
y = -5/2 x – 1

WRITING EQUATIONS Write an equation of the line that passes through the given points.
Question 17.
(-1, 4), (0, 2)
Answer:
m = (y2 – y1)/(x2 – x1)
= (2 – 4)/(0 – (-1))
= -2/1
= -2
Because y = 2 when x = 0, the y – intercept is 2
y = mx + b
y = -2x + 2

Question 18.
(-1, 0), (0, 0)
Answer:
m = (y2 – y1)/(x2 – x1)
= (0 – 0)/(0 – (-1))
= 0/1
= 0
Because y = 0 when x = 0, the y – intercept is 0
y = mx + b
y = 0

Question 19.
(0, 4), (0, -3)
Answer:
Both points belong to the y-axis. Therefore the equation of the line passing through them is
x = 0

Question 20.
YOU BE THE TEACHER
Your friend writes an equation of the line shown. Is your friend correct? Explain your reasoning.

Answer:
Because in the given graph, y = -2 when x = 0, so the y – intercept is -2. The equation of the line should be: y = 1/2 x – 2
No my friend is NOT correct.

Question 21.
MODELING REAL LIFE
A boa constrictor is 18 inches long at birth and grows 8 inches per year. Write an equation in slope y-intercept form that represents the length (in feet) of a boa constrictor that is x years old.

Answer:
Given,
A boa constrictor is 18 inches long at birth and grows 8 inches per year.
Length after x years = birth length + number of years . Growth per year
y = 18 + 8x
y = 8x + 18
Convert it into feet
y = 2/3 x + 3/2

Question 22.
MODELING REAL LIFE
The table shows the speeds y (in miles per hour) of a car after x seconds of braking. Write an equation of the line that passes through the points in the table. Interpret the slope and the y-intercept of the line.

Answer:
m = (y2 – y1)/(x2 – x1)
= (60 – 70)/(1 – 0)
= -10/1
= -10
Because y = 70 when x = 0, the y – intercept is 70
y = mx + b
y = -10x + 70
Slope = -10 represents the decrease in the speed of the car each seconds after breaking.
The y – intercept of 70 represents the initial speed of the car.

Question 23.
MODELING REAL LIFE
A dentist charges a flat fee for an office visit, plus an additional fee for every tooth removed. The graph shows the total cost y (in dollars) for a patient when the dentist removes x teeth. Interpret the slope and the y-intercept.

Answer:
(2, 500), (4, 900)
m = (900 – 500)/(4 – 2)
m = 400/2
m = 200
y = mx + b
500 = 200(2) + b
500 = 400 + b
b = 500 – 400
b = 100
The slope shows that the amount charged for each removed tooth is $200.
The y – intercept shows that the flat fee for an office visit is $100.

Question 24.
MODELING REAL LIFE
One of your friends gives you $10 for a charity walkathon. Another friend gives you an amount per mile. After 5 miles, you have raised $13.50 total. Write an equation that represents the amount y of money you have raised after x miles.
Answer:
Given,
One of your friends gives you $10 for a charity walkathon.
Another friend gives you an amount per mile. After 5 miles, you have raised $13.50 total.
y = mx + b
b = 10
13.50 = 5m + 10
13.50 – 10 = 5m
3.50 = 5m
m = 3.50/5
m = 0.7
y = 0.7x + 10

Question 25.
PROBLEM SOLVING
You have 500 sheets of notebook paper. After 1 week, you have 72% of the sheets left. You use the same number of sheets each week. Write an equation that represents the number y of sheets remaining after x weeks.
Answer:
y = mx + b
500 – 0.72 × 500 = 500 – 360 = 140 sheets
m = -140
b = 500
y = -140x + 500

Question 26.
DIG DEEPER!
The palm tree on the left is 10 years old. The palm tree on the right is 8 years old. The trees grow at the same rate.

a. Estimate the height y (in feet) of each tree.
b. Plot the two points (x, y), where x is the age of each tree and y is the height of each tree.
c. What is the rate of growth of the trees?
d. Write an equation that represents the height of a palm tree in terms of its age.
Answer:
a. estimate
left: 18
right: 12
plot y = 1.8x

Lesson 4.7 Writing Equations in Point-Slope Form

EXPLORATION 1

Deriving an Equation
Work with a partner. Let (x₁, y₁) represent a specific point on a line. Let (x, y) represent any other point on the line.

a. Write an equation that represents the slope m of the line. Explain your reasoning.
b. Multiply each side of your equation in part(a) by the expression in the denominator. What does the resulting equation represent? Explain your reasoning.
Answer:

EXPLORATION 2

Writing an Equation
Work with a partner.
For 4 months, you saved $25 a month. You now have $175 in your savings account.

a. Draw a graph that shows the balance in your account after t months.
b.Use your result from Exploration 1 to write an equation that represents the balance A after t months.
Answer:

Try It
Write an equation in point -slope form of the line that passes through the given point and has the given slope.
Question 1.
(1, 2); m = -4
Answer:
y – y1 = m(x – x1)
y – 2 = -4(x – (1))
y – 2 = -4(x – 1)

Question 2.
(7, 0); m = 1
Answer:
y – y1 = m(x – x1)
y – 0 = 1(x – (7))
y – 0 = 1(x – 7)

Question 3.
(-8, -5); m = –\(\frac{3}{4}\)
Answer:
y – y1 = m(x – x1)
y – (-5) = –\(\frac{3}{4}\)(x – (-8))
y + 5 = –\(\frac{3}{4}\)(x + 8)

Write an equation in slope-intercept form of the line that passes through the given points.
Question 4.
(-2, 1), (3, -4)
Answer:
Slope(m) = (-4 – 1)/(3 – (-2))
= -5/5
m = -1
y – y1 = m(x – x1)
y – 1 = -1(x – (-2))
y – 1 = -1(x + 2)
y – 1 = -x – 2
y = -x – 1

Question 5.

Answer:
Slope(m) = (3 – 5)/(-3 – (-5))
= -2/2
m = -1
y – y1 = m(x – x1)
y – 1 = -1(x – (-1))
y – 1 = -1(x + 1)
y – 1 = -x – 1
y = -x – 1 + 1
y = -x

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING AN EQUATION Write an equation in point-slope form of the line that passes through the given point and has the given slope.
Question 6.
(2, 0); m = 1
Answer:
y – y1 = m(x – x1)
y – 0 = 1(x – (2))
y – 0 = 1(x – 2)

Question 7.
(-3, -1); m = –\(\frac{1}{3}\)
Answer:
y – y1 = m(x – x1)
y – (-1) = –\(\frac{1}{3}\)(x – (-3))
y + 1 = –\(\frac{1}{3}\)(x + 3)

Question 8.
(5, 4); m = 3
Answer:
y – y1 = m(x – x1)
y – 4 = 3(x – (5))
y – 4 = 3(x – 5)

Question 9.
WRITING AN EQUATION
Write an equation of the line that passes through the points given in the table.

Answer:
Slope(m) = (-2 – 1)/(5 – 3)
= -3/2
m = -1
y – y1 = m(x – x1)
y – (-5) = -3/2(x – 7)
y + 5 = -3/2(x – 7)
y + 5 = -3/2 x + 21/2
y = -3/2 x + 11/2

Question 10.
DIFFERENT WORDS, SAME QUESTION
Which is different? Sketch “both” graphs.

Answer:
y – 7 = 4x – 4
y = 4x + -4 + 7
y = 4x + 3
Graph line passes through the points (4, 5) and (5, 9)

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 11.
A writer finishes a project that a coworker started at a rate of 3 pages per hour. After 3 hours,25% of the project is complete.
a. The project is 200 pages long. Write and graph an equation for the total number y of pages that have been finished after the writer works for x hours.
b. The writer has a total of 45 hours to finish the project. Will the writer meet the deadline? Explain your reasoning.
Answer:
m = 3
y = 3x + b
b + 9 = 25%(200)
b + 9 = 0.25(200)
b + 9 = 50
b = 50 – 9
b = 41
y = 3x + 41

y = 3x + 41
y = 3(45) + 41 = 176 pages
As 176 < 200, the writer will not meet the deadline.

Question 12.
DIG DEEPER!
You and your friend begin to run along a path at different constant speeds.After 1 minute,your friend is 45 meters ahead of you. After 3 minutes, your friend is 105 meters ahead of you.
a. Write and graph an equation for the distance y (in meters) your friend is ahead of you after x minutes. Justify your answer.

Answer:
y = mx + b
45 = m + b
105 = 3m + b
105 – 45 = (3m + b) – (m + b)
60 = 2m
m = 30
45 = 30 + b
b = 45 – 30
b = 15
y = 30x + 15

b. Did you and your friend start running from the same spot? Explain your reasoning.
Answer:
The distance between you and your friend in the initial moment is b = 15 meters. So you are ahead your friend by 15 meters at the starting point.

Writing Equations in Point-Slope Form Homework & Practice 4.7

Review & Refresh

Write an equation in slope-intercept form of the line that passes through the given points.
Question 1.

Answer:
Slope(m) = (5 – 4)/(0 – (-2))
= 1/2
m = 1/2
Because y = 5 when x = 0, the y – intercept is 5.
y = mx + b
y = 1/2 x + 5

Question 2.

Answer:
Slope(m) = (5 – (-1))/(2 – (-2))
= (5 + 1)/(2 + 2)
m = 6/4
m = 3/2
From the graph, the line crosses the y – axis at (0, 2)
y = mx + b
y = 3/2 x + 2

Solve the equation. Check your solution, if possible.
Question 3.
2x + 3 = 2x
Answer:
Given the equation
2x + 3 = 2x
3 = 2x – 2x
3 ≠ 0

Question 4.
6x – 7 = 1 – 3x
Answer:
Given the equation
6x – 7 = 1 – 3x
6x + 3x = 1 + 7
9x = 8
x = 8/3

Question 5.
0.1x – 1 = 1.2x – 5.4
Answer:
Given the equation
0.1x – 1 = 1.2x – 5.4
0.1x – 1.2x = 1 – 5.4
-1.1x = -4.4
x = 4

Concepts, Skills, &Problem Solving

WRITING AN EQUATION The value of a new car decreases $4000 each year. After 3 years, the car is worth $18,000. (See Exploration 2, p. 179.)
Question 6.
Draw a graph that shows the value of the car after t years.
Answer:

Question 7.
Write an equation that represents the value V of the car after t years.
Answer:
y = -4000t + b
where b is the original price
18,000 = -4000(3) + b
18,000 + 12,000 = b
b = 30,000
y = -4000t + 30,000

WRITING AN EQUATION Write an equation in point-slope form of the line that passes through the given point and has the given slope.
Question 8.
(3, 0); m = –\(\frac{2}{3}\)
Answer:
y – y1 = m(x – x1)
y – (0) = -2/3(x – 3)
y – 0 = -2/3(x – 3)

Question 9.
(4, 8); m = \(\frac{3}{4}\)
Answer:
y – y1 = m(x – x1)
y – (8) = 3/4(x – 4)
y – 8 = 3/4(x – 4)

Question 10.
(1, -3); m = 4
Answer:
y – y1 = m(x – x1)
y – (-3) = 4(x – 1)
y + 3 = 4(x – 1)

Question 11.
(7, -5); m = –\(\frac{1}{7}\)
Answer:
y – y1 = m(x – x1)
y – (-5) = –\(\frac{1}{7}\)(x – 7)
y + 5 = –\(\frac{1}{7}\)(x – 7)

Question 12.
(3, 3); m = \(\frac{5}{3}\)
Answer:
y – y1 = m(x – x1)
y – (3) = \(\frac{5}{3}\)(x – 3)
y – 3 = \(\frac{5}{3}\)(x – 3)

Question 13.
(-1, -4); m = -2
Answer:
y – y1 = m(x – x1)
y – (-4) = -2(x – (-1))
y + 4 = -2(x + 1)

WRITING AN EQUATION Write an equation in slope-intercept form of the line that passes through the given points.
Question 14.
(-1, -1), (1, 5)
Answer:
Slope(m) = (5 – (-1))/(2 – (-1))
= (5 + 1)/(1 + 1)
m = 6/2
m = 3
y – y1 = m(x – x1)
y – (5) = 3(x – (1))
y – 5 = 3x – 3
y = 3x + 2

Question 15.
(2, 4), (3, 6)
Answer:
Slope(m) = (6 – 4)/(3 – 2)
m = 2/1
m = 2
y – y1 = m(x – x1)
y – (4) = 2(x – (2))
y – 4 = 2x – 4
y = 2x

Question 16.
(-2, 3), (2, 7)
Answer:
Slope(m) = (7 – (3))/(2 – (-2))
= (7 – 3)/(2 + 2)
m = 4/4
m = 1
y – y1 = m(x – x1)
y – (3) = 1(x – (-2))
y – 3 = x + 2
y = x + 5

Question 17.
(4, 1), (8, 2)
Answer:
Slope(m) = (2 – (1))/(8 – (4))
= (2 – 1)/(8 – 4)
m = 1/4
y – y1 = m(x – x1)
y – (1) = 1/4(x – (4))
y – 1 = 1/4 x – 1
y = 1/4 x

Question 18.
(-9, 5), (-3, 3)
Answer:
Slope(m) = (3 – (5))/(-3 – (-9))
= (3 – 5)/(-3 + 9)
m = -2/6
m = -1/3
y – y1 = m(x – x1)
y – (3) = -1/3(x + 3)
y – 3 = -1/3 x – 1
y = -1/3 x + 2

Question 19.
(1, 2), (-2, -1)
Answer:
Slope(m) = (2 – (1))/(8 – (4))
= (-1 – 2)/(-2 – 1)
m = -3/-3
m = 1
y – y1 = m(x – x1)
y – (2) = 1(x – (1))
y – 2 = x – 1
y = x + 1

Question 20.
MODELING REAL LIFE
At 0° C, the volume of a gas is 22 liters. For each degree the temperature T (in degrees Celsius) increases, the volume V (in liters) of the gas increases by \(\frac{2}{25}\). Write an equation that represents the volume of the gas in terms of the temperature.
Answer:
The equation modeling the situation has the form:
V = mT + b
m = 2/25
22 = 2/25(0) + b
b = 22
V = 2/25 T + 22

WRITING AN EQUATION Write an equation of the line that passes through the given points in any form. Explain your choice of form.
Question 21.

Answer:
m = (y2 – y1)/(x2 – x1)
= (2.5 – 1.5)/(0 – (-1))
= 1/1
= 1
Because the line crosses the y – axis at (0, 2.5), the y – intercept is 2.5
y = mx + b
y = (1)x + 2.5
y = x + 2.5

Question 22.

Answer:
m = (y2 – y1)/(x2 – x1)
= (3.5 – 1.5)/(2 – (1))
= 2/1
= 2
y – y1 = m(x – x1)
y – (1.5) = 2(x – (1))
y – 1.5 = 2x – 2
y = 2x – 0.5

Question 23.

Answer:
m = (y2 – y1)/(x2 – x1)
= (-1.5 – 4.5)/(1 – (-1))
= -6/2
= -3
y – y1 = m(x – x1)
y – (-1.5) = -3(x – (1))
y + 1.5 = -3x + 3
y = -3x + 1.5

Question 24.

Answer:
m = (y2 – y1)/(x2 – x1)
= (-0.5 – 3.5)/(1 – (-1))
= -4/2
= -2
y – y1 = m(x – x1)
y – (-0.5) = -2(x – (1))
y + 0.5 = -2x – 2
y = -2x – 2.5

Question 25.

Answer:
m = (y2 – y1)/(x2 – x1)
= (1 – (-1))/(0 – (-3))
= (1 + 1)/(0 + 3)
= 2/3
Because y = 1 when x = 0, the y – intercept is 1.
y = mx + b
y = 2/3 x + 1

Question 26.

Answer:
m = (y2 – y1)/(x2 – x1)
= (4 – 6)/(-3 – (-7))
= -2/4
= -1/2
y – y1 = m(x – x1)
y – (2) = -1/2(x – (1))
y – 2 = -1/2x + 1/2
y = -1/2 x + 5/2

Question 27.
REASONING
Write an equation of the line that passes through the point (8, 2) and is parallel to the graph of the equation y = 4x – 3.
Answer:
y = 4x – 3
Comparing the given equation with y = mx + b, we get
m = 4
y – y1 = m(x – x1)
y – 2 = 4(x – 8)
y – 2 = 4x – 32
y = 4x – 32 + 2
y = 4x – 30

Question 28.
MODELING REAL LIFE
The table shows the amount y (in fluid ounces) of carpet cleaner in a tank after x minutes of cleaning.

a. Write an equation that represents the amount of cleaner x in the tank after minutes.
b. How much cleaner is in the tank when the cleaning begins?
c. After how many minutes is the tank empty? Justify your answer.
Answer:

Question 29.
DIG DEEPER!
According to Dolbear’s law, you can predict the temperature T (in degrees Fahrenheit) by counting the number x of chirps made by a snowy tree cricket in 1 minute.When the temperature is 50°F, a cricket chirps 40 times in 1 minute. For each rise in temperature of 0.25°F, the cricket makes an additional chirp each minute.
a. You count 100 chirps in 1 minute. What is the temperature?
b. The temperature is 96°F.How many chirps do you expect the cricket to make? Justify your answer.
Answer:

Question 30.
PROBLEM SOLVING
The Leaning Tower of Pisa in Italy was built between 1173 and 1350.
a. Write an equation that represents the yellow line.
b. The tower is 56 meters tall. How far from the center is the top of the tower? Justify your answer.

Answer:

Graphing and Writing Linear Equations Connecting Concepts

Using the Problem-Solving Plan
Question 1.
Every item in a retail store is on sale for 40% off. Write and graph an equation that represents the sale price of an item that has an original price of x dollars.

Understand the problem.
You know the percent discount of items in a retail store.You are asked to write and graph an equation that represents the sale price of an item that has an original price of x dollars.
Make a plan.
Selling an item for 40% off is the same as selling an item for 60% of its original price. Use this information to write and graph an equation that represents the situation.
Solve and check.
Use the plan to solve the problem. Then check your solution.
Answer:
40% = 0.40 and to find a percent of a number you multiply the number by the percent in decimal form.
So, the equation is d = 0.4p

Question 2.
Two supplementary angles have angle measures of x° and y°. Write and graph an equation that represents the relationship between the measures of the angles.
Answer:

Question 3.
A mechanic charges a diagnostic fee plus an hourly rate. The table shows the numbers of hours worked and the total costs for three customers.A fourth customer pays $285. Find the number of hours that the mechanic worked for the fourth customer.

Answer:

Performance Task

Anatomy of a Hurricane
At the beginning of this chapter, you watched a STEAM Video called “Hurricane!” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.

Graphing and Writing Linear Equations Chapter Review

Review Vocabulary
Write the definition and give an example of each vocabulary term.

Graphic Organizers
You can use a Definition and Example Chart to organize information about a concept. Here is an example of a Definition and Example Chart for the vocabulary term linear equation.

Choose and complete a graphic organizer to help you study the concept.

1. slope
2. slope of parallel lines
3. proportional relationship
4. slope-intercept form
5. standard form
6. point-slope form

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.

4.1 Graphing Linear Equations (pp. 141–146)
Learning Target: Graph linear equations.Graph the linear equation.

Question 1.
y = \(\frac{3}{5}\)x
Answer:

Question 2.
y = -2
Answer:

Question 3.
y = 9 – x
Answer:

Question 4.
y = -0.25x + 4
Answer:

Question 5.
y = \(\frac{2}{3}\)x + 2
Answer:

Question 6.
x = -5
Answer:

Question 7.
The equation y = 0.53x + 3 represents the cost y (in dollars) of riding in a taxi x miles.

a. Use a graph to estimate how much it costs to ride 5.25 miles in a taxi.
b. Use the equation to find exactly how much it costs to ride 5.25 miles in a taxi.
Answer:

y = 0.5x + 3
y = 0.5(5.25) + 3
y ≈ 5.6

Question 8.
The equation y = 9.5x represents the earnings y (in dollars) of an aquarium gift shop employee that works x hours.

a. Graph the linear equation.
b. How much does the employee earn for working 40 hours?
Answer:

Determine y for x = 40:
y = 9.5x
y = 9.5(40) = 380

Question 9.
Is y = x² a linear equation? Explain your reasoning.
Answer:
y = x²
The graph of the given equation passes through the origin, but is not linear, therefore it is not a linear equation.
So, the answer is no.

Question 10.
The sum S of the exterior angle measures of a polygon with n sides is S = 360°.
a. Plot four points (n, S) that satisfy the equation. Is the equation a linear equation? Explain your reasoning.
b. Does the value n = 2 make sense in the context of the problem? Explain your reasoning.
Answer:

The value n = 2 does not make sense in the context of the problem because a polygon has at least 3 sides.

4.2 Slope of a Line (pp. 147–154)
Learning Target: Find and interpret the slope of a line.

Describe the slope of the line. Then find the slope of the line.
Question 11.

Answer:
(x1, y1) = (3, 1)
(x2, y2) = (-3, -3)
m = (y2 – y1)/(x2 – x1)
m = (-3 – 1)/(-3 – 3)
m = -4/-6
m = 2/3

Question 12.

Answer:
(x1, y1) = (0, 4)
(x2, y2) = (2, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 4)/(2 – 0)
m = -6/2
m = -3
The slope is negative

Find the slope of the line through the given points.
Question 13.
(-5, 4), (8, 4)
Answer:
(x1, y1) = (-5, 4)
(x2, y2) = (8, 4)
m = (y2 – y1)/(x2 – x1)
m = (4 – 4)/(8 – (-5))
m = 0/13
m = 0

Question 14.
(-3, 5), (-3, 1)
Answer:
(x1, y1) = (-3, 5)
(x2, y2) = (-3, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – 5)/(-3 + 3)
m = -4/0
m = undefined

The points in the table lie on a line. Find the slope of the line.
Question 15.

Answer:
(x1, y1) = (0, -1)
(x2, y2) = (1, 0)
m = (y2 – y1)/(x2 – x1)
m = (0 – (-1))/(1 – 0)
m = 1/1
m = 1

Question 16.

Answer:
(x1, y1) = (-2, 3)
(x2, y2) = (0, 4)
m = (y2 – y1)/(x2 – x1)
m = (4 – 3)/(0 – (-2))
m = 1/2

Question 17.
How do you know when two lines are parallel? Use an example to justify your answer.
Answer:
Two lines are parallel when their slopes are the same. In order for the two lines not to coincide, we must add the condition that their y – intercepts.
Example 1:
d1: y = 3x – 6
d2: 3x – y = 6
The lines d1 and d2 have the same slope and the same y – intercept, therefore they coincide.

Question 18.
Draw a line through the point (-1, 2) that is parallel to the graph of the line in Exercise 11.
Answer:
y = 2/3 x – 1
A (-1, 2)
y = 2/3 x + b
y = 2/3 (-1) + b
b = 8/3
The equation of d1 is:
y = 2/3 x + 8/3
Determine the x intercept of d1:
0 = 2/3 x + 8/3
0 = 2x + 8
2x = -8
x = -8/2
x = -4

4.3 Graphing Proportional Relationships (pp. 155–160)
Learning Target: Graph proportional relationships.

Tell whether x and y are in a proportional relationship. Explain your reasoning. If so, write an equation that represents the relationship.
Question 19.

Answer:
x and y are not in a proportional relationship because the line does not pass through the origin.

Question 20.

Answer:
x and y are in a proportional relationship because the line does passes through the origin.
Determine the slope k using two points from the graph
k = (10 – 0)/(2 – 0)
k = 10/2
k = 5x

Question 21.
The cost y (in dollars) to provide food for guests at a dinner party is proportional to the number x of guests attending the party. It costs $30 to provide food for 4 guests.
a. Write an equation that represents the situation.
b. Interpret the slope of the graph of the equation.
c. How much does it cost to provide food for 10 guests? Justify your answer.
Answer:
y = kx
30 = 4k
k = 30/4
k = 7.5
y = 7.5x
b. The slope 7.5 represents the unit cost for a guest.
y = 7.5 × 10
y = 75
c. Determine y for x = 10
So it costs $75 to provide food for 10 guests.

Question 22.
The distance y (in miles) you run after weeks is represented by the equation y =8x. Graph the equation and interpret the slope.
Answer:
y = 8x

Question 23.
You research that hair grows 15 centimeters per year on average. The table shows your friend’s hair growth.

a. Does your friend’s hair grow faster than average? Explain.

Answer:
The rate of growth on average is
15/12 = 1.25 cm/month
The slope/rate of growth for your friend is
(6 – 3)/(4 – 2) = 3/2 = 1.5 cm/month
As 1.5 > 1.25, your friends hair grows faster than average.

b. In the same coordinate plane, graph the average hair growth and the hair growth of your friend. Compare and interpret the steepness of each of the graphs.
Answer:
The equation for the average growth is
y = 1.25x
The equation for the friends growth is
y = 1.5 x

4.4 Graphing Linear Equations in Slope-Intercept Form (pp. 161–166)
Learning Target: Graph linear equations in slope-intercept form.

Find the slope and the -intercept of the graph of the linear equation.
Question 24.
y = -4x + 1
Answer:
y = mx + b
slope = -4 and y – intercept = 1

Question 25.
y = \(\frac{2}{3}\)x – 12
Answer:
y = mx + b
slope = \(\frac{2}{3}\) and y – intercept = -12

Question 26.
y – 7 = 0.5x
Answer:
Given the equation
y – 7 = 0.5x
y = 0.5x + 7
slope = 0.5 and y – intercept = 7

Graph the linear equation. Identify the -intercept.
Question 27.
y = 2x – 6
Answer:
Given the equation
y = 2x – 6
Comparing the above equation with slope – intercept equation
slope = 2, y – intercept = -6
Slope = rise/run = 2/1
Plot the point that is 1 unit right and 2 units up from (0, -6) = (1, -4)

The line crosses the x – axis at (3, 0)
So, the x – intercept is 3.

Question 28.
y = -4x + 8
Answer:
y = -4x + 8
slope = -4 and y – intercept = 8
So plot (0, 8)
Slope = rise/run = -4/1
plot the point that is 1 unit right and 4 units down from (0, 8) = (1, 4)

The line crosses the x- axis at (2, 0)
So the x – intercept is 2.

Question 29.
y = -x – 8
Answer:
Given the equation
y = -x – 8
comparing the above equation with sloope – intercept equation.
Slope = -1 and y – intercept = -8
Slope = rise/run = -1/1
Plot the point that is 1 unit right and 1 unit down from (0, -8) = (1, -9)

The line crosses the x-axis at (-8, 0)
So, the intercept is -8.

Question 30.
The cost y (in dollars) of one person buying admission to a fair and going on x rides is y = x + 12.

a. Graph the equation.
b. Interpret the y-intercept and the slope.
Answer:
y = x + 12
Comparing the above equation with slope – intercept equation.
Slope = 1 and y – intercept = 12
So plot (0, 20)
Slope = rise/run = 1/1
Plot the point that is 1 unit right and 1 unit up from (0, 12) = (1, 13)

The y – intercept is 12 so the initial cost of admission is $12.
The slope is 1 so for each ride the cost of the person increases $1 per ride.

Question 31.
Graph the linear equation with slope -5 and y-intercept 0.
Answer:
y – intercept = 0. So plot (0, 0)
Plot the point that is 1 unit right and 5 unit down from (0, 0) = (1, -5)

4.5 Graphing Linear Equations in Standard Form (pp. 167–172)
Learning Target: Graph linear equations in standard form.

Write the linear equation in slope-intercept form.
Question 32.
4x + 2y = -12
Answer:
4x + 2y = -12
2y = -12 – 4x
y = -6 – 2x
y = -2x – 6

Question 33.
x – y = \(\frac{1}{4}\)
Answer:
Given the equation
x – y = \(\frac{1}{4}\)
y = x – \(\frac{1}{4}\)

Graph the linear equation.
Question 34.
\(\frac{1}{4}\)x + y = 3
Answer:
\(\frac{1}{4}\)x + y = 3
y = 3 – \(\frac{1}{4}\)x
y = –\(\frac{1}{4}\)x + 3
Slope = –\(\frac{1}{4}\) and y – intercept = 3
So plot (0, 3)
Slope = rise/run = –\(\frac{1}{4}\)
Plot the point that is 4 units right and 1 unit down from (0, 3) = (4, 2)

Question 35.
-4x + 2y = 8
Answer:
-4x + 2y = 8
2y = 8 + 4x
y = 2x + 4
Slope = 2 and y – intercept = 4
So plot (0, 4)
Slope = rise/run = 2/1
Plot the point that is 1 unit right and 2 units up from (0, 4) = (1, 6)

Question 36.
x + 5y = 10
Answer:
x + 5y = 10
5y = -x + 10
y = -1/5 x + 2
Slope = -1/5 and y – intercept = 2
So, plot (0, 2)
Slope = rise/run = -1/5
Plot the point that is 5 unit right and 1 unit down from (0, 2) = (5, 1)

Question 37.
–\(\frac{1}{2}\)x + \(\frac{1}{8}\)y = \(\frac{3}{4}\)
Answer:
–\(\frac{1}{2}\)x + \(\frac{1}{8}\)y = \(\frac{3}{4}\)
\(\frac{1}{8}\)y = \(\frac{3}{4}\)  + \(\frac{1}{2}\)x
y = 4x + 6
Slope = 4 and y – intercept = 6
So plot (0, 6)
Slope = rise/run = 4/1
Plot the point that is 1 unit right and 4 units up from (0, 6) = (1, 10)

Question 38.
A dog kennel charges $30 per night to board your dog and $6 for each hour of playtime. The amount of money you spend is given by 30x + 6y = 180, where x is the number of nights and y is the number of hours of playtime. Graph the equation and interpret the intercepts.
Answer:
Given,
A dog kennel charges $30 per night to board your dog and $6 for each hour of playtime.
The amount of money you spend is given by 30x + 6y = 180,
where x is the number of nights and y is the number of hours of playtime.
30x + 6y = 180
6y = -30x + 180
y = -5x + 30

The x – intercept is 6, which means that the dog can stay for 6 nights when there is no playtime.
The y – intercept is 30, which means the dog can play for 30 hours when he does not spend any night at the kennel.

4.6 Writing Equations in Slope-Intercept Form (pp. 173–178)
Learning Target: Write equations of lines in slope-intercept form.

Write an equation in slope-intercept form of the line that passes through the given points.
Question 39.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (1 – (-2))/(3 – 0)
m = (1 + 2)/(3 – 0)
m = 3/3
m = 1
We have to find the y – intercept because the line crosses the y – axis at (0, -2)
y = mx + b
y = x – 2

Question 40.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (4 – 2)/(0 – 4)
m = 2/-4
m = -1/2
We have to find the y – intercept because the line crosses the y – axis at (0, 4)
y = mx + b
y =-1/2 x + 4

Question 41.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (-2 – 1)/(2 – 0)
m = -3/2
We have to find the y – intercept because y = 1 when x = 0, the y – intercept is 1.
y = mx + b
y = -3/2 x + 1

Question 42.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (-1 – (-3))/(1 – 0)
m = 2/1
m = 2
We have to find the y – intercept because y = -3 when x = 0, the y – intercept is -3.
y = mx + b
y = 2x + (-3)
y = 2x – 3

Question 43.
Write an equation of the line that passes through (0, 8) and (6, 8).
Answer:
m = (y2 – y1)/(x2 – x1)
m = (8 – 8)/(6 – 0)
m = 0/6
m = 0
We have to find the y – intercept because y = 8 when x = 0, the y – intercept is 8.
y = mx + b
y = (0) x + 8
y = 8

Question 44.
Write an equation of the line that passes through (0, -5) and (-5, -5).
Answer:
m = (y2 – y1)/(x2 – x1)
m = (-5 – (-5))/(-5 – 0)
m = 0/-5
m = 0
We have to find the y – intercept because y = -5 when x = 0, the y – intercept is -5
y = mx + b
y = (0) x + (-5)
y = -5

Question 45.
A construction crew is extending a highway sound barrier that is 13 miles long. The crew builds \(\frac{1}{2}\) of a mile per week. Write an equation in slope -intercept form that represents the length y (in miles) of the barrier after x weeks.
Answer:
Given,
A construction crew is extending a highway sound barrier that is 13 miles long.
The crew builds \(\frac{1}{2}\) of a mile per week.
y = mx + b
m = \(\frac{1}{2}\)
b = 13
y = \(\frac{1}{2}\)x + 13

4.7 Writing Equations in Point-Slope Form (pp. 179–184)
Learning Target: Write equations of lines in point-slope form.

Write an equation in point-slope form of the line that passes through the given point and has the given slope.
Question 46.
(4, 4); m = 3
Answer:
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – (4) = 3(x – 4)
y – 4 = 3(x – 4)

Question 47.
(2, -8); m = –\(\frac{2}{3}\)
Answer:
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – (-8) = –\(\frac{2}{3}\)(x – 2)
y + 8 = –\(\frac{2}{3}\)(x – 2)

Write an equation in slope-intercept form of the line that passes through the given points.
Question 48.
(-4, 2), (6, -3)
Answer:
m = (y2 – y1)/(x2 – x1)
m = (-3 – 2)/(6 – (-4))
m = -5/10
m = -1/2
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – 2 = –\(\frac{1}{2}\)(x – (-4))
y – 2 = –\(\frac{1}{2}\)(x + 4)
y = –\(\frac{1}{2}\)x

Question 49.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (5 – 1)/(3 – 2)
m = 4/1
m = 4
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – (-3) = 4(x – 1)
y + 3 = 4x – 4
y = 4x – 7

Question 50.
The table shows your elevation y (in feet) on a ski slope after x minutes.

a. Write an equation that represents your elevation after x minutes.

Answer:
m = (600 – 800)/(2 – 1)
m = -200
800 = -200(1) + b
800 = -200 + b
800 + 200 = b
b = 1000 feet

b. What is your starting elevation?

Answer:
The starting elevation is the y – intercept
b = 1000 feet

c. After how many minutes do you reach the bottom of the ski slope? Justify your answer.
Answer:
0 = -200x + 1000
0 – 1000 = -200x
-1000 = -200x
200x = 1000
x = 5 minutes

Question 51.
A company offers cable television at$29.95 per month plus a one-time installation fee. The total cost for the first six months of service is $214.70. a. Write an equation in point-slope form that represents the total cost you pay for cable television after x months.
b. How much is the installation fee? Justify your answer.
Answer:
y – y1 = m(x – x1)
m = 29.95
y – 214.70 = 29.95(x – 6)
y – 214.70 + 214.70 = 29.95x – 179.97 + 2147.70
y = 29.95x + 35
b = 35

Question 52.
When might it be better to represent an equation in point-slope form rather than slope-intercept form? Use an example to justify your answer.
Answer:
When we are given the slope and a point that is the y – intercept, then the easiest way is to use the slope – intercept form y = mx + b
Example:
m = 2
(0, 5)
y = 2x + 5
m = 2
(1, 3)
y – 3 = 2(x – 1)
Easier when given the slope and a point that is not the y – intercept.

Graphing and Writing Linear Equations Practice Test

Find the slope and the -intercept of the graph of the linear equation.
Question 1.
y = 6x – 5
Answer:
y = 6x – 5
Slope = 6 and y – intercept = -5

Question 2.
y – 1 = 3x + 8.4
Answer:
Given the equation
y – 1 = 3x + 8.4
y = 3x + 8.4 + 1
y = 3x + 9.4
Slope = 3 and y – intercept = 9.4

Question 3.
–\(\frac{1}{2}\)x + 2y = 7
Answer:
Given the equation
–\(\frac{1}{2}\)x + 2y = 7
y = \(\frac{1}{4}\)x + \(\frac{7}{2}\)
Slope = \(\frac{1}{4}\) and y – intercept = \(\frac{7}{2}\)

Graph the linear equation.
Question 4.
y = –\(\frac{1}{2}\)x – 5
Answer:
Given the equation
y = –\(\frac{1}{2}\)x – 5
Slope = –\(\frac{1}{2}\) and y – intercept = -5
So plot (0, -5)
Plot the point that is 2 units right and 1 unit down from (0, -5) = (2, -6)
Draw a line through the two points.

Question 5.
-3x + 6y = 12
Answer:
Given the equation
-3x + 6y = 12
6y = 3x + 12
y = \(\frac{1}{2}\)x + 2
Slope = \(\frac{1}{2}\), y – intercept = 2
Slope = rise/run = \(\frac{1}{2}\)
Plot the point that is 2 units right and 1 unit up from (0, 2) = (2, 3)

Question 6.
y = \(\frac{2}{3}\)x
Answer:
Given the equation
y = \(\frac{2}{3}\)x
Slope = \(\frac{2}{3}\), y – intercept = 0
Slope = rise/run = \(\frac{2}{3}\)
Plot the point that is 3 units right and 2 unit up from (0, 0) = (3, 2)

Question 7.
Which lines are parallel? Explain.

Answer:
Red line:
(x1, y1) = (-4, 1)
(x2, y2) = (2, 4)
m = (y2 – y1)/(x2 – x1)
m = (4 – 1)/(2 – (-4))
m = 3/6
m = 1/2
Blue line:
(x1, y1) = (-4, -1)
(x2, y2) = (2, 0.5)
m = (y2 – y1)/(x2 – x1)
m = (0.5 – (-1))/(2 – (-4))
m = 1.5/6
m = 1/4
Green Line:
(x1, y1) = (-2, -4)
(x2, y2) = (2, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – (-4))/(2 – (-2))
m = 2/4
m = 1/2
Red lines and Green lines are parallel because both have same slope = 1/2

Question 8.
The points in the table lie on a line. Find the slope of the line.

Answer:
(x1, y1) = (-1, -4)
(x2, y2) = (0, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – (-4))/(0 – (-1))
m = 3/1
m = 3

Write an equation in slope-intercept form of the line that passes through the given points.
Question 9.

Answer:
(x1, y1) = (-4, 1)
(x2, y2) = (2, 4)
m = (y2 – y1)/(x2 – x1)
m = (-5 – (-1))/(3 – 0)
m = -4/3
Because the line crosses the y – axis at (0, -1), the y – intercept is -1.
y = mx + b
y = -4/3x – 1

Question 10.

Answer:
m = (y2 – y1)/(x2 – x1)
m = (2 – 2)/(0 – (-2))
m = 0/2
m = 0
Because y = 2 when x =0, the y – intercept is 2.
y = mx + b
y = 2

Question 11.
Write an equation in point-slope form of the line that passes through (-4, 1) and (4, 3).
Answer:
m = (y2 – y1)/(x2 – x1)
m = (3 – 1)/(4 – (-4))
m = 2/8
m = 1/4
y – y1 = m(x – x1)
y – 1 = 1/4(x – (-4))
y – 1 = 1/4(x + 4)

Question 12.
The number y of new vocabulary words that you learn after x weeks is represented by the equation y = 15x.
a. Graph the equation and interpret the slope.
b. How many new vocabulary words do you learn after 5 weeks?
c. How many more vocabulary words do you learn after 6 weeks than after 4 weeks?
Answer:
a.
b. y = 15 . 5
y = 75 words
c. 15 . 6 – 15 . 4 = 90 – 60 = 30 words

Question 13.
You used $90 worth of paint for a school float. The amount of money you spend is given by 18x + 15y = 90, where x is the number of gallons of blue paint and y is the number of gallons of white paint. Graph the equation and interpret the intercepts.
Answer:
Given,
18x + 15y = 90
15y = -18x + 90
y = -6/5 x + 6

The x – intercept is 5 and shows that 5 gallons of blue paint might be bought when no gallon of the white pants is bought.
The y – intercept is 6 and shows that 6 gallons of white paint might be bought when no gallon of blue is bought.

Graphing and Writing Linear Equations Cumulative Practice

Question 1.
Which equation matches the line shown in the graph?

A. y =2x – 2
B. y = 2x + 1
C. y = x – 2
D. y = x + 1
Answer:
m = (-2 – 0)/(0 – 1)
m = 2
y = 2x – 2
Thus the correct answer is option A.

Question 2.
Which point lies on the graph of 6x – 5y = 14?
F. (-4, -1)
G. (-2, 4)
H. (-1, -4)
I. (4, -2)
Answer:
6x – 5y = 14
F. 6(-4) – 5(-1) = 14
-24 + 5 = 14
-19 ≠ 14
G. 6(-2) – 5(4) = 14
-12 – 20 = 14
-32 ≠ 14
H. 6(-1) – 5(-4) = 14
-6 + 20 = 14
14 = 14
Thus the correct answer is option H.

Question 3.
You reflect the triangle in the x-axis. What are the coordinates of the image?

A. X'(4, 1), Y'(2, 3), Z'(-2, 1)
B. X'(4, -1), Y'(2, -3), Z'(-2, -1)
C. X'(-4, -1), Y(-2, -3), Z'(2, -1)
D. X'(1, 4), Y'(3, 2), Z'(1, -2)
Answer:

Thus the correct answer is option C.

Question 4.
Which of the following is the equation of a line parallel to the line shown in the graph?

Answer:
m = (-4 – 2)/(6 – 4)
m = -6/2
m = -3
Two lines parallel if they have the same slope.
From the given equations, the one having the slope -3 is y = -3x + 5
Thus the correct answer is option H.

Question 5.
What is the value of x?

Answer:
122 = 47 + x
47 + x = 122
x = 122 – 47
x = 75

Question 6.
An emergency plumber charges $49.00 plus $70.00 per hour of the repair. A bill to repair your sink is $241.50. This can be modeled by 70.00 h + 49.00 = 241.50, where h represents the number of hours for the repair. How many hours did it take to repair your sink?
A. 2.75 hours
B. 3.45 hours
C. 4.15 hours
D. 13,475 hours
Answer:
70.00 h + 49.00 = 241.50
70h = 241.50 – 49
70h = 192.5
h = 2.75 hours
Thus the correct answer is option A.

Question 7.
It costs $40 to rent a car for one day. In addition, the rental agency charges you for each mile driven, as shown in the graph.

Part A Determine the slope of the line joining the points on the graph.
Part B Explain what the slope represents.
Answer:
m = (50 – 40)/(100 – 0)
m = 10/100
m = 0.1

Question 8.
What value of makes the equation true?

7 + 2x = 4x – 5
Answer:
7 + 2x = 4x – 5
2x – 4x = -5 – 7
-2x = -12
x = 6

Question 9.
Trapezoid KLMN is graphed in the coordinate plane shown.

Rotate Trapezoid KLMN 90° clockwise about the origin. What are the coordinates of point M’, the image of point M after the rotation?
F. (-3, -2)
G. (-2, -3)
H. (-2, 3)
I. (3, 2)
Answer: M'(-3, -2)
Thus the correct answer is option F.

Question 10.
Solve the formula K = 3M – 7.
A. M = K + 7
B. M = \(\frac{K+7}{3}\)
C. M = \(\frac{K}{3}\) + 7
D. M = \(\frac{K-7}{3}\)
Answer:
K = 3M – 7
K + 7 = 3M
M = \(\frac{K+7}{3}\)
Thus the correct answer is option B.

Question 11.
What is the distance across the canyon?

F. 3.6 ft
G. 12 ft
H. 40 ft
I. 250 ft
Answer:
100/30 = d/12
3d = 12 × 10
3d = 120
d = 40 feet
Thus the correct answer is option H.

Conclusion:

All the solutions in the above article are beneficial for all the students of middle school students. All the solutions are prepared by the math professionals. The solutions are given clearly with step by step explanations. If you have any doubts regarding the chapter we are always ready to clarify your doubts. All you have to do is to post the comments in the below comment box.

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Big Ideas Math Book 1st Grade Answer Key Chapter 9 Add Two-Digit Numbers

Start your preparation by taking help from Big Ideas Math Book 1st Grade Solution Key Chapter 9 Add Two-Digit Numbers and learn the concepts from practice tests, cumulative tests, and others. Big Ideas Math Answers Grade 1 Chapter 9 Add Two-Digit Numbers helps both Students and Teachers out there to get all the concepts underlying. Score good marks in your exams with the help of Big Ideas Math Book 1st Grade Answer Key Chapter 9 Add Two-Digit Numbers.

Vocabulary

• Add Two-Digit Numbers Vocabulary

Lesson: 1 Add Tens and Ones

• Lesson 9.1 Add Tens and Ones
• Add Tens and Ones Practice 9.1

Lesson: 2 Add Tens and Ones Using a Number Line

• Lesson 9.2 Add Tens and Ones Using a Number Line
• Add Tens and Ones Using a Number Line Practice 9.2

Lesson: 3 Make a 10 to Add

• Lesson 9.3 Make a 10 to Add
• Make a 10 to Add Practice 9.3

Lesson: 4 Add Two-Digit Numbers

• Lesson 9.4 Add Two-Digit Numbers
• Add Two-Digit Numbers Practice 9.4

Lesson: 5 Practice Addition Strategies

• Lesson 9.5 Practice Addition Strategies
• Practice Addition Strategies Practice 9.5

Lesson: 6 Problem Solving: Addition

• Lesson 9.6 Problem Solving: Addition
• Problem Solving: Addition Practice 9.6

Chapter-9: Add Two-Digit Numbers

• Add Two-Digit Numbers Performance Task
• Add Two-Digit Numbers Change Practice

Add Two-Digit Numbers Vocabulary

Organize It

Review Words:
120 chart
column
ones
row
tens

Answer:
Tens and ones.

Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 3 lines 3 x 10 = 30.
In ones row there 7 boxes 7 x 1 = 7
so, total there are 37

Define It

Use the review words to complete the puzzle.

Answer:
1) 120Chart
2) Column
3)Row

Explanation:
First figure shows the numbers from 1 to 120.
The vocabulary for the figure is 120 chart which is represented in the figure horizontally.
Second figure shows columns colored
The vocabulary for second figure is column
Third figure shows rows colored
The vocabulary for the third figure is row

Lesson 9.1 Add Tens and Ones

Explore and Grow

Show how you can use a model to solve.

32 + 7 = _________

Answer:
32 + 7= 39 Three tens and 9 ones.
Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 3 lines 3 x 10 = 30.
In ones row there 9 boxes 9 x 1 = 9
so, total there are 39

Show and Grow

Question 1.
25 + 12 = _________

Answer:
25 + 12 = 37 Three tens and 7 ones
Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 2 lines 2 x 10 = 20.
In ones row there 5 boxes 5 x 1 = 5
so, total there are 25.
The first row represents the tens and second row represents the ones
In tens row there are 1 lines 1 x 10 = 10.
In ones row there 2 boxes 2 x 1 = 2
so, total there are 12
By adding both the numbers the answer is 37.

Question 2.
36 + 3 = __________

Answer:
36 +3= 39, three tens and 9 ones
Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 3 lines 3 x 10 = 30.
In ones row there 6 boxes 6 x 1 = 6
so, total there are 36.
The first row represents the tens and second row represents the ones
In tens row there are 0 lines 0 x 10 = 0.
In ones row there 3 boxes 3 x 1 = 3
so, total there are 3
By adding both the numbers the answer is 39.

Question 3.
21 + 8 = __________

Answer:
21+ 8 = 29, 2 tens and 9 ones.
Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 2 lines 2 x 10 = 20.
In ones row there 1 boxes 1 x 1 = 1
so, total there are 1.
The first row represents the tens and second row represents the ones
In tens row there are 0 lines 0 x 10 = 0.
In ones row there 8 boxes 8 x 1 = 8
so, total there are 8
By adding both the numbers the answer is 29.

Question 4.
22 + 24 = __________

Answer:
22 + 24 = 46, 4 tens and 6 ones
Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 2 lines 2 x 10 = 20.
In ones row there 2 boxes 2 x 1 = 2
so, total there are 22.
The first row represents the tens and second row represents the ones
In tens row there are 2 lines 2 x 10 = 20.
In ones row there 4 boxes 4 x 1 = 4
so, total there are 24
By adding both the numbers the answer is 46.

Apply and Grow: Practice

Question 5.
34 + 4 = __________

Answer:
34 + 4 = 38, 3 tens and 8 ones.
Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 3 lines 3 x 10 = 30.
In ones row there 4 boxes 4 x 1 = 4
so, total there are 34.
The first row represents the tens and second row represents the ones
In tens row there are 0 lines 0 x 10 = 0.
In ones row there 4 boxes 4 x 1 = 4
so, total there are 4
By adding both the numbers the answer is 38.

Question 6.
43 + 15 = __________

Answer:
43 + 15 = 58, 5 tens and 8 ones.

Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 4 lines 4 x 10 = 40.
In ones row there 3 boxes 3 x 1 = 3
so, total there are 43.
The first row represents the tens and second row represents the ones
In tens row there are 1 lines 1 x 10 = 10.
In ones row there 5 boxes 5 x 1 = 5.
so, total there are 15
By adding both the numbers the answer is 58.

Question 7.
71 + 20 = __________
Answer:
71 + 20 = 91,
9 tens and 1 ones.

Explanation:
The number of choclates with rim is 71
The number of choclates with jim is 20
Total number of choclates is 91.

Question 8.
93 + 6 = __________
Answer:
93 + 6 = 99,
9 tens and 9 ones.

Explanation:
The number of choclates with rim is 93
The number of choclates with jim is 6
Total number of choclates is 99

Question 9.
55 + 23 = __________
Answer:
55 + 23 = 78, 7 tens and 8 ones.

Explanation:
The number of choclates with rim is 55
The number of choclates with jim is 23
Total number of choclates is 78.

Question 10.
62 + 32 = __________
Answer:
62 + 32 = 94, 9 tens and 4 ones.

Explanation:
The number of choclates with rim is 62
The number of choclates with jim is 32
Total number of choclates is 94.

Question 11.
MP Reasoning
Circle the number to complete the equation.

Answer:
41 + 5 = 46,
Circled the number 5.

Think and Grow: Modeling Real Life

You watch television for 24 minutes in the morning and 32 minutes at night. How many minutes do you spend watching television in all?

Addition equation:

Model:

___________ minutes
Answer: 56

Explanation:
Models: television,
24 minutes in the morning and 32 minutes in the night
24 + 32 = 56.

Show and Grow

Question 12.
You do 42 jumping jacks in the morning and 46 at night. How many jumping jacks do you do in all?

Addition equation:

Model:

___________ jumping jacks
Answer: 88

Explanation:
Models : Jumping jacks,
42 jumping jacks in the morning and 46 in the night
42 + 46 = 88. 88 Jumping jacks

Add Tens and Ones Practice 9.1

Question 1.
42 + 7 = _________

Answer:
42 + 7 = 49, 4 tens and 9 ones.

Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 4 lines 4 x 10 = 40.
In ones row there 2 boxes 2 x 1 = 2.
so, total there are 42.
The first row represents the tens and second row represents the ones
In tens row there are 0 lines 0 x 10 = 0.
In ones row there 7 boxes 7 x 1 = 7
so, total there are 7
By adding both the numbers the answer is 49.

Question 2.
61 + 35 = __________

Answer:
61 + 35 = 96, 9 tens and 6 ones.

Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 6 lines 6 x 10 = 60.
In ones row there 1 boxes 1 x 1 = 1
so, total there are 61.
The first row represents the tens and second row represents the ones
In tens row there are 3 lines 3 x 10 = 30.
In ones row there 5 boxes 5 x 1 = 5
so, total there are 35
By adding both the numbers the answer is 96.

Question 3.
74 + 11 = __________
Answer:
74 + 11 = 85, 8 tens and 5 ones.

Explanation:
The number of choclates with rim is 74
The number of choclates with jim is 11
Total number of choclates is 85.

Question 4.
86 + 2 = ___________
Answer:
86 + 2 = 88, 8 tens and 8 ones.

Explanation:
The number of choclates with rim is 86
The number of choclates with jim is 2
Total number of choclates is 88.

Question 5.
MP Reasoning
Circle the number to complete the equation.

Answer:
22 + 70 =  92, circled the 70 to complete the number.

Question 6.

Modeling Real Life
You eat 33 grapes. Your friend eats 23 grapes. How many grapes do you and your friend eat in all?

___________ grapes
Answer: 55

Explanation:
Me ate 33 and my friend ate 23 in all
33 + 23 = 55.
We bote ate 55 in all.

Review & Refresh

Question 7.

__________ tens and _________ ones is _________ .
Answer:
8 tens and 2 ones is 82

Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 8 lines 8 x 10 = 80.
In ones row there 2 boxes 2 x 1 = 2
so, total there are 82.

Question 8.

__________ tens and _________ ones is _________ .
Answer:
4 tens and 6 ones is 46

Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 4 lines 4 x 10 = 40.
In ones row there 6 boxes 6 x 1 = 6
so, total there are 46.

Lesson 9.2 Add Tens and Ones Using a Number Line

Explore and Grow

Color to show how you can use the hundred chart to find the sum.

23 + 34 = ___________

Answer:
23 + 34 = 57,
5 tens and 7 ones.

Explanation:
In the below figure 120 chart is shown in that 23 and 34 are colored
that two numbers are added
to get the total sum.

Show and Grow

Question 1.
22 + 7 = __________

Answer: 29

Explanation:
22 + 7 = 29,
the addition is represented on number line

Question 2.
35 + 41 = __________

Answer:
35 + 41 = 76.

Explanation:
35 + 41 is represented on the number line.

Apply and Grow: practice

Question 3.
53 + 40 = _________

Answer:
53 + 40 = 93.

Question 4.
82 + 12 = __________

Answer: 82 + 12 = 94.

Question 5.
48 + 31 = ___________

Answer: 48 + 31 = 79.
From the above equation 48 is marked and from that point
3 points with 10 are taken and 1 with one is taken
48 + 31 = 79

Question 6.
MP Structure
Write an equation that matches the number line.

Answer: 37 + 23 = 60

Explanation:
From the point 37 to 60
2 points with 10 are taken and 3 points with 1 are taken
so, the total equation is 37 + 23 = 60.

Think and Grow: Modeling Real Life

The home team scores 37 points. The visiting team scores 22 more. How many points does the visiting team score?

Addition equation:

Model:

__________ points
Answer: 37 + 22 = 59

Explanation:
A point on the number line is taken as 37
From there 2 poins with 10 that is 20
From there 2 more points with 2 are taken
total it makes 22

Show and Grow

Question 7.
Your friend scores 63 points. You score 25 more than your friend. How many points do you score?

Addition equation:

Model:

__________ points
Answer: 63 + 25 = 88
Explanation:
A point on the number line is taken as 63
From there 2 poins with 10 that is 20
From there 5 more points with 5 are taken
total it makes 88.

Add Tens and Ones Using a Number Line Practice 9.2

Question 1.
13 + 60 = __________

Answer: 73

Question 2.
81 + 8 = _____________

Answer: 89

Question 3.
56 + 42 = ____________

Answer: 98

Question 4.
MP Structure
Write an equation that matches the number line.

Answer: 17 + 41= 58

Question 5.
Modeling Real Life
There are 36 black keys on a piano. There are 16 more white keys than black keys. How many white keys are there?

____________ white keys
Answer: 52 white keys

Review & Refresh

Question 6.
3 + 7 + 4 = ___________
Answer: 14

Question 7.
4 + 5 + 6 = ___________
Answer: 15

Lesson 9.3 Make a 10 to Add

Explore and Grow

How can you use the model to solve?

38 + 6 = ___________

Answer:
38 + 6 = 44

Explanation:
The first row represents the tens and second row represents the ones
In tens row there are 3 lines 3 x 10 = 30.
In ones row there 8 boxes 8 x 1 = 8
so, total there are 38.
The first row represents the tens and second row represents the ones
In tens row there are 0 lines 0 x 10 = 0.
In ones row there 6 boxes 6 x 1 = 6
so, total there are 6
By adding both the numbers the answer is 44.
Count one numbers from both the sides 8 + 6 = 14
It has 10 in it

Show and Grow

Question 1.
41 + 7 = ___________

Make a 10?           Yes         No
Answer: No

Explanation:
By adding ones place from both the sides
1 + 7 = 8
So, the answer is no

Question 2.
56 + 8 = ___________

Make a 10?           Yes         No
Answer: Yes

Explanation:
By adding ones place from both the sides
6 + 8 = 14
So, there is 10 in it.

Apply and Grow: Practice

Question 3.
72 + 4 = _________

Make a 10?           Yes         No
Answer: No

Explanation:
By adding ones place from both the sides
2 + 4 = 6
So, There is no 10 in it.

Question 4.
63 + 9 = ___________

Make a 10?           Yes         No
Answer: Yes

Explanation:
By adding ones place from both the sides
3  + 9 = 11
So, there is 10 in it.

Question 5.
14 + 6 = __________
Make a 10?           Yes         No
Answer: Yes

Explanation:
By adding ones place from both the sides
4 + 6 = 10
so, there is 10 in it.

Question 6.
27 + 5 = __________
Make a 10?           Yes         No
Answer: Yes

Explanation:
By adding ones place from both the sides
7 + 5 = 12
So, there is 10 in it.

Question 7.
46 + 7 = ___________
Make a 10?           Yes         No
Answer: Yes

Explanation:
By adding ones place from both the sides
6 + 7 = 13
So, there is 10 in it

Question 8.
81 + 8 = ___________
Make a 10?           Yes         No
Answer: No

Explanation:
By adding ones place from both the sides
1 + 8 = 9
so, there is 10 in it.

MP Logic
Complete.
Question 9.

Answer: 62

Explanation:
56 + 6 = 62
56 + 4 makes 60 and 4 + 2 = 6
6 is divided as 4 + 2
60 + 2

Question 10.

Answer: 48

Explanation:
39 + 9 = 48
39 + 1 = 40, 9 is divided to 1 and 8.

Think and Grow: Modeling Real Life

You put 17 puzzle pieces together. There are 7 left. How many puzzle pieces are there in all?

Addition equation:

Model:

Make a 10?           Yes         No

___________ Puzzle pieces
Answer: Yes.

Explanation:
17 puzzle pieces are together
7 added to it
17 + 7 = 24
The ones place of both sides are added 7 + 7 = 14
so, there is ten in it.

Show and Grow

Question 11.
You color 46 states. There are 4 left. How many states are there in all?

Addition equation:

Model:

Make a 10?           Yes         No

___________ States
Answer: 50

Explanation:
Colored states are 46
uncolored states are 4
Total states are 46 + 4 = 50
6 + 4 = 10
Both the sides ones places are added

Make a 10 to Add Practice 9.3

Question 1.
66 + 5 = __________

Make a 10?           Yes         No
Answer: yes

Explanation:
Number of lines which represents 10 is 6 that in tens place
in ones place both the sides 6 + 5 = 11
so, there is 10 in it

Question 2.
74 + 3 = ____________

Make a 10?           Yes         No
Answer: No

Explanation:
Number of lines which represents 10 is 7 that in tens place
in ones place both the sides 4 + 3 = 7.
so, there is no 10 in it.

Question 3.
28 + 8 _____________
Make a 10?           Yes         No
Answer: Yes

Explanation:
2 represents tens place 2 x 10 = 20
In ones place from both the sides 8 + 8 = 16
so, There is ten in it.

Question 4.
52 + 9 = ______________
Make a 10?           Yes         No
Answer: Yes

Explanation:
5 represents tens place 5 x 10 = 50
In ones place from both the sides 2 + 9 = 11
so, There is ten in it.
52 + 9 = 61.

Question 5.
26 + 7 = ____________
Make a 10?           Yes         No
Answer: yes

Explanation:
2 represents tens place 2 x 10 = 20
In ones place from both the sides 6 + 7 = 13
so, There is ten in it.
26 + 7 = 33.

Question 6.
41 + 6 = ____________
Make a 10?           Yes         No
Answer: No

Explanation:
4 represents tens place 4 x 10 = 40
In ones place from both the sides 1 + 6 = 7
so, There is no ten in it.
41 + 6 = 47.

MP Logic
Complete.
Question 7.

Answer: 41

Explanation:
37 + 4 = 41
4 is divided to 3 and 1
37 + 3 = 40
40 plus 1 is 41

Question 8.

Answer: 54

Explanation:
48 + 6 = 54
6 is divided in to 2 and 4
48 + 2 = 50 So, there is ten in it
50 + 4 = 54

Question 9.
Modeling Real Life
A snake lays 24 eggs. Another snake lays 9 eggs. How many eggs are there in all?

Answer:
33 eggs

Explanation:
A snake lays 24 eggs
another snake lays 9
there are 33 eggs in all.

Review & Refresh

Question 10.
Color the shapes that have 4 vertices.

Answer:  Square and rectangle

Explanation:
For a square there are 4 vertices and for a rectangle there are 4 vertices. vertices are Nothing but corners or joining of the two lines.

Lesson 9.4 Add Two-Digit Numbers

Explore and Grow

Show how you can use a model to solve.

43 + 28 = _________

Answer: 43 + 28 = 61

Explanation:
4 tens and 3 ones
2 tens and 8 ones
The figure shows lines represents tens and
circls represent ones

Show and Grow

Question 1.
39 + 45 = ?

___________ tens __________ ones
Answer: 7 tens and 14 ones

Explanation:
The figure shows 39 represents 3 tens and 9 ones
and 4 tens 5 ones
3 + 4 = 7 and 9 + 5 = 14.

Apply and Grow: Practice

Question 2.
19 + 35 = ?

Answer: 1 + 3 = 4 and 9 + 5 = 14

Explanation:
First figure shows 1 tens and 9 ones
3 tens and 5 ones
The second figure shows 1 + 3 = 4
and 9 + 5 = 14

Question 3.

Answer: 60
Explanation:
43 + 17 = 60
In the 4 and 3 are in tens place and 3 and 7 are in ones place
My friend has 43 choclates and me has 17 more of it.
Total choclates is 60.

Question 4.

Answer: 81

Explanation:
67 + 41 = 81
my friend jim has 67 lollipops
and rim has 41 lollipops
in total jim and rim has 81 lollipops.

Question 5.
YOU BE THE TEACHER
Is the sum correct? Explain.

Answer: 76
Explanation:

58 + 28 = 76
The line represents 10 and circles represents 1
5 tens and 8 ones
2 tens and 8 ones
by adding 8 from the both places we get 16 and there is 10 in it.

Think and Grow: Modeling Real Life

You earn a sticker for every 10 pages you read. You read 34 pages one week and 37 the next. How many stickers do you earn?
Addition problem:

Model:

Write the missing numbers:
_________ tens _________ one
__________ stickers
Answer: 71

Explanation:
I earn a sticker for every 10 pages
first I read 34 pages and then 37 pages
34 + 37 = 71
so, total 71 pages for every 10 pages 1 sticker 7 stickers

Show and Grow

Question 6.
You earn a coin for every lo cans you recycle. You recycle 18 cans one week and 25 the next. How many coins do you earn?

Addition problem:

Model:

Write the missing numbers:
_________ tens _________ one
__________ coins
Answer:  4 tens and 3 ones 4 coins
Explanation:
If I recycle 10 cans I get 1 coin
in first week i recycled 18 cans and in next week 25 cans
18 + 25 = 43.

Add Two-Digit Numbers Practice 9.4

Question 1.
57 + 15 = ?

Answer: 57 + 15 = 72

Explanation:
In 57 5 tens and 7 ones
in 15 1 tens and 5 ones
If we add both we get 7 tens and 2 ones.

Question 2.

Answer: 76

Explanation:
In 40 4 tens and 0 ones
in 36 3 tens and 6 ones
if we add both we get 7 tens and 6 ones.

Question 3.

Answer: 81
Explanation:
In 29 2 tens and 9 ones
in 52 5 tens and  2 ones
If we add both 29 and 52 is 81
8 tens and 1 ones.

Question 4.
DIG DEEPER!
Do you need to use the make a 10 strategy to find each sum?
28 + 34 = ?      Yes      No
56 + 15 = ?      Yes      No
42 + 21 = ?      Yes      No
68 + 11 = ?      Yes      No
Answer:
28 + 34 = 62      Yes
56 + 15 = 71    Yes
42 + 21 = 63      No
68 + 11 = 79     No
Explanation:
In first sum 28 + 34
if we take 8 and 4 and add we get 12 taken from ones place
we get 10 in it.
In second sum 56 + 15
if we take 6 and 5 and add we get 11 taken from ones place
we get 10 in it.
In third sum 42 + 21
if we take 2 and 1 and add we get 3 taken from ones place
we didn’t get 10 in it
In fourth sum 68 + 11
if we take 8 and 1 and add we get 9 taken from ones place
we didn’t get 10 in it.

Question 5.
You need a box for every 10 muffins you make. You make 33 blueberry muffins and 47 banana muffins. How many boxes do you need?

__________ boxes
Answer: 8 boxes
Explanation:
We need a box for 10 muffins that we make
We made 33 blueberry muffins
and 47 banana muffins
if we add we get 33 + 47 = 80
so, I get 8 boxes.

Review & Refresh

Question 6.
Is the equation true or false?

Answer: 15 = 16 false

Explanation:
6+ 9 = 15 and 17 – 1= 16
15 is not equal to 16

Lesson 9.5 Practice Addition Strategies

Explore and Grow

Show two ways you can find the sum.

23 + 39 = __________

23 + 39 = __________

Answer:  23 + 39 = 62
Explanation: one method

second method

Show and Grow

Question 1.
47 + 24 = ________
Answer: 72

Explanation:
4 tens and 7 ones
2 tens and 4 ones
tens represents the lines and ones represents the circles

Question 2.
38 + 43 = ________
Answer: 81

Explanation:
3 tens and 8 ones
4 tens and 3 ones
tens represents the lines and ones represents circles

Apply and Grow: Practice

Question 3.
22 + 18 = ________
Answer: 40

Explanation:
2 tens and 2 ones
1 tens and 8 ones
tens represents the lines and ones represents the circles

Question 4.
57 + 34 = __________
Answer: 91

Explanation:
5 tens and 7 ones
3 tens and 4 ones
lines represents the tens and circles represents the ones

Question 5.
73 + 19 = __________
Answer: 92

Explanation:
lines represents the tens and circles represents the ones
7 ten and 3 ones
1 tens and 9 ones

Question 6.
81 + 11 = __________
Answer: 92

Explanation:
the lines represents tens and the circles represents ones
8 tens and 1 ones
1 tens and 1 ones
If we add both we get
92

Question 7.
YOU BE THE TEACHER
Is the sum correct? Explain.

Answer: No

Explanation:
no the sum is not correct.
But figure shown is correct
17 + 26 = 43

Think and Grow: Modeling Real Life

You hove 48 songs. Your friend has 27 more than you. How many songs does your friend have?

Addition equation:

Model:

____________ songs
Answer: 75 songs

Explanation:
Model : songs
I have 48 songs
and my friend had 27 more than me
Total songs that my friend had is 48 + 27 = 75 songs

Show and Grow

Question 8.
Your friend sells 56 candles. You sell 35 more than your friend. How many candles do you sell?

Addition equation:

Model:

____________ candles
Answer: 91 candles

Explanation:
Models: candles
My friend has 56 candles
and me has 35 more than my friend
so, I had 91 candles with me

Practice Addition Strategies Practice 9.5

Question 1.
62 + 29 = _______
Answer: 99

Explanation:
6 tens and 2 ones
2 tens and 9 ones
by adding both the numbers i get is 99

Question 2.
84 + 8 = ________
Answer: 92

Explanation:
8 tens and 4 ones
0 tens and 8 ones
by adding both the numbers i get is 92

Question 3.
75 + 17 = ________
Answer: 92

Explanation:
7 tens and 5 ones
1 tens and 7 ones
by adding both the numbers i get is 92

Question 4.
YOU BE THE TEACHER
Is the sum correct? Explain.

Answer:

Question 5.
Modeling Real Life You collect 12 leaves. Your friend collects 26 more than you. How many leaves does your friend collect?

____________ leaves
Answer: 38 leaves

Explanation:
I collect 12 leaves and my friend collect 26 more than me
total leaves my friend had is
12 + 26 = 38 leaves.

Review & Refresh

Question 6.
Circle the measurable attributes of the table.

Answer: length or height.

Explanation:
For table length or height is the measuring attribute
cubes represents to measure the length and height.

Lesson 9.6 Problem Solving: Addition

Explore and Grow

Model the story.

Newton has 15 dog bones. Descartes gives him 8 more. How many dog bones does Newton have now?

Answer: 23 bones

Explanation:
Newton has 15 bones and Descartes gave him 8 more
The total bones newton had is 23 bones.

Show and Grow

Question 1.
You have 49 toy soldiers. You buy some more. Now you have 84. How many toy soldiers did you buy?

Circle what you know:

Underline what you need to find.

Solve:

_____________ toy soldiers
Answer: 35 toy soldiers

Explanation:
I have 49 toy soldiers
If i buy 35 more
Then i will have 84 in total.

Apply and Grow: Practice

Question 2.
You have 55 pounds of dog food and some cat food. You have 63 pounds of pet food in all. How many pounds of cat food do you have?

Circle what you know.

Underline what you need to find.

Solve:

___________ pounds
Answer:  8 pounds

Explanation:
I have 55 pound of dog food
If i had 8 pounds of cat food
In total the pet food had is 63 pounds

Question 3.
A teacher has 34 erasers. There are 46 fewer erasers than pencils. How many pencils are there?

____________ pencils
Answer:

Question 4.
DIG DEEPER!
You have 25 toys. Your friend has more than you. There are more than 60 toys in all. How many toys can your friend have?
29       33       24      38
Answer: 35 toys

Explanation:
I have 25 toys
and if had 35 then the total toys will be 60
according to the given options it might be less than 35
29, 33 or 24

Think and Grow: Modeling Real Life

You need 60 invitations. You have 36 and buy 36 more. Do you have enough invitations?

Circle what you know.

Underline what you need to find.

Solve:

Compare: ___________ ○ 60        Yes          No
Answer: yes

Explanation:
66 > 60
I had 6 more invitations
Yes they have enough invitations.

Show and Grow

Question 5.
You need 84 bottles of water. You have 48 and buy 32 more. Do you have enough bottles of water?

Circle what you know.

Underline what you need to find.

Solve:

Compare: ___________ ○ 84        Yes          No
Answer:  No

Explanation:
The water bottles I had 80
The water bottles needed is 84
80 < 84
so, there are no enough waterbottles

Problem Solving: Addition Practice 9.6

Question 1.
You have 31 stuffed animals. You and your friend have 60 stuffed animals in all. How many stuffed animals does your friend have?

______________ stuffed animals
Answer:  29

Explanation:
31 + 29 = 60,
There are 29 stuffed animals with my friend.

Question 2.
A store has 56 shirts. There are 28 fewer shirts than pairs of pants. How many pairs of pants are there?

______________ Pairs of pants
Answer: 28

Explanation:
Total shirts is 56 and 56 – 28 = 28,
The pairs of paints are 28.
28 + 28 = 56.

Question 3.
YOU BE THE TEACHER
You have 25 movies. You have 18 more video games than movies. Your friend says you have (43 movies and video games in all. Is your friend correct? Explain.

___________________________________________________

___________________________________________________
Answer: 43:

Explanation
Me having 25 movies and 18 videogames 25 + 18 = 43,
Yes my friend is correct I have 43 movies and  videogames in all.

Question 4.
Modeling Real Life
Newton needs 9o chairs for a party. He has 51. He rents 3 more. Does Newton have enough chairs?

Circle:         Yes          No
Answer: No,

Explanation:
Newton doesn’t have enough chairs.
He needed is 90 he has 51 + 3 = 54.

Review & Refresh

Circle the longer object.

Question 5.

Answer:
The scissor is longer than the brick
circled the scissor

Question 6.

Answer:
The first pencil is longer than the second pencil
First pencil is circled

Add Two-Digit Numbers Performance Task

Question 1.
You play a game. Each red ball you collect is worth 10 points. Each yellow ball you collect is worth 1 point.

a. You collect 3 red balls and 13 yellow balls. How many points do you have?

___________ points
Answer: 30

Explanation
Red ball = 10
yellow = 1.
I have collected 3 red and 13 yellow,
3 × 10 = 30

39 + 13 = 52. Total points are 52.

b. Your teammates score 38 points and 24 points. How many points do your teammates have in all?

___________ points
Answer:
38 + 24 = 62,
62 points our teammates have in all.

c. Your learn wants to have 100 points. Does your team reach its goal?

Yes No
Answer: No,
we didn’t reach the goal.

d. Why do you think a red ball is worth more points?
Answer:
Because it is small in size and easy to handle to kids by seeing in the figure.

Add Two-Digit Numbers Change Practice

Add Tens and Ones Homework & Practice 9.1

Question 1.
56 + 3 = _________

Answer:
53 + 3 = 56,
5 tens and 6 ones.

Question 2.
22 + 54 = __________

Answer:
24 + 54 = 78,
7 tens and 8 ones.

Add Tens and Ones Using a Number Line Homework & Practice 9.2

Question 3.
62 + 25 = ___________

Answer: 62 + 25 = 87

Explanation:
6 tens and 2 ones
2 tens and 5 ones
by adding both we get 87

Question 4.
38 + 51 = ___________

Answer: 38 + 51 = 89

Explanation:
3 tens and 8 ones
5 tens and 1 ones
by adding both we get 89.

Question 5.
MP Structure
Write an equation that matches the number line.

Answer: 16 + 52 = 68

Make a 10 to Add Homework & Practice 9.3

Question 6.
42 + 6 = ___________

Make a 10?       Yes           No
Answer: No

Explanation:
lines represents the tens and circles represents the ones
4 tens and 2 ones
0 tens and 6 ones
By adding ones place from both the sides 2 + 6 = 8
so there is no 10 in it.

Question 7.
27 + 7 = ___________

Make a 10?       Yes           No
Answer: yes

Explanation:
lines represents the tens and circles represents the ones
2 tens and 7 ones
0 tens and 7 ones
By adding ones place from both the sides 7 + 7 = 14
so there is 10 in it

MP Logic
Complete.
Question 8.

Answer:
34 + 7 = 41.

Question 9.

Answer:
59 + 8 = 67

Add Two-Digit Numbers Homework & Practice 9.4

Make quick sketches to find the sum.

Question 10.

Answer:
28 + 33 = 61, 6 tens and 1 ones.

Question 11.

____________ tens ___________ one
Answer:
49 + 24 =73, 7 tens and 3 ones.

Question 12.
Modeling Real Life
Your club earns a badge for every 10 trees planted. Your club plants 25 trees in the fall and 25 in the spring. How many badges does your club earn?

_______________ badges
Answer:
One badge for 10, 25 in spring and 25 in fall.
Total trees planted is 25 + 25 = 50.
So, 5 badges.

Practice Addition Strategies Homework & Practice 9.5

Question 13.
19 + 43 = __________
Answer: 19 + 43 = 62, 6 tens and 2 ones.

Question 14.
66 + 28 = __________
Answer: 66 + 28 = 94, 9 tens and 4 ones.

Problem Solving: Addition Homework & Practice 9.6

Question 15.
Your friend has 5 marbles. You have 23 more than your friend. How many marbles do you have?

_______________ marbles
Answer:  28 marbles

Explanation
My friend has 5 and me having 23 more
than my friend 23 + 5 = 28.

Question 16.
Modeling Real Life
You need 50 party hats. You have 24. You buy 16 more. Do you have enough party hats?

Yes           No
Answer: No

Explanation:
Me having 24 caps and bought 16 more
24 + 16 = 40,
No, I don’t have enough caps.

Conclusion:

Download Big Ideas Math Answers Grade 1 Chapter 9 Add Two-Digit Numbers pdf and start your preparation for the exams. Big Ideas Math Answer is the best platform for the students to improve their math skills. If you have any queries about Big Ideas Math Answers Grade 1 Chapters you can post the comments in the below comment box. All the Best!!

Are you searching for the Big Ideas Math Grade 2 Chapter 8 Count and Compare Numbers to 1,000 Answer Key? If yes, then stay on this page. Have a look at the detailed explanation for all the questions of Big Ideas Math 2nd Grade 8th Chapter Count and Compare Numbers to 1,000 Book. The BIM Grade 2 Answer Key for Chapter 8 Count and Compare Numbers to 1,000 will help you to become a pro in solving questions. Hence Download Big Ideas Math Answers Grade 2 Chapter 8 Count and Compare Numbers to 1,000 PDF and begin the preparation as early as possible.

Big Ideas Math Book Grade 2 Answer Key Chapter 8 Count and Compare Numbers to 1,000

Every student has to prepare all the topics provided in BIM Grade 2 Answers Chapter 8 Count and Compare Numbers to 1,000 to score good marks in the exam. This answer key is also helpful to complete the homework and assignments within the given time. After practicing chapter 8, you can be able to understand counting. The different topics included in Big Ideas Math Book 2nd Grade Answers Chapter 8 Count and Compare Numbers to 1,000 are Count and Compare Numbers to 1,000 Vocabulary, Count to 120 in Different Ways, Count to 1,000 in Different Ways, Place Value Patterns, Find More or Less, Compare Numbers Using Symbols, and Compare Numbers Using a Number Line.

Vocabulary

• Count and Compare Numbers to 1,000 Vocabulary

Lesson 1 Count to 120 in Different Ways

• Lesson 8.1 Count to 120 in Different Ways
• Count to 120 in Different Ways Homework & Practice 8.1

Lesson 2 Count to 1,000 in Different Ways

• Lesson 8.2 Count to 1,000 in Different Ways
• Count to 1,000 in Different Ways Homework & Practice 8.2

Lesson 3 Place Value Patterns

• Lesson 8.3 Place Value Patterns
• Place Value Patterns Homework & Practice 8.3

Lesson 4 Find More or Less

• Lesson 8.4 Find More or Less
• Find More or Less Homework & Practice 8.4

Lesson 5 Compare Numbers Using Symbols

• Lesson 8.5 Compare Numbers Using Symbols
• Compare Numbers Using Symbols Homework & Practice 8.5

Lesson 6 Compare Numbers Using a Number Line

• Lesson 8.6 Compare Numbers Using a Number Line
  Compare Numbers Using a Number Line Homework & Practice 8.6
• Count and Compare Numbers to 1,000 Performance Task
• Count and Compare Numbers to 1,000 Activity
• Count and Compare Numbers to 1,000 Chapter Practice
• Count and Compare Numbers to 1,000 Cumulative Practice 1 – 8
  

Count and Compare Numbers to 1,000 Vocabulary

Organize It
Use the review words to complete the graphic organizer.

Answer:

Define It

Use your vocabulary cards to complete the puzzle.

Chapter 8 Vocabulary cards

Lesson 8.1 Count to 120 in Different Ways

Explore and Grow

Start at 5. Skip count by fives. Circle the numbers you count. Start at 10. Skip count by tens. Color the numbers you count.

What patterns do you notice?
____________________
____________________
Answer:

In column 5 we notice the pattern of count by tens and in column 10 we notice the pattern of count by tens.

Show and Grow

Question 1.
Count by ones.
35, 36, 37, ___, ____, ___, ___, ___
Answer:
35, 36, 37, 38, 39, 40, 41, 42 .

Question 2.
Count by fives.
55, 60, 65, __, ___, ___, ___, ___
Answer:
55, 60, 65, 70, 75, 80, 85, 90

Question 3.
Count by tens.
21, 31, 41, ___, __, ___, ___, ___
Answer:
21, 31, 41, 51, 61, 71, 81, 91 .

Apply and Grow: Practice

Count by ones.

Question 4.
57, 58, 59, __, ___, ___, ___, ___
Answer:
57, 58, 59, 60, 61, 62, 63, 64

Count by fives.

Question 6.
35, 40, 45, __, ___, ___, __, ___
Answer:
35, 40, 45, 50, 55, 60, 65, 70

Question 5.
__, 106, __, 108, __, ___, ___
Answer:
105, 106, 107, 108, 109, 110, 111

Count by fives.

Question 6.
35, 40, 45, __, ___, ___, ___, ___
Answer:
35, 40, 45, 50, 55, 60, 65, 70

Question 7.
__, 80, __, 90, __, __, ___
Answer:
70, 80, 90, 100, 110, 120 .

Count by tens.

Question 8.
12, 22, 32, __, ___, __, ___, __
Anwer:
12, 22, 32, 42, 52, 62, 72, 82 .

Question 9.
__, 50, __, 70, __, __, ___
Answer:
40, 50, 60, 70, 80, 90, 100 .

Question 10.
Number Sense
Newton counts by ones from 47 to 53. Which numbers does he count?

Answer:
counts by ones from 47 to 53 are :
47, 48, 49, 50, 51, 52, 53 .
numbers does he count are 49 and 52 numbers

Question 11.
Number Sense
Descartes counts by fives from 90 to 115. Which numbers does he count?

Answer:
counts by fives from 90 to 115 are :
90, 95, 100, 105, 110, 115 .
numbers does he count are 105 and 110 .

Think and Grow: Modeling Real Life

Newton has 65 points. He captures small aliens worth 5 points. Descartes has 25 points. He captures large aliens worth 10 points.


Answer:
Number of points Newton have = 65
Number of points for small aliens = 5.
Number of aliens did Newton have = 65 / 5 = 13.

Number of points Descartes have = 25
Number of points for Large aliens = 10.
Number of aliens did Descartes have = 25 / 10 = 2.5
Newton have more aliens than Descartes.so Descartes needs more aliens .

Show and Grow

Question 12.
Newton has 55 points. He collects gold coins worth 10 points. Descartes has 70 points. He collects silver coins worth 5 points. Who needs to collect more coins to reach 100 points?


Answer:
Number of points Newton have = 55 points
he collects gold coins worth = 10 points each .
for 100 points newton requires 45 points
Number of coins required for 45 points for Newton= 45/ 10 = 4.5 coins.

Number of points Descartes have = 70 points
he collects silver coins worth = 5 points each .
for 100 points Descartes requires 30 points
Number of coins required for 30 points for Descartes= 30/ 5 = 6 coins.
Descartes requires more 6 coins to reach 100 points .

Question 13.
You and your friend count from 30 to 70. You count by fives. Your friend counts by tens. Who says more numbers? Explain.
Answer:
Count by fives:( by me)
30, 35, 40, 45, 50, 55, 60, 65, 70 = total 9 numbers.
Count by tens:( my friend)
30, 40, 50, 60, 70 = total 5 numbers.
I say more numbers than my friend .

Count to 120 in Different Ways Homework & Practice 8.1

Count by ones.

Question 1.
63, 64, 65, __, __, ___, ___, ___
Answer:
63, 64, 65, 66, 67, 68, 69, 70 .

Question 2.
__, 112, __, 114, __, __, ___
Answer:
111, 112, 113, 114, 115, 116, 117 .

Count by fives.

Question 3.
10, 15, 20, __, __, ___, __, ___
Answer:
10, 15, 20, 25, 30, 35, 40, 45 .

Question 4.
__, 95, __, 105, __, __, ___
Answer:
90, 95, 100, 105, 110, 115, 120 .

Count by tens.

Question 5.
44, 54, 64, __, __, __, ___, __
Answer:
44, 54, 64, 74, 84, 94, 104 .

Question 6.
__, 20, __, 40, __, __, __
Answer:
10, 20, 30, 40, 50, 60, 70 .

Question 7.
YOU BE THE TEACHER
Newton counts by fives. Is he correct? Explain.


Answer:
No, Count by fives means 5, 10, 15, 20, 25, 30 and so on …..
As per the above count 5, 15, 25, 35, 45, 55 they are count by tens as there is 10 differences between the sequence of numbers.

Question 8.
Modeling Real Life
Who needs to make more shots to earn 50 points?

Answer:
Newton
Number of points with Newton = 30 .
Newton requires more 20 points to make it 50 points
Points for one shot = 5 .
Number of shots required for Newton = 20 / 5 = 4 shots.

Descartes
Number of points with Descartes = 20 .
Descartes requires more 30 points to make it 50 points
Points for one shot = 10 .
Number of shots required for Descartes = 30 / 10 = 3 shots.

Therefor Newton needs to make more shots than Descartes .

Review & Refresh

Question 9.
34 – 16 = ?

Answer:

Question 10.
75 – 32 = ?

Answer:

Question 11.
93 – 28 = ?

Answer:

Lesson 8.2 Count to 1,000 in Different Ways

Explore and Grow

Count by hundreds to 1,000.

Answer:

Count by tens to 1,000.

Answer:

Count by fives to 1,000.

Answer:

Count by ones to 1,000.

Answer:

Show and Grow

Question 1.
Count by fives.
675, 680, 685, __, __, __, __, ___
Answer:
675, 680, 685, 690, 695, 700, 705, 710 .

Question 2.
Count by tens.
850, 860, 870, __, __, __, __, ___
Answer:
850, 860, 870, 880, 890, 900, 910, 920 .

Question 3.
Count by hundreds.
100, 200, 300, __, __, __, __, ___
Answer:
100, 200, 300, 400, 500, 600, 700, 800 .

Apply and Grow: Practice

Count by fives.

Question 4.
520, 525, 530, __, __, __, __, ___
Answer:
520, 525, 530, 535, 540, 545, 550, 555 .

Question 5.
875, 880, __, __, __, __, ___
Answer:
875, 880, 885, 890, 895, 900, 905 .

Count by tens.

Question 6.
600, 610, 620, __, __, __, __, ___
Answer:
600, 610, 620, 630, 640, 650, 660, 670 .

Question 7.
460, 470, __, __, __, __, ___
Answer:
460, 470, 480, 490, 500, 510, 520 .

Count by hundreds.

Question 8.
200, 300, 400, __, __, __, __, ___
Answer:
200, 300, 400, 500, 600, 700, 800, 900 .

Question 9.
400, 500, __, __, __, __, __
Answer:
400, 500, 600, 700, 800, 900, 1000 .

Question 10.
DIG DEEPER!
Newton counts by hundreds. Find the missing number. Think: How do you know?

Answer:
The missing number is 0  as the count start from 0, 100, 200, 300, 400, 500 Each number is 100 more than the previous number.

Question 11.
Structure
Did Descartes count by tens or by hundreds? Think: How do you know?

Answer:
It is count by tens as Each number is 10 more than the previous number .

Think and Grow: Modeling Real Life

A summer camp leader has 240 T-shirts. He buys 6 more colors with 10 shirts in each color. How many T-shirts does he have now?


Answer:
Number of T-Shirts = 240.
Number of T-shirts were bought = 6 X 10 ( 6 different colors ) = 60 T-shirts .
Total Number of T-Shirts = 240 + 60 = 300 .

Show and Grow

Question 12.
You have 100 bracelets. You buy 5 more boxes with 100 bracelets in each box. How many bracelets do you have now?


Answer:
Number of Bracelets = 100.
Number of Boxes bought = 5 x 100 = 500 bracelets.
Total Number of Bracelets = 100 + 500 = 600 bracelets.

Question 13.
You and your friend count from 370 to 420. You count by tens. Your friend counts by fives. Who says more numbers? Explain.
_________________________
_________________________
Answer:
Count by Tens:( by me)
370, 380, 390, 400, 410, 420 . = total 6 numbers.
Count by fives:( my friend)
370, 375, 380, 385, 390, 395, 400, 405, 410, 415, 420 = total 11 numbers.
My friends say more numbers than me .

Count to 1,000 in Different Ways Homework & Practice 8.2

Count by fives.

Question 1.
445, 450, 455, ___, __, __, __, ___
Answer:
445, 450, 455, 460, 465, 470, 475, 480 .

Question 2.
770, 775, __, __, __, __, ___
Answer:
770, 775, 780, 785, 790, 795, 800,

Count by tens.

Question 3.
520, 530, 540, __, __, __, __, __
Answer:
520, 530, 540, 550, 560, 570, 580, 590 .

Question 4.
660, 670, __, __, __, __, ___
Answer:
660, 670, 680, 690, 700, 710, 720 .

Count by Hundreds.

Question 5.
300, 400, 500, __, __, __, __, ___
Answer:
300, 400, 500, 600, 700, 800, 900, 1000.

Question 6.
200, 300, __, __, __, __, ___
Answer:
200, 300, 400, 500, 600, 700, 800 .

Review & Refresh

Question 7.
DIG DEEPER!
Newton starts at 950 and counts to 1,000 by fives. Complete the number line to show the last 6 numbers he counts.

Answer:

Question 8.
Modeling Real Life
A carnival worker has 380 stuffed animals. She buys 6 more boxes with 5 stuffed animals in each box. How many stuffed animals does she have now?

Answer:
Number of stuffed animals = 380 .
Number of animals bought = 6 x 5 = 30.
Total Number of stuffed animals = 380 + 30 = 410 Animals .

Question 9.
Modeling Real Life
A water park shop owner has 100 goggles. He buys 4 more colors with 100 goggles in each color. How many goggles does the shop owner have now?

Answer:
Number of Goggles = 100.
Number of goggles bought = 4 x 100 = 400
Total Number of Goggles = 100 + 400 = 500 Goggles .

Question 10.
You see 14 geese in a pond. 17 more join them. Then you see 11 more fly to the pond. How many geese do you see in all?

Answer:
Number of geese in a pond = 14.
Number of geese joined = 17 .
Total geese’s = 14 + 17 = 31 .
Number of geese’s flew to pond = 11 .
Total Geese’s = 31 + 11 = 42 .

Lesson 8.3 Place Value Patterns

Explore and Grow

What patterns do you see in the shaded row and column? Use the patterns to complete the chart.

Answer:
The pattern in Rows = Count by ones.
The pattern in Columns = Count by Tens.

Show and Grow

Use place value to find the missing numbers.

Question 1.
485, 486, 487, __, __, ___, ___
Answer:
485, 486, 487, 488, 489, 490, 491 .

Question 2.
612, 622, 632, __, __, ___, ___
Answer:
612, 622, 632, 642, 652, 662, 672 .

Question 3.
267, 277, __, 297, ___, 317, __
Answer:
267, 277, 287, 297, 307, 317, 327 .

Question 4.
101, 201, ___, 401, __, __, ___
Answer:
101, 201, 301, 401, 501, 601, 701 .

Apply and Grow: Practice

Use place value to find the missing numbers.

Question 5.
324, 325, ___, 327, __, __, ___
Answer:
324, 325, 326, 327, 328, 329, 330 .

Question 6.
194, 294, __, 494, __, __, ___
Answer:
194, 294, 394, 494, 594, 694, 794 .

Question 7.
463, 473, __, 493, __, __, __
Answer:
463, 472, 483, 493, 503, 513, 523 .

Question 8.
232, 332, __, 532, __, ___, __
Answer:
232, 332, 432, 532, 632, 732, 832 .

Question 9.
985, 986, __, 988, __, __, __
Answer:
985, 986, 987, 988, 989, 990 .

Question 10.
751, 761, __, 781, __, __, __
Answer:
751, 761, 771, 781, 791, 801, 811 .

Question 11.
606, 607, __, 609, __, __, __
Answer:
606, 607, 608, 609, 700, 701, 702 .

Question 12.
Repeated Reasoning
Use place value to describe each pattern.

Answer:

Think and Grow: Modeling Real Life

There are 273 tickets in a bin. Some more are put in the bin. Now there are 973. How many groups of 100 tickets were put in the bin?


Answer:

Number of tickets in bin = 273.
Number of tickets added in bin = X.
Total Number tickets in bin now = 973 = 273 + X
X = 973 – 273 = 700 .
Number of groups = 700/100 = 7 (Each group contains 100 tickets) .
Therefore 7 groups of 100 tickets were put in bin .

Show and Grow

Question 13.
You have 338 pennies in a jar. You put more in the jar. Now there are 388. How many groups of 10 pennies were put in the jar?

Answer:
Number of pennies =338.
Number of pennies added = X.
Total Number of pennies = 388 = 338 + X.
X= 388 – 328 = 50.
Number of groups of 10 pennies were put in the jar = 50 / 10 = 5.

Question 14.
DIG DEEPER!
There are 410 people at a show. 8 more rows of seats get filled. Now there are 490 people. How many people can sit in each row?

Explain how you solved.
______________
______________
Answer:
Number of people = 410 .
Number of people added = X.
Total Number of people = 490.
490=410 + X.
X = 490 – 410 = 80 .
Number of people added = 80 .
Number of Rows = 8.
Number of people in each row = 80 /8 = 10.

Place Value Patterns Homework & Practice 8.3

Use place value to find the missing numbers.

Question 1.
710, 711, 712, __, __, __, ___
Answer:
710, 711, 712, 713, 714, 715, 716 .

Question 2.
822, 832, 842, __, __, __, __
Answer:
822, 832, 842, 852, 862, 872, 882 .

Question 3.
325, 425, 525, __, __, __, __
Answer:
325, 425, 525, 625, 725, 825, 925 .

Question 4.
669, 679, __, 699, __, __, ___
Answer:
669, 679, 689, 699, 709, 719, 729 .

Question 5.
534, 535, __, 537, __, __, __
Answer:
534, 535, 536, 537, 538, 539 , 540

Question 6.
368, 468, __, 668, __, __, ___
Answer:
368, 468, 568, 668, 768, 868, 968 .

Question 7.
YOU BE THE TEACHER
Newton says the hundreds digit in the numbers shown increases by 1. Is he correct? Explain.

Answer:
No, There is increase in the tens place not in hundreds place .Each Number is 10 more than the previous number so it is count by tens.

Question 8.
Modeling Real Life
A farmer has 467 cornstalks. The farmer grows some more. Now there are 967 cornstalks. How many groups of 100 cornstalks did the farmer add?

Answer:
Number of cornstalks = 467 .
Number of cornstalks grown =X
Total Number of cornstalks = 967.
467 + X =967.
X = 967- 467 = 500.
Number of cornstalks grown = 500 .
Number of groups of 100 cornstalks added = 500/100 = 5.

Question 9.
DIG DEEPER!
There are 250 people at a party. 3 more tables get filled. Now there are 280 people. How many people can sit at each table?

Answer:
Number of people in the party = 250 .
Number of more people added = X.
Total Number of people in the party = 280 .
250 + X = 280 .
X = 280- 250 = 30 .
Number of more people added = 30 .
Number of tables got filled with added people = 3.
Number of people in each table = 30 / 3 = 10 .

Review & Refresh

Question 10.
8 + 4 = __
Answer:
12.

Question 11.
15 – 8 = __
Answer:
7.

Question 12.

Answer:
10.

Question 13.

Answer:
6

Question 14.

Answer:
13.

Question 15.

Answer:
9 .

Lesson 8.4 Find More or Less

Explore and Grow

Model 253. Use your model to complete the sentences.

1 more than 253 is __.
1 less than 253 is __.
10 more than 253 is __.
10 less than 253 is __.
100 more than 253 is __.
100 less than 253 is __.
Answer:
1 more than 253 is 254.
1 less than 253 is 252.
10 more than 253 is 263.
10 less than 253 is 243.
100 more than 253 is 353.
100 less than 253 is 153.

Show and Grow

Question 1.
10 more than 452 is __.
Answer:
10 more than 452 is 462.

Question 2.
10 less than 813 is __.
Answer:
10 less than 813 is 803..

Question 3.
100 less than 729 is __.
Answer:
100 less than 729 is 629.

Question 4.
100 more than 386 is __.
Answer:
100 more than 386 is 486.

Apply and Grow: Practice

Question 5.
10 more than 571 is __.
Answer:
10 more than 571 is 581.

Question 6.
10 less than 333 is __.
Answer:
10 less than 333 is 323.

Question 7.
100 more than 604 is __.
Answer:
100 more than 604 is 704.

Question 8.
100 less than 592 is __.
Answer:
100 less than 592 is 492.

Question 9.
1 more than 934 is __.
Answer:
1 more than 934 is 935.

Question 10.
1 less than 101 is __.
Answer:
1 less than 101 is 100.

Question 11.
10 less than 286 is __.
Answer:
10 less than 286 is 276.

Question 12.
1 more than 467 is __.
Answer:
1 more than 467 is 468.

Question 13.
10 more than 763 is __.
Answer:
10 more than 763 is 773.

Question 14.
100 less than 846 is __.
Answer:
100 less than 846 is 746.

Question 15.
1 less than 999 is __.
Answer:
1 less than 999 is 998.

Question 16.
100 more than 28 is __.
Answer:
100 more than 28 is 128.

Question 17.
100 less than 135 is __.
Answer:
100 less than 135 is 35.

Question 18.
100 more than 900 is __.
Answer:
100 more than 900 is 1000.

Question 19.
Number Sense
Complete each sentence.

Answer:

Think and Grow: Modeling Real Life

An orange tree has 639 oranges. A lemon tree has 100 fewer lemons. How many lemons does the tree have?

Answer:
Number of Oranges = 639.
Number of Lemons = 639 – 100 =539.

Show and Grow

Question 20.
A history book has 197 pictures. A science book has 10 more pictures. How many pictures are in the science book?

Answer:
Number of pictures in history books = 197 .
Number of pictures in science book = 197 + 10  = 207.

Question 21.
DIG DEEPER!
A boat puzzle has 525 pieces. A bird puzzle has 100 more than the boat puzzle. A space puzzle has 10 fewer than the bird puzzle. How many puzzle pieces does the space puzzle have?


Answer:
Number of puzzles in boat puzzle = 525 .
Number of puzzles in bird puzzle = 100 + 525 = 625.
Number of puzzles in space puzzle = 625 – 10 = 615 .

Question 22.
DIG DEEPER!
You have 398 points. Newton has 100 fewer than you. Descartes has 10 more than Newton. How many points does Descartes have?

Answer:
Number of points = 398 .
Number of points Newton have = 398 – 100 = 298 .
Number of points Descartes have = 298 + 10 = 308 .

Find More or Less Homework & Practice 8.4

Question 1.
10 more than 106 is __.
Answer:
10 more than 106 is 116.

Question 2.
10 less than 467 is __
Answer:
10 less than 467 is 457.

Question 3.
100 more than 321 is __.
Answer:
100 more than 321 is 421.

Question 4.
100 less than 945 is __.
Answer:
100 less than 945 is 845.

Question 5.
1 more than 513 is __.
Answer:
1 more than 513 is 514.

Question 6.
1 less than 899 is __.
Answer:
1 less than 899 is 898.

Question 7.
1 less than 264 is __.
Answer:
1 less than 264 is 263.

Question 8.
100 more than 555 is __.
Answer:
100 more than 555 is 655.

Question 9.
1 more than 852 is __.
Answer:
1 more than 852 is 853.

Question 10.
100 less than 573 is __.
Answer:
100 less than 573 is 473.

Question 11.
10 less than 314 is __
Answer:
10 less than 314 is 304.

Question 12.
10 more than 687 is __
Answer:
10 more than 687 is 697.

Question 13.
Structure
Complete the table.

Answer:

Question 14.
Number Sense
Complete the sentence.
__ is 10 less than 546 and 10 more than __.
Answer:
536 is 10 less than 546 and 10 more than 556.

Question 15.
Modeling Real Life
Your magic book has 163 tricks. Your friend’s magic book has 100 more tricks than yours. How many tricks does your friend’s magic book have?

Answer:
Number of tricks in my magic books = 163 .
Number of tricks in my friends magic books = 100+163 = 263.

Question 16.
DIG DEEPER!
You have 624 songs. Newton has 100 fewer than you. Descartes has 10 more than Newton. How many songs does Descartes have?

Answer:
Number of songs = 624.
Number of songs Newton have = 624 – 100 = 524.
Number of songs Descartes have = 524 + 10  = 534 .

Review & Refresh

Question 17.
A bookcase has 5 shelves. There are 2 stuffed animals on each shelf. How many stuffed animals are there in all?


Answer:
Number of shelves = 5.
Number of stuffed animals in each shelf = 2
Total Number of stuffed animals in each 5 shelves = 2 added 5 times.
2+2+2+2+2 = 10 animals.

Lesson 8.5 Compare Numbers Using Symbols

Explore and Grow

Make a quick sketch of each number. Circle the greater number.

How do you know which number is greater?
____________________
____________________
Answer:

When comparing the values of two numbers, you can use a number line to determine which number is greater. The number on the right is always greater than the number on the left. In the example Above, you can see that 472 is greater than 439 because 472 is to the right of 439 on the number line.

Show and Grow

Question 1.
Compare.

Answer:
652 > 614.

Apply and Grow: Practice

Compare

Question 2.

Answer:
324 > 317.

Question 3.

Answer:
26 < 206 .

Question 4.

Answer:
546 < 564 .

Question 5.

Answer:
931 > 842.

Question 6.

Answer:
700 + 30 + 5 = 735.
735 = 735 .

Question 7.

Answer:
400 + 20 = 420.
412 < 420.

Question 8.
Reasoning
Find the number that will make all three comparisons true.

Answer:
105.

Question 9.
YOU BE THE TEACHER
Is Newton correct? Explain.

Answer:
625 < 631.
The given number is a 3-digit number .In 3-digit number Comparison First check the hundred place both the numbers having 6 in hundred place . so then go to the tens place in 625 we have 2 in tens place and in 631 we have 3 in tens place so 3 is greater than 2. so 631 is greater than 625.

Question 10.
There are 125 kids in a taekwondo club. There are 135 kids in a soccer club. Which club has fewer kids?

Answer:
Number of kids in taekwondo club = 125 .
Number of kids in soccer club = 135.
Kids in taekwondo club are 10 fewer than soccer club .

Think and Grow: Modeling Real Life

Newton reads 200 pages on Monday, 70 on Tuesday, and 9 on Wednesday. Descartes reads 297 pages. Who reads more pages in all?


Answer:
Number of pages read by Newton on Monday = 200.
Number of pages read by Newton on Tuesday = 70.
Number of pages read by Newton on Wednesday =9.
Total pages read by Newton all 3 days = 200 + 70 + 9 = 279.
Total pages read by Descartes = 297.
Descartes reads more pages.
279 < 297 .

Show and Grow

Question 11.
Newton counts train cars. The train has 100 boxcars, 40 tank cars, and 4 locomotives. Descartes counts a train with 142 cars. Who counts more cars in all?


Answer:
Total cars counted by Newton = 100 + 40 = 140.
Total cars counted by Descartes = 142.
142 > 140 .
Descartes counts more cars .

Question 12.
You have 221 coins in your piggy bank. Your friend has 219 coins. Who has fewer coins?

Answer:
Number of coins in my piggy bank = 221.
Number of coins my friend have = 219.
My friend have 2 coins fewer than me .

Question 13.
DIG DEEPER!
652 people go to a play on Friday. 625 people go on Saturday. 655 people go on Sunday. On which day are there fewer than 650 people at the play?
Answer:
People played on Friday = 652.
People played on Saturday =625
People played on Sunday =655.
On Saturday 25 people are fewer than 650 people at the play .

Compare Numbers Using Symbols Homework & Practice 8.5

Compare

Question 1.

Answer:
923 > 854.

Question 2.

Answer:
386 < 389.

Question 3.

Answer:
406 = 406 .

Question 4.

Answer:
621 > 63.

Question 5.

Answer:
746 < 752 .

Question 6.

Answer:
235 > 130.

Question 7.

Answer:
500 + 60 + 1 = 561.
562 > 561 .

Question 8.

Answer:
100 + 10 = 110.
110 = 110.

Question 9.
DIG DEEPER!
What is Descartes’s number?

• It is less than 300.
• It is greater than 200.
• The ones digit is 6 less than 10.
• The tens digit is 2 more than the ones digit.

Answer:
lies between 200 – 300.
ones digit = 10 – 6 = 4.
Tens digit = 4 + 2 = 6.
264.

Question 10.
There are 428 pages in a science book. There are 424 pages in a math book. Which book has more pages?

Answer:
Number of pages in science book = 428
Number of pages in Math book = 424 .
science – 428 > 424 – math
Science book has more pages.

Question 11.
Modeling Real Life
A concession stand sells 300 bags of popcorn on Saturday, 50 on Sunday, and 4 on Monday. They sell 345 drinks. Did they sell more bags of popcorn or drinks?


Total Number of popcorn sold all 3 days = 300 + 50 + 4 = 354.
Number of drinks sold = 345 .
354 > 345 .
popcorn bags are sold more.

Question 12.
DIG DEEPER!
Newton climbs 136 stairs on Friday. He climbs 132 on Saturday. He climbs 128 on Sunday. On which day does he climb more than 134 stairs?

Answer:
Friday = 136.
Saturday = 132.
Sunday = 128 .
On Friday he climbed 136 stairs more than 134 stairs.

Review & Refresh

Find the difference. Use addition to check your answer.

Question 13.

Answer:

Question 14.

Answer:

Lesson 8.6 Compare Numbers Using a Number Line

Explore and Grow

Identify a number that is less than 538. Identify a number that is greater than 538. Model the numbers on the number line.

Explain how you know.
____________________
____________________
____________________
Answer:
The number on the right is always greater than the number on the left.

Show and Grow

Compare

Question 1.

Answer:
527 > 525.

Question 2.

Answer:
521 < 524 .

Question 3.

Answer:
528 = 528 .

Question 4.

Answer:
530 > 520

Question 5.

Answer:
522 < 523

Question 6.

Answer:
529 > 526

Write a number that makes the statement true.

Question 7.
372 < __
Answer:
372 < 373

Question 8.
195 > __
Answer:
195 > 190

Apply and Grow: Practice

Compare.

Question 9.

Answer:
714 = 714

Question 10.

Answer:
720 > 710

Question 11.

Answer:
718 > 717

Question 12.

Answer:
711 < 713

Write a number that makes the statement true.

Question 13.
736 = __
Answer:
736 = 736 .

Question 14.
461 > __
Answer:
461 > 460

Question 15.
__ < 295
Answer:
290 < 295

Question 16.
__ > 573
Answer:
574> 573

Question 17.
Logic
Is greater than or less than ? Explain.

Answer:
The is greater than .As the values goes towards 100. The is close to 100 than .so has higher value than . > .

Question 18.
DIG DEEPER!
What number might Newton be thinking?

Answer:
342<350<356

Think and Grow: Modeling Real Life

Order the race numbers from least to greatest.

Model:

Order from least to greatest: __, __, __, __
Your race number is greater than all of the other numbers but less than 900. What is a possible race number for you?

Answer:
Order from least to greatest : 856 , 865 , 868 , 876 .
Possible race number = 878.

Show and Grow

Question 19.
Order the race times from least to greatest.

DIG DEEPER!
Your time is less than all of the other times but greater than 320 seconds. What is a possible time for you?

Answer:
The Race times from least to greatest = 329,  335, 340, 342 .
320<329.
Possible time 325

Compare Numbers Using a Number Line Homework & Practice 8.6

Compare.

Question 1.

Answer:
450 < 460

Question 2.

Answer:
459 > 457

Question 3.

Answer:
455 > 451

Question 4.

Answer:
456 = 456

Question 5.

Answer:
455 > 454

Question 6.

Answer:
452 < 453

Write a number that makes the statement true.

Question 7.
529 > __
Answer:
529 > 528

Question 8.
815 < __
Answer:
815 < 820

Question 9.
__ < 142
Answer:
140 < 142

Question 10.
__ = 364
Answer:
364 = 364

Question 11.
YOU BE THE TEACHER
Is Descartes correct? Explain.
______________
______________
______________

Answer:
986 < 987 .
The numbers which move to the right of the number line will increase in value .
The number on the right is always greater than the number on the left.
so the number 987 is greater than 986 .

Question 12.
DIG DEEPER!
I am not greater than 243. I am not less than 243. What number am I? Explain how you know.
Answer:
243

Question 13.
Modeling Real Life
Order the numbers from least to greatest.

Answer:
The numbers from least to greatest = 675 , 679, 689, 698 .

DIG DEEPER!
Your car’s number is greater than all of the others but less than 705. What is a possible number for your car?

Review & Refresh

Question 14.
Circle the shapes with flat surfaces that are circles.

Answer:

Count and Compare Numbers to 1,000 Performance Task

The table shows the number of each type of fish in a tank.

Question 1.
Complete the table.

Answer:

Question 2.
Compare the numbers of fish.

Answer:

Question 3.
All of the purple, green, and pink fish are moved to a new exhibit. How many fish are left in the tank?

Total Number of fishes = 351 + 529 + 312 + 458 + 312 = 1962.
Number of fishes moved to new exhibit = 312 + 529 + 351 = 1192
Number of fishes left = 1962 – 1192 = 770.

Question 4.
A school of 24 fish swim in an array. Draw an array for the fish.
Answer:

Count and Compare Numbers to 1,000 Activity

Number Boss

To Play: Place Number Cards 0–9 in a pile. Each player flips 3 cards and makes a three-digit number. Compare the numbers. The player with the greater number takes both sets of cards. If the numbers are equal, flip cards again. The person with the greater number takes all of the cards. Repeat until all of the cards have been used.

Count and Compare Numbers to 1,000 Chapter Practice

8.1 Count to 120 in Different Ways

Question 1.
Count by ones.
113, 114, 115, __, __, __, __, ___
Answer:
113, 114, 115, 116, 117, 118, 119, 120

Question 2.
Count by fives.
25, 30, 35, __, __, __, __, ___
Answer:
25, 30, 35, 40, 45, 50, 55, 60

Question 3.
Count by tens.
33, 43, 53, __, __, __, __, __
Answer:
33, 43, 53, 63, 73, 83, 93, 103

8.2 Count to 1,000 in Different Ways

Question 4.
Count by fives.
210, 215, 220, ___, __, __, __, ___
Answer:
210, 215, 220, 225, 230, 235, 240, 245

Question 5.
Count by tens.
740, 750, 760, __, __, __, __, __, ___
Answer:
740, 750, 760, 770, 780, 790, 800, 810

Question 6.
Count by hundreds.
300, 400, 500, __, __, __, __, __
Answer:
300 , 400, 500, 600, 700, 800, 900, 1000

8.3 Place Value Patterns

Use place value to find the missing numbers.

Question 7.
854, 855, 856, __, __, __, __
Answer:
854, 855, 856, 857, 858, 859, 860

Question 8.
940, 950, 960, __, __, __, __
Answer:
940, 950, 960, 970, 980, 990, 1000

Question 9.
275, 375, 475, ___, __, __, __
Answer:
275, 375, 475, 575, 675, 775, 875

Question 10.
Repeated Reasoning
Use place value to describe each pattern.

Answer:

8.4 Find More or Less

Question 11.
10 more than 813 is ___
Answer:
10 more than 813 is 823.

Question 12.
10 less than 976 is __
Answer:
10 less than 976 is 966.

Question 13.
100 more than 254 is __.
Answer:
100 more than 254 is 354.

Question 14.
100 less than 531 is __.
Answer:
100 less than 531 is 431.

Question 15.
1 more than 444 is ___
Answer:
1 more than 444 is 445.

Question 16.
1 less than 622 is __.
Answer:
1 less than 622 is 621

Question 17.
Modeling Real Life
Your craft book has 110 ideas. Your friend’s craft book has 10 fewer ideas than yours. How many ideas does your friend’s craft book have?

Answer:
Number of Ideas in my craft book = 110 .
Number of ideas in my friends craft book =110 – 10 =100 .

Question 18.
Modeling Real Life
You have 324 beads. Newton has 100 more than you. Descartes has10 fewer than Newton. How many beads does Descartes have?

Answer:
Number of beads = 324.
Number of beads newton have = 324 + 100 = 424 .
Number of beads Descartes have = 424 – 10 = 414 .

8.5 Compare Numbers Using Symbols

Compare

Question 19.

Answer:
583 = 583

Question 20.

Answer:
626 < 725

Question 21.

Answer:
932 > 910

Question 22.

Answer:
49 < 411

Question 23.

Answer:
300 + 40 + 6 = 346
328 <346

Question 24.

Answer:
200 + 10 + 8 = 218
280 > 218

Question 25.
There are 318 kids in a gymnastics club. There are 219 kids in a swim club. Which club has fewer kids?

Answer:
Number of kids in Gymnastics club = 318.
Number of kids in Swim club =219.
Swim club has 99 fewer kids than Gymnastics club .

8.6 Compare Numbers Using a Number Line

Compare

Question 26.

Answer:
683 < 687

Question 27.

Answer:
689 > 688

Question 28.

Answer:
681 = 681

Question 29.

Answer:
690 > 680

Write a number that makes the statement true.

Question 30.
324 < __
Answer:
324 < 325

Question 31.
136 > __
Answer:
136 > 133

Question 32.
__ = 750
Answer:
750 = 750

Question 33.
__ < 871
Answer:
771 < 871

Question 34.
Number Sense
What number might Descartes be thinking?

Answer:
238<___>325.
238<315>325

Count and Compare Numbers to 1,000 Cumulative Practice 1 – 8

Question 1.
Which equation can you use to check your answer to 32 − 18?

Answer:
32 – 18 =14.
Equation
18 + 14 = 32

Question 2.
Which number does not belong?

Answer:
324, 334, 344, 354, 364, 374
345 does not belong

Question 3.
Find the missing digits.

Answer:

Question 4.
Use the number cards to decompose to subtract.

Answer:

Question 5.
Which choice does not show 124?

Answer:

Question 6.
Which quick sketch shows 43 − 25?

Answer:
43 – 25 = 18

Question 7.
You have 4 fewer gel pens than your friend. You have 6 gel pens. Which picture shows how many gel pens your friend has?

Answer:
Number of gel pens with me = 6.
Number of gel pens with my friend = 6 + 4 = 10.

Question 8.
Which number does not belong?

Answer:
563<_____<567 . It can be 564, 565, 566

Question 9.
Your friend uses compensation to add. Complete the equation to show what numbers he added after using compensation.

Answer:

Question 10.
Which choices show 238?

Answer:

Question 11.
Which picture shows 2 groups of 3?

Answer:

Question 12.
Descartes wants to use addition to subtract 51 − 25. Help him complete the number line and equations.

Answer:

Conclusion:

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