{"id":90173,"date":"2022-03-30T14:56:38","date_gmt":"2022-03-30T09:26:38","guid":{"rendered":"https:\/\/bigideasmathanswers.com\/?p=90173"},"modified":"2022-04-07T14:32:29","modified_gmt":"2022-04-07T09:02:29","slug":"envision-math-common-core-grade-7-answer-key-topic-4","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/envision-math-common-core-grade-7-answer-key-topic-4\/","title":{"rendered":"enVision Math Common Core Grade 7 Answer Key Topic 4 Generate Equivalent Expressions"},"content":{"rendered":"

Practice with the help of enVision Math Common Core Grade 7 Answer Key<\/a> Topic 4 Generate Equivalent Expressions<\/strong> regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 7th Grade Answers Key Topic 4 Generate Equivalent Expressions<\/h2>\n

Topic 4 Essential Question<\/strong>
\nHow can properties of operations help to generate equivalent expressions that can be used in solving problems?<\/p>\n

3-ACT MATH<\/strong>
\n\"Envision<\/p>\n

Topic 4 enVision STEM Project<\/h3>\n

Did You Know?<\/strong>
\nIn 2013, just over 30% of American consumers knew about activity trackers. By 2015, about 82% recognized them.
\n\"Envision<\/p>\n

Continued research and development leads to technological advances and breakthroughs, such as the use of biosensing apparel to track activity.
\n\"Envision<\/p>\n

Your Task: Analyze Activity Tracker Data
\nThe ways that data are communicated and presented to the user are just as important as the types of data collected. You and your classmates will continue your exploration of activity trackers and use data to develop models based on individual fitness goals.
\n\"Envision<\/p>\n

Topic 4 Get Ready<\/p>\n

Review What You Know<\/strong><\/p>\n

Vocabulary<\/strong><\/p>\n

Choose the best term from the box to complete each definition.
\nevaluate
\nexpression
\nfactor
\norder of operations
\nsubstitute
\nterm<\/p>\n

Question 1.
\nWhen you __________ an expression, you replace each variable with a given value.<\/p>\n

Answer:
\nWhen you evaluate an expression, you replace each variable with a given value.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nwhen we evaluate an expression, we replace each variable with a given value.
\nfor example:
\nEvaluate 3a-2b.
\nfor a = 6 and b = 4.
\n3(6) – 2(4).
\n18 – 8.
\n10.<\/p>\n

Question 2.
\nTo evaluate a + 3 when a = 7, you can _________ 7 for a in the expression.<\/p>\n

Answer:
\nTo evaluate a + 3 when a = 7, you can substitute 7 for a in the expression.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nif we evaluate a + 3 when a = 7.
\nwe will substitute 7 for a in the expression.
\na + 3.
\n7 + 3.
\n10.<\/p>\n

Question 3
\nThe set of rules used to determine the order in which operations are performed is called the _________<\/p>\n

Answer:
\nThe set of rules used to determine the order in which operations are performed is called the order of operations.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe set of rules used to determine the order in which operations are performed is called the order of operations.
\nfor example:
\n3 + [6(11 + 1 – 4)]\/8 x 2.
\n3+[6(8)]\/8 x 2.
\n3 + 48 \/ 8 x 2.
\n3 + 6 x 2.
\n3 + 12.
\n15.<\/p>\n

Question 4.
\nEach part of an expression that is separated by a plus or minus sign is a(n) __________.<\/p>\n

Answer:
\nEach part of an expression that is separated by a plus or minus sign is the term.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nEach part of an expression that is separated by a plus or minus sign is the term.
\nfor example:
\n2x + 4y – 9.
\nwhere x and y are variables.
\n9 is the constant.
\n2 and 4 are coefficients.
\nterms are 2x, 4y, and 9.<\/p>\n

Question 5.
\nA(n) __________ is a mathematical phrase that can contain numbers, variables, and operation symbols.<\/p>\n

Answer:
\nAn expression is a mathematical phrase that can contain numbers, variables, and operation symbols.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAn expression is a mathematical phrase that can contain numbers, variables, and operation symbols.
\nfor example:
\nn + 7 = 10.
\nx – 5 = 3.
\n3p = 15.
\ny\/2 = 5.<\/p>\n

Question 6.
\nWhen two numbers are multiplied to get a product, each number is called a(n) _________.<\/p>\n

Answer:
\nWhen two numbers are multiplied to get a product, each number is called a factor.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nWhen two numbers are multiplied to get a product, each number is called a factor.
\nfor example:
\n3 x 5 = 15.
\n3 and 5 are the factors.
\n15 is the product.<\/p>\n

Order of Operations<\/strong><\/p>\n

Evaluate each expression using the order of operations.
\nQuestion 7.
\n3(18 \u2013 7) + 2<\/p>\n

Answer:
\n3(18 – 7) + 2 = 35.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3(18 – 7) + 2.
\n3(11) + 2.
\n33 + 2.
\n35.
\n3(18 – 7) + 2 = 35.<\/p>\n

Question 8.
\n(13 + 2) \u00f7 (9 – 4)<\/p>\n

Answer:
\n(13 + 2) \u00f7 (9 – 4) = 3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(13 + 2) \u00f7 (9 – 4).
\n(13 + 2) \u00f7 (9 – 4).
\n15 \/ 5.
\n3.
\n(13 + 2) \u00f7 (9 – 4) = 3.<\/p>\n

Question 9.
\n24 \u00f7 4 \u2022 2 – 2<\/p>\n

Answer:
\n24 \u00f7 4 \u2022 2 – 2 = 10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n24 \u00f7 4 \u2022 2 – 2.
\n6 . 2 – 2.
\n12 – 2.
\n10.
\n24 \u00f7 4 \u2022 2 – 2 = 10.<\/p>\n

Equivalent Expressions<\/strong><\/p>\n

Evaluate each expression when a = -4 and b = 3.
\nQuestion 10.
\nab<\/p>\n

Answer:
\nab = -12.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\na = -4 and b = 3.
\n– 4 x 3.
\n-12.
\nab = -12.<\/p>\n

Question 11.
\n2a + 3b<\/p>\n

Answer:
\n2a + 3b = 1.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\na = -4 and b = 3.
\n2(-4) + 3(3).
\n-8 + 9.
\n1.
\n2a + 3b = 1.<\/p>\n

Question 12.
\n2(a – b)<\/p>\n

Answer:
\n2(a – b) = -14.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\na = -4 and b = 3.
\n2(-4 – 3).
\n2(-7).
\n-14.
\n2(a – b) = -14.<\/p>\n

Question 13.
\nExplain the difference between evaluating 3 \u2022 7 – 4 \u00f7 2 and evaluating 3(7 – 4) \u00f7 2.<\/p>\n

Answer:
\nThe two expressions are different.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3 . 7 – 4 \u00f7 2.
\n3 . 7 – 2.
\n21 – 2.
\n19.
\n3(7 – 4) \u00f7 2.
\n3(3) \/ 2.
\n9 \/ 2.<\/p>\n

Language Development<\/strong><\/p>\n

Complete each math statement using the word bank.
\n\"Envision<\/p>\n

To evaluate an algebraic expression, substitute a __________ for the variable in the expression.<\/p>\n

Answer:
\nTo evaluate an algebraic expression, substitute properties of operations for the variable in the expression.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTo evaluate an algebraic expression, substitute properties of operations for the variable in the expression.
\nfor example:
\nn + 7 = 10.
\nx – 5 = 3.
\n3p = 15.
\ny\/2 = 5.<\/p>\n

In the algebraic expression 3(x – 2), 3 and x – 2 are ___________<\/p>\n

Answer:
\nIn the algebraic expression 3(x – 2), 3, and x – 2 are coefficients.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nIn the algebraic expression 3(x – 2), 3 and x – 2 are coefficients.
\nfor example:
\n3(x – 2).
\n3 and x-2 are coefficients.<\/p>\n

To generate equivalent expressions, you can use the __________<\/p>\n

Answer:
\nTo generate equivalent expressions, you can use the order of operations.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTo generate equivalent expressions, you can use the order of operations.
\nfor example:
\n5(x – 1) + 7.
\n5(x) + 5(-7) + 7.
\n5x – 5 + 7.
\n5x + 2.<\/p>\n

In the expression 4x + 2x – 6y, you first need to __________<\/p>\n

Answer:
\nIn the expression 4x + 2x – 6y, you first need to add.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nIn the expression 4x + 2x – 6y, you first need to add.
\n4x + 2y – 6y.
\n4x – 4y.<\/p>\n

You can use the Distributive Property to __________ the algebraic expression 5(x – 7).<\/p>\n

Answer:
\nYou can use the Distributive property to find the algebraic expression.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nwe can use the distributive property to find the algebraic expression.
\nfor example:
\n5(x – 7).
\n5x – 35.<\/p>\n

In the algebraic expression, 6x + 10, x is the ________ , 6 is the ________ and 10 is the ___________<\/p>\n

Answer:
\nIn the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nIn the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.
\nfor example:
\n6x + 10.
\nwhere 6 is the variable.
\nx is coefficient.
\n10 is constant.<\/p>\n

Four words that describe operations that can be used with expressions are _________, and ________, _________ and __________.<\/p>\n

Answer:
\nThe words that describe the operations are constants, terms, variables, and coefficients.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe words that describe the operations are constants, terms, variables, and coefficients.
\nfor example:
\n2x + 4y – 9.
\nwhere x and y are variables.
\n9 is the constant.
\n2 and 4 are coefficients.
\nterms are 2x, 4y, and 9.<\/p>\n

In the algebraic expression 5x + 4 + 6x – 3, you use the Commutative Property to _________ like terms next to each other and the Associative Property to _________ like terms together.<\/p>\n

Answer:<\/p>\n

Pick A Project<\/strong><\/p>\n

PROJECT 4A
\nWhich emojis would you use to tell the story of your day so far?
\nPROJECT: WRITE AND ILLUSTRATE A CHILDREN’S BOOK
\n\"Envision<\/p>\n

PROJECT 4B
\nHow many different ways can you represent a dollar?
\nPROJECT: GENERATE EQUIVALENCE
\n\"Envision<\/p>\n

PROJECT 4C
\nIf you wrote a song, what would it sound like?
\nPROJECT: COMPOSE A SONG
\n\"Envision<\/p>\n

PROJECT 4D
\nWhat was your favorite structure at a playground when you were younger?
\nPROJECT: BUILD A MODEL PLAYGROUND
\n\"Envision<\/p>\n

Lesson 4.1 Write and Evaluate Algebraic Expressions<\/h3>\n

Solve & Discuss It!<\/strong>
\nMr. Ramirez’s class was playing a game in which students need to match sticky notes that have equivalent expressions. How can you sort the expressions into groups?
\nI can… write and evaluate algebraic expressions.
\n\"Envision<\/p>\n

Focus on math practices
\nReasoning is there more than one way to group the expressions? Give an example.<\/p>\n

Answer:
\nYes, there are more than one way to group the expressions.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA numerical expression in mathematics can be a combination of numbers, integers combined using.
\nfor example:
\n16 is an numerical expression.<\/p>\n

Essential Question<\/strong>
\nHow can algebraic expressions be used to represent and solve problems?<\/p>\n

Answer:
\nWe can use algebra to solve mathematical problems.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nwe can also interpret the solution in the context of the original problem.
\nfor example:
\n2x + 5 = 43.
\nwhere 43 is the constant.
\nalways has an equal symbol.
\n2x + 5 = algebraic expression.<\/p>\n

Try It!<\/strong><\/p>\n

Misumi started with $217 in her bank account. She deposits $25.50 each week and never withdraws any money. What expression can Misumi use to determine her account balance after w weeks?
\n\"Envision<\/p>\n

Answer:
\nThe expression can Misumi use to determine her account balance after w weeks = 8.5 weeks.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nMisumi started with $217 in her bank account.
\nShe deposits $25.50 each week and never withdraws any money.
\n$217 \/ 25.50 = 8.5.
\nso the expression can Misumi use to determine her account balance after w weeks = 8.5.<\/p>\n

Convince Me! How did you determine which value to use for the constant and which value to use for the coefficient?<\/p>\n

Answer:
\nx is coefficient and 3 is constant.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n2x + 3 is the expression.
\nx is the coefficient.
\n3 is the constant.<\/p>\n

Try It!<\/strong><\/p>\n

The cost to rent a scooter is $15.50 per hour and the cost to rent a watercraft is $22.80 per hour. Use the expression 15.5s + 22.8w to determine how much it would cost to rent a scooter for 3\\(\\frac{1}{2}\\) hours and a watercraft for 1\\(\\frac{3}{4}\\) hours.<\/p>\n

Answer:
\nThe cost would cost to rent a scooter for 3(1\/2) hours and watercraft for 1(3\/4) hours = $54.25 and $40.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe cost to rent a scooter is $15.50 per hour and the cost to rent a watercraft is $22.80 per hour.
\nUse the expression 15.5s + 22.8w to determine how much.
\n15.5s + 22.8w.
\n3. 1\/2 = 7\/2.
\n1. 3\/4 = 7\/4.
\n15.5(7\/2) + 22.8(7\/4).
\n108.5\/2 + 159.6\/4.
\n54.25 + 39.9.<\/p>\n

Try It!<\/strong><\/p>\n

Emelia earns $8.74 per hour plus a gas allowance of $3.50 per day at her job. How much does Emelia’s job pay in a day when she works 5\\(\\frac{1}{2}\\) hours? Write an expression and evaluate for 5\\(\\frac{1}{2}\\) hours.<\/p>\n

Answer:
\nEmelia’s job pay in a day when she works 5(1\/2) hours = $67.32.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nEmelia earns $8.74 per hour plus a gas allowance of $3.50 per day at her job.
\n$8.74 + $3.50.
\n$12.24.
\n5(1\/2) = 5.5.
\n5.5 x $12.24 = $67.32.
\nso Emelia’s job pay in a day when she works 5(1\/2) hours = $67.32.<\/p>\n

KEY CONCEPT<\/strong>
\nAlgebraic expressions can be used to represent problems with unknown or variable values. Values can be substituted for variables to evaluate the expression.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question How are algebraic expressions used to represent and solve problems?<\/p>\n

Answer:
\nAlgebraic expressions are used to represent problems with unknowns or variable values.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAlgebraic expressions are used to represent problems with unknowns or variable values.
\nValues can be substituted for variables to evaluate the expression.
\nfor example:
\n2x + 3y = a.
\nwhere x = 2 and y = 3.
\n2 x 2 + 3 x 3 = a.
\n4 + 9 = a.
\n13 = a.<\/p>\n

Question 2.
\nUse Structure How is a constant term different than a variable term for an expression that represents a real-world situation?<\/p>\n

Answer:
\na = 13.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n2x + 3y = a.
\nwhere x = 2 and y = 3.
\n2 x 2 + 3 x 3 = a.
\n4 + 9 = a.
\n13 = a.<\/p>\n

Question 3.
\nLook for Relationships Explain why you can have different values when evaluating an algebraic expression.<\/p>\n

Answer:
\nTo evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTo evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.
\nfor example:
\nx + 6.
\nwhere x = 6.
\n6 + 6 = 12.
\nif we know the value of our variables, we can replace the variables with their values and then evaluate the expression.<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nA tank containing 35 gallons of water is leaking at a rate of \\(\\frac{1}{4}\\) gallon per minute. Write an expression to determine the number of gallons left in the tank after m minutes.<\/p>\n

Answer:
\nThe number of gallons left in the tank after m minutes = 8.75 gallons.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA tank containing 35 gallons of water is leaking at a rate of \\(\\frac{1}{4}\\) gallon per minute.
\n35 x 1\/4 = 35\/4.
\n8.75.
\nso the number of gallons left in the tank after m minutes = 8.75 gallons.<\/p>\n

Question 5.
\nWrite an algebraic expression that Marshall can use to determine the total cost of buying a watermelon that weighs w pounds and some tomatoes that weigh t pounds. How much will it cost to buy a watermelon that weighs 18\\(\\frac{1}{2}\\) pounds and 5 pounds of tomatoes?
\n\"Envision<\/p>\n

Answer:
\nThe much will it cost to buy a watermelon that weighs 18(1\/2) pounds and 5 pounds of tomatoes = $29.25 and $3.4.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe cost of tomatoes is $3.25 per lb.
\nthe cost of watermelons is $0.68 per lb.
\n18\/2 = 9.
\n$3.25 x 9 = $29.25.
\n0.68 x 5 = $3.4.
\nso the much will it cost to buy a watermelon that weighs 18(1\/2) pounds and 5 pounds of tomatoes = $29.25 and $3.4.<\/p>\n

Question 6.
\nWhat is the value of \\(\\frac{3}{8}\\)x – 4.5 when x = 0.4?<\/p>\n

Answer:
\n(3\/8)x – 4.5 = 4.35.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(3\/8)x – 4.5.
\nwhere x = 0.4.
\n(3\/8)0.4 – 4.5.
\n0.375 x 0.4 – 4.5.
\n0.15 – 4.5.
\n4.35.
\n(3\/8)x – 4.5 = 4.35.<\/p>\n

Question 7.
\nWhat is the value of 8.4n – 3.2p when n = 2 and p = 4?<\/p>\n

Answer:
\n8.4n – 3.2p = 4.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n8.4n – 3.2p.
\n8.4 (2) – 3.2 (4).
\n16.8 – 12.8.
\n4.
\n8.4n – 3.2p = 4.<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice For 8-10, fill in the boxes to complete the problems.
\nQuestion 8.
\nEvaluate 10.2x + 9.4y when x = 2 and y = 3.
\n10.2 (_______) + 9.4 (_______)
\n= _______ + 28.2
\n= _______<\/p>\n

Answer:
\n10.2 x + 9.4 y = 48.6.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n10.2 x + 9.4 y.
\nwhere x = 2 and y = 3.
\n10.2 x 2 + 9.4 x 3.
\n20.4 + 28.2.
\n48.6.
\n10.2 x + 9.4 y = 48.6.<\/p>\n

Question 9.
\nEvaluate \\(\\frac{1}{2}\\)t + \\(\\frac{3}{8}\\) when t = \\(\\frac{1}{4}\\)
\n\\(\\frac{1}{2}\\)(________) + \\(\\frac{3}{8}\\)
\n= ______ + \\(\\frac{3}{8}\\)
\n= ______<\/p>\n

Answer:
\n1\/2 x 1\/4 + 3\/8 = 0.5.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n1\/2 = 0.5.
\n1\/4 = 0.25.
\n0.5 x 0.25 + 3\/8.
\n0.125 + 0.375.
\n0.5.
\n1\/2 x 1\/4 + 3\/8 = 0.5.<\/p>\n

Question 10.
\nWrite an expression that represents the height of a tree that began at 6 feet and increases by 2 feet per year. Let y represent the number of years.
\n_____ + ______ y<\/p>\n

Answer:
\n6x + 2y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe height of a tree that began at 6 feet and increases by 2 feet per year.
\n6x + 2y.
\nwhere y represents the number of years.
\nso the expression is 6x + 2y.<\/p>\n

For 11-14, evaluate each expression for the given value of the variable(s).
\nQuestion 11.
\n3d – 4
\nd = 1.2<\/p>\n

Answer:
\n3d – 4 = 0.4.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3d – 4.
\nwhere d = 1.2.
\n3(1.2) – 4.
\n3.6 – 4.
\n0.4.
\n3d – 4 = 0.4.<\/p>\n

Question 12.
\n0.5f – 2.39
\nf = 12, 9 = 2<\/p>\n

Answer:
\n0.5f – 2.39 = 3.68.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n0.5f – 2.39.
\nwhere f = 12 and 9 = 2.
\n0.5 x 12 – 2.32.
\n6 – 2.32.
\n3.68.
\n0.5f – 2.39 = 3.68.<\/p>\n

Question 13.
\np + 3
\np = \\(\\frac{3}{5}\\)<\/p>\n

Answer:
\np + 3 = 3.6.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\np + 3.
\nwhere p = 3\/5.
\n3\/5 = 0.6.
\n0.6 + 3.
\n3.6.
\np + 3 = 3.6.<\/p>\n

Question 14.
\n34 + \\(\\frac{4}{9}\\)w
\nw = –\\(\\frac{1}{2}\\)<\/p>\n

Answer:
\n34 + 4\/9x w = 33.9.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n34 + 4\/9x w.
\nwhere w = -1\/2.
\n34 + 4\/9(-1\/2).
\n34 + 0.4(-0.5).
\n34 – 0.1.
\n33.9.<\/p>\n

Question 15.
\nModel with Math What expression can be used to determine the total cost of buying g pounds of granola for $3.25 per pound and f pounds of flour for $0.74 per pound?<\/p>\n

Answer:
\n$3.25g and $0.74f.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe total cost of buying g pounds of granola for $3.25 per pound.
\nf pounds of flour for $0.74 per pound.
\n$3.25g + $0.74f<\/p>\n

Question 16.
\nModel with Math Which expression can be used to determine the total weight of a box that by itself weighs 0.2 kilogram and contains p plaques that weigh 1.3 kilograms each?
\n\"Envision
\nA. 1.3p +0.2
\nB. 0.2p + 1.3
\nC. 0.2 – 1.3p
\nD. 1.2p<\/p>\n

Answer:
\nOption A is correct.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\ntotal weight of a box that by itself weighs 0.2 kilogram.
\nand contains p plaques that weigh 1.3 kilograms each.
\n0.2 + 1.3p.
\nso option A is correct.<\/p>\n

Question 17.
\nThe expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level and ascended at a rate of 13 feet per minute. What was the depth of the submarine after 6 minutes?<\/p>\n

Answer:
\nThe depth of the submarine after 6 minutes = – 42 feet.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level.
\nascended at a rate of 13 feet per minute.
\n-120 + 13(6).
\n-120 + 78.
\n-42.
\nso the depth of the submarine after 6 minutes = -42 feet.<\/p>\n

Question 18.
\nBe Precise<\/strong> A full grain silo empties at a constant rate. Write an expression to determine the amount of grain left after s seconds.
\n\"Envision<\/p>\n

Answer:
\nThe amount of grain left after 5 seconds = 2982.5 cubic feet.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA full grain silo empties at a constant rate.
\nthe capacity of food grain is 3000 cubic feet.
\n3000 – 3.5\/s.
\n3000 – 3.5(5).
\n3000 – 17.5.
\n2982.5.
\nso the amount of grain left after 5 seconds = 2982.5 cubic feet.<\/p>\n

Question 19.
\nHigher Order Thinking<\/strong> For the expression 5 – 5x to have a negative value, what must be true about the value of x?<\/p>\n

Answer:
\nThe value of x = 4.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 5 – 5x.
\nwhere x = 4.
\n5 – 5(4).
\n5 – 20.
\n-15.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 20.
\nJoe bought g gallons of gasoline for $2.85 per gallon and c cans of oil for $3.15 per can.
\nPART A
\nWhat expression can be used to determine the total amount Joe spent on gasoline and oil?<\/p>\n

Answer:
\nThe total amount joe spend on gasoline and oil = $2.85g + $3.15c.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nJoe bought g gallons of gasoline for $2.85 per gallon.
\nc cans of oil for $3.15 per can.
\n$2.85g + $3.15c.
\nso the total amount joe spends on gasoline and oil = $2.85g + $3.15c.<\/p>\n

PART B
\nJoe spent $15. He bought 2 cans of oil. About how many gallons of gasoline did he buy?
\nA. 2.5
\nB. 3
\nC. 3.5
\nD. 4<\/p>\n

Answer:
\nThe gallons of gasoline did he buy = 3 gallons.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nJoe spent $15. He bought 2 cans of oil.
\n1.5 x 2.
\n3.
\nthe gallons of gasoline did he buy = 3.
\nso option B is correct.<\/p>\n

Question 21.
\nThe outside temperature was 73\u00b0F at 1 P.M. and decreases at a rate of 1.5\u00b0F each hour. What expression can be used to determine the temperature h hours after 1 P.M.?<\/p>\n

Answer:
\nThe expression can be used to determine the temperature h hours after 1 P.M = 71.5\u00b0F.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe outside temperature was 73\u00b0F at 1 P.M. and decreases at a rate of 1.5\u00b0F each hour.
\n73 – 1.5.
\n71.5\u00b0F.
\nso the expression can be used to determine the temperature h hours after 1 P.M = 71.5\u00b0F.<\/p>\n

Lesson 4.2 Generate Equivalent Expressions<\/h2>\n

Explore It!<\/strong>
\nA shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
\nI can… write equivalent expressions for given expressions.
\n\"Envision<\/p>\n

A. How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram.<\/p>\n

Answer:
\n1 dozen + 1\/2 dozen eggs.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
\n1 dozen = 12.
\n1\/2 doxen = 12\/2.
\n12\/2 = 6.
\n12 + 1\/2 eggs.<\/p>\n

B. How can you represent the total number of eggs in the shipment using expressions? What variables do you use? What do they represent?<\/p>\n

Answer:
\n1 dozen + 1\/2 dozen eggs.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
\n1 dozen = 12.
\n1\/2 doxen = 12\/2.
\n12\/2 = 6.
\n12 + 1\/2 eggs.<\/p>\n

Focus on math practices
\nConstruct Arguments How do the two representations compare? How are they different?<\/p>\n

Essential Question<\/strong>
\nWhat are equivalent expressions?<\/p>\n

Try It!<\/strong><\/p>\n

Nancy wrote the expression 3x – 12 to represent the relationship in a table of values. Use properties of operations to write two equivalent expressions.
\n3(x – _____)
\n_____ + 3x<\/p>\n

Answer:
\nThe two equivalent expressions are 36 + 3x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3x – 12.
\n3(x – 12).
\n3x – 36.
\n36 + 3x.<\/p>\n

Convince Me! What property can you use to write an equivalent expression for -5(x – 2)? Explain.<\/p>\n

Answer:
\n-5(x – 2) = -5x – 10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-5(x – 2).
\n-5x – 10.
\nwe can use the distributive property.
\n-5x -10.<\/p>\n

Try It!<\/strong><\/p>\n

Use properties of operations to write two expressions that are equivalent to \\(\\frac{3}{4}\\)n + (8 + \\(\\frac{1}{3}\\)z).<\/p>\n

Answer:
\n3\/4n + (8 + {1\/3}) = 0.75n + 8.3z.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3\/4n + (8 + {1\/3})z.
\n3\/4n + 8 + 0.3z.
\n0.75n + 8.3z.
\n3\/4n + (8 + {1\/3}) = 0.75n + 8.3z.<\/p>\n

Try It!<\/strong><\/p>\n

Write two expressions that are equivalent to –\\(\\frac{5}{4}\\)x – \\(\\frac{3}{4}\\)<\/p>\n

Answer:
\n-5\/4 – 3\/4 = 2.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-5\/4 – 3\/4.
\n-5\/4 = 1.25.
\n3\/4 = 0.75.
\n-1.25 – 0.75.
\n2.<\/p>\n

KEY CONCEPT<\/strong>
\nYou can use properties of operations to write equivalent expressions.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nWhat are equivalent
\nQuestion 1.
\nEssential Question expressions?<\/p>\n

Answer:
\n-1\/2(x + 8), -1\/2x + (-4) and -4 +(-1\/2x) are equivalent.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-1\/2(x + 8).
\n-1\/2x + (-1\/2) . 8.
\n-1\/2x + (-4).
\n-4 + (-1\/2x).
\nthe three expressions are true.<\/p>\n

Question 2.
\nMake Sense and Persevere For which operations is the Commutative Property true?<\/p>\n

Answer:
\n(-4) + -1\/2x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(-4) + -1\/2x.
\nwe can use the commutative property for the expression.
\n-1\/2x + (-4).<\/p>\n

Question 3.
\nHow can the Associative Property be applied when writing equivalent expressions with variables?<\/p>\n

Answer:<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nWrite an expression equivalent to -3 + \\(\\frac{2}{3}\\)y – 4 – \\(\\frac{1}{3}\\)y.<\/p>\n

Answer:
\n-3 + (2\/3)y – 4 – (1\/3)y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-3 + \\(\\frac{2}{3}\\)y – 4 – \\(\\frac{1}{3}\\)y.
\n-3 + (2\/3)y – 4 – (1\/3)y.
\n-3 – 4 + (2\/3)y – (1\/3)y.
\n-7 + 1\/3y.
\n-3 + (2\/3)y – 4 – (1\/3)y = -7 + 1\/3y.<\/p>\n

Question 5.
\nComplete the tables to determine if the expressions are equivalent. If the expressions are equivalent, name the property or properties that make them equivalent.
\n\"Envision<\/p>\n

Answer:
\n3(x – 5) = 3x – 15.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3(x – 5).
\n3x – 15.
\nx = 1.
\n3 – 15 = -12.
\nx = 2.
\n6 – 15 = -9.
\nx = 3.
\n9 – 15 = -6.<\/p>\n

Question 6.
\nUse the properties of operations to write an expression equivalent to 4x + \\(\\frac{1}{2}\\) + 2x – 3.<\/p>\n

Answer:
\n4x +[{1\/2}] + 2x – 3 = 3x -3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4x + (1\/2) + 2x – 3.
\n4x + 2x + (1\/2) – 3.
\n6x + (1\/2) -3.
\n3x – 3.
\n4x +[{1\/2}] + 2x – 3 = 3x -3.<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

For 7-9, write an equivalent expression.
\nQuestion 7.
\n-3(7 + 5g)<\/p>\n

Answer:
\n-36g.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-3(7 + 5g).
\n-3 x 7 = -21.
\n-3 x 5 = 15.
\n-21 + (-15g).
\n-36g.<\/p>\n

Question 8.
\n(x + 7) + 3y<\/p>\n

Answer:
\n24xy.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(x + 7) + 3y.
\n3y x X + 3y x 7.
\n3xy + 21y.
\n24xy.<\/p>\n

Question 9.
\n\\(\\frac{2}{9}\\) – \\(\\frac{1}{5}\\) \u2022 x<\/p>\n

Answer:
\n2\/9 – 1\/5 . X =<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n2\/9 x X – (1\/5)x.
\n2\/9 x – 1\/5 x.<\/p>\n

Question 10.
\nWhich expression is equivalent to t + 4 + 3 – 2t?
\nA. t + 7
\nB. -t + 7
\nC. 6t
\nD. 10t<\/p>\n

Answer:
\nt + 4 + 3 – 2t = -t + 7.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nt + 4 + 3 – 2t.
\nt + 7 – 2t.
\nt – 2t + 7.
\n-t + 7.
\nt + 4 + 3 – 2t = -t + 7.<\/p>\n

Question 11.
\nThe distance in feet that Karina swims in a race is represented by 4d – 4, where d is the distance for each lap. What is an expression equivalent to 4d – 4?
\n\"Envision<\/p>\n

Answer:
\nThe expression equal to 4d – 4 = 4(d – 4).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4d – 4.
\n4(d – 4).
\n4d – 16.
\n4d – 4 = 4d – 16.<\/p>\n

Question 12.
\nUse the Associative Property to write an expression equivalent to (w + 9) + 3.<\/p>\n

Answer:
\nThe expression is (3 + w) + 9.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is (w + 9) + 3.
\n(3 + w) + 3.
\n6 + w.<\/p>\n

Question 13.
\nNigel is planning his training schedule for a marathon over a 4-day period. He is uncertain how many miles he will run on two d
\n\"Envision<\/p>\n

Answer:
\nThe number of miles he will run on two days = 14.5 miles.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nNigel is planning his training schedule for a marathon over a 4-day period.
\non day 1 he will run 12 miles.
\non day 2 he will run 14.5 miles.
\non day 3 he will run 17 miles.
\n12 + 17 = 29.
\n29\/2 = 14.5.
\non day 2 he will run 14.5 miles.<\/p>\n

Question 14.
\nMaria said the expression -4n+ 3 + 9n – 4 is equivalent to 4n. What error did Maria likely make?<\/p>\n

Answer:
\n-4n + 3 + 9n – 4 = 5n – 1.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-4n + 3 + 9n – 4.
\n-4n + 9n = 5n.
\n5n + 3 – 4.
\n5n – 1.
\n-4n + 3 + 9n – 4 = 5n – 1.<\/p>\n

Question 15.
\nWrite an expression equivalent to x – 3y + 4.<\/p>\n

Answer:
\nx – 3y + 4 = 4 + x – 3y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nx – 3y + 4.
\n4 + x – 3y.
\nx + 4 – 3y.<\/p>\n

Question 16.
\nAndre wrote the expression -2 + 4x = 3 to represent the relationship shown in the table.
\n\"Envision
\nWrite two other expressions that also represent the relationship shown in the table.<\/p>\n

Answer:
\n-2 + 4x = 3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-2 + 4x.
\nx = 0.
\n-2 + 0.
\n-2.
\nx = 6.
\n-2 + 4(6).
\n-2 + 24.
\n22.
\nx = 12.
\n-2 + 4(12).
\n-2 + 48.
\n46.<\/p>\n

Question 17.
\nHigher Order Thinking<\/strong> to rent a car for a trip, four friends are combining their money. The group chat shows the amount of money that each puts in. One expression for their total amount of money is 189 plus p plus 224 plus q.
\n\"Envision
\na. Use the Commutative Property to write two equivalent expressions.<\/p>\n

Answer:
\nThe expressions are 189 + p + 224q.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n189 + p + 224q.
\np + 189 + 224q.
\n224q + p + 189.<\/p>\n

b. If they need $500 to rent a car, find at least two different pairs of numbers that p and q could be.<\/p>\n

Answer:
\n$500 + p + 224q.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n500 – 224 = 276.
\n276 + p + 224q.
\np + 276 + 224q.
\n224q + 276 + p.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 18.
\nSelect all expressions equivalent to \\(\\frac{3}{5}\\)x + 3.
\n\"Envision<\/p>\n

Answer:
\nThe expressions equivalent to (3\/5)x + 3 = 1 + 2\/5x + 3 and 4\/5x – 1\/5x + 3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(3\/5)x + 3.
\n1 + 2\/5x + 3.
\n4\/5x – 1\/5x + 3.
\n3\/5x + 3.
\nso the expressions equivalent to (3\/5)x + 3 = 1 + 2\/5x + 3 and 4\/5x – 1\/5x + 3.<\/p>\n

Lesson 4.3 Simplify Expressions<\/h3>\n

Solve & Discuss It!<\/strong>
\nHow can the tiles below be sorted?
\nI can… use properties of operations to simplify expressions.
\n\"Envision<\/p>\n

Focus on math practices
\nReasoning Would sorting the tiles with positive coefficients together and tiles with negative coefficients together help to simplify an expression that involves all the tiles? Explain.<\/p>\n

Answer:
\nThe positive coefficients are 4.25, 3\/5y, 3\/8, 1\/5, 2.1x, and 1\/2x.
\nThe negative coefficients are -0.5, -0.5x, -2.1y, and -2.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe coefficients are 4.25, 3\/5y, 3\/8, 1\/5, 2.1x, -0.5, -0.5x, -2.1y, -2, and 1\/2x.
\nthe positive coefficients are 4.25, 3\/5y, 3\/8, 1\/5, 2.1x, and 1\/2x.
\nthe negative coefficients are -0.5, -0.5x, -2.1y, and -2.<\/p>\n

Essential Question<\/strong>
\nHow are properties of operations used to simplify expressions?<\/p>\n

Try It!<\/strong><\/p>\n

Simplify the expression – 6 – 6f + 7 \u2013 3f – 9.
\n______ – 3f – _____ + 7 – ______
\n_____ – ______<\/p>\n

Answer:
\n-6 – 6f + 7 – 3f – 9 = -9f – 23.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-6 – 6f + 7 – 3f – 9.
\n-6f\u00a0 – 3f – 6 + 7 – 9.
\n-9f – 13 – 9.
\n-9f – 23.
\n-6 – 6f + 7 – 3f – 9 = -9f – 23.<\/p>\n

Convince Me! How do you decide in what way to reorder the terms of an expression when simplifying it?<\/p>\n

Try It!<\/strong><\/p>\n

Simplify each expression.
\na. 59.95m – 30 + 7.95m + 45 + 9.49m<\/p>\n

Answer:
\n59.95m – 30 + 7.95m + 45 + 9.49m = 52m – 5.51.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n59.95m – 30 + 7.95m + 45 + 9.49.
\n59.95m + 7.95m – 30 + 45 + 9.49.
\n52m – 15 + 9.49.
\n52m – 5.51.<\/p>\n

b. -0.5p + \\(\\frac{1}{2}\\)p – 2.75 + \\(\\frac{2}{3}\\)p<\/p>\n

Answer:
\n-0.5p + (1\/2)p – 2.75 + (2\/3)p = 0.6p – 2.75.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-0.5p + (1\/2)p – 2.75 + (2\/3)p.
\n-0.5p + 0.5p – 2.75 + 0.6p.
\n0.6p – 2.75.
\n-0.5p + (1\/2)p – 2.75 + (2\/3)p = 0.6p – 2.75.<\/p>\n

Try It!<\/strong><\/p>\n

Simplify the expression -3.7 +59 + 4k + 11.1 – 10g.
\n(______ – 10g) + 4k + (______ + 11.1)
\n= _______ + 4k + ______
\nThe simplified expression is ________.<\/p>\n

Answer:
\n-3.7 + 59 + 4k + 11.1 – 10g = 55.3 – 10g.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-3.7 + 59 + 4k + 11.1 – 10g.
\n-3.7 + 11.1 – 11.1 + 59 – 10g.
\n-3.7 + 59 – 10g.
\n55.3 – 10g.<\/p>\n

KEY CONCEPT<\/strong>
\nWhen simplifying algebraic expressions, use properties of operations to combine like terms.
\nTo simplify the expression below, group like terms.
\n\\(\\frac{3}{10}\\)y – 3.5x \u2013 \\(\\frac{3}{8}\\) +0.53x + 5.25 \u2013 2.75y – 12
\n(-3.5x + 0.53x) + (\\(\\frac{3}{10}\\)y – 2.75y) + (-\\(\\frac{3}{8}\\) + 5.25 \u2013 12)
\nThen combine like terms.
\n-2.97x \u2013 2.45y \u2013 7.125<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question How are properties of operations used to simplify expressions?<\/p>\n

Answer:
\nThe properties of operations are used to combine like terms.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe properties of operations used to combine like terms.
\nfor example:
\n\\(\\frac{3}{10}\\)y – 3.5x \u2013 \\(\\frac{3}{8}\\) +0.53x + 5.25 \u2013 2.75y – 12
\n(-3.5x + 0.53x) + (\\(\\frac{3}{10}\\)y – 2.75y) + (-\\(\\frac{3}{8}\\) + 5.25 \u2013 12)
\nThen combine like terms.
\n-2.97x \u2013 2.45y \u2013 7.125<\/p>\n

Question 2.
\nMake Sense and Persevere Explain why constant terms expressed as different rational number types can be combined.<\/p>\n

Answer:
\nThe constant terms remain the same.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nconstant terms expressed as different rational number types can be combined.
\nfor example:
\n\\(\\frac{3}{10}\\)y – 3.5x \u2013 \\(\\frac{3}{8}\\) +0.53x + 5.25 \u2013 2.75y – 12
\n(-3.5x + 0.53x) + (\\(\\frac{3}{10}\\)y – 2.75y) + (-\\(\\frac{3}{8}\\) + 5.25 \u2013 12)
\nThen combine like terms.
\n-2.97x \u2013 2.45y \u2013 7.125<\/p>\n

Question 3.
\nReasoning<\/strong> How do you know when an expression is in its simplest form?<\/p>\n

Answer:
\nThe expression is in its simplest form when it has only limited expressions.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nfor example:
\n-3.7 + 59 + 4k + 11.1 – 10g.
\n-3.7 + 11.1 – 11.1 + 59 – 10g.
\n-3.7 + 59 – 10g.
\n55.3 – 10g.<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nSimplify -4b + (-9k) \u2013 6 \u2013 3b + 12.<\/p>\n

Answer:
\n-7b – 9k + 6.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-4b + (-9k) \u2013 6 \u2013 3b + 12.
\n-4b – 3b -9k – 6 + 12.
\n-7b – 9k + 6.<\/p>\n

Question 5.
\nSimplify -2 + 6.45z – 6+ (-3.25z).<\/p>\n

Answer:
\n-8 + 3.2z.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-2 + 6.45z – 6+ (-3.25z).
\n-2 + 6.45z – 6 – 3.25z.
\n-8 + 3.2z.<\/p>\n

Question 6.
\nSimplify \u20139 + (-\\(\\frac{1}{3}\\)y) +6 – \\(\\frac{4}{3}\\)y.<\/p>\n

Answer:
\n-3 – 5\/3y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\u20139 + (-\\(\\frac{1}{3}\\)y) +6 – \\(\\frac{4}{3}\\)y.
\n-9 – 1\/3y + 6 – 4\/3y.
\n-9 + 6 – 5\/3y.
\n-3 – 5\/3y.<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

In 7-10, simplify each expression.
\nQuestion 7.
\n\u20132.8f +0.96 \u2013 12 – 4<\/p>\n

Answer:
\n-2.8f + 0.96 – 12 – 4 = -2.8f – 15.04.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-2.8f + 0.96 – 12 – 4.
\n-2.8f + 0.96 – 16.
\n-2.8f – 15.04.
\n-2.8f + 0.96 – 12 – 4 = -2.8f – 15.04.<\/p>\n

Question 8.
\n3.2 – 5.1n – 3n + 5<\/p>\n

Answer:
\n3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3.2 – 5.1n – 3n + 5.
\n3.2 – 8.1n + 5.
\n8.2 – 8.1n.
\n3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n.<\/p>\n

Question 9.
\n2n + 5.5 – 0.9n – 8 + 4.5p<\/p>\n

Answer:
\n2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n2n + 5.5 – 0.9n – 8 + 4.5p.
\n2n – 0.9n + 5.5 – 8 + 4.5p.
\n1.1n – 2.5 + 4.5p.
\n4.5p – 2.5 + 1.1n.
\n2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n.<\/p>\n

Question 10.
\n12 + (-4) – \\(\\frac{2}{5}\\)j – \\(\\frac{4}{5}\\)j + 5<\/p>\n

Answer:
\n12 + (-4) – 2\/5j – 4\/5j + 5 = 13 – 6\/5j.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n12 + (-4) – 2\/5j – 4\/5j + 5.
\n12 -4 – 6\/5j + 5.
\n8 – 6\/5j + 5.
\n13 – 6\/5j.
\n12 + (-4) – 2\/5j – 4\/5j + 5 = 13 – 6\/5j.<\/p>\n

Question 11.
\nWhich expression is equivalent to -5v + (-2) + 1 + (-2v)?
\nA. -9v
\nB. -4v
\nC. -7v – 1
\nD. -7V + 3<\/p>\n

Answer:
\nOption C is correct.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-5v + (-2) + 1 + (-2v).
\n-5v -2 + 1 -2v.
\n-7v -1.
\nso option C is correct.<\/p>\n

Question 12.
\nWhich expression is equivalent to \\(\\frac{2}{3}\\)x + (-3) + (-2) \u2013 \\(\\frac{1}{3}\\)x?
\nA. x + 5
\nB. –\\(\\frac{1}{3}\\)x + 5
\nC. \\(\\frac{1}{3}\\)x – 1
\nD. \\(\\frac{1}{3}\\)x – 5<\/p>\n

Answer:
\nOption D is correct.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\\(\\frac{2}{3}\\)x + (-3) + (-2) \u2013 \\(\\frac{1}{3}\\)x.
\n2\/3x -3 -2 -1\/3x.
\n1\/3x -5
\nso option D is correct.<\/p>\n

Question 13.
\nThe dimensions of a garden are shown. Write an expression to find the perimeter.
\n\"Envision<\/p>\n

Answer:
\nThe perimeter of the garden = x – 7.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe length of the garden = x.
\nthe width of the garden = 1\/2x – 7.
\narea of the garden = l x b.
\nx + 1\/2x – 7.
\n2\/2x – 7.
\nx – 7.
\nso the perimeter of the garden = x – 7.<\/p>\n

Question 14.
\nSimplify the expression 8h + (-7.3d) \u2013 14 + 5d – 3.2h.<\/p>\n

Answer:
\n8h + (-7.3d) \u2013 14 + 5d – 3.2h = 5.2h – 2.3d – 14.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n8h + (-7.3d) \u2013 14 + 5d – 3.2h.
\n8h – 7.3d – 14 + 5d – 3.2h.
\n5.2h – 2.3d – 14.
\n8h + (-7.3d) \u2013 14 + 5d – 3.2h = 5.2h – 2.3d – 14.<\/p>\n

Question 15.
\nSimply 4 – 2y + (-8y) + 6.2.<\/p>\n

Answer:
\n4 – 2y + (-8y) + 6.2 = 10.2 – 10y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4 – 2y + (-8y) + 6.2.
\n4 – 2y – 8y + 6.2.
\n4 – 10y + 6.2.
\n10.2 – 10y.
\n4 – 2y + (-8y) + 6.2 = 10.2 – 10y.<\/p>\n

Question 16.
\nSimplify \\(\\frac{4}{9}\\)z – \\(\\frac{3}{9}\\)z + 5 – \\(\\frac{5}{9}\\)z – 8.<\/p>\n

Answer:
\n4\/9z – 3\/9z + 5 – 5\/9z – 8 = -4\/9z – 3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4\/9z – 3\/9z + 5 – 5\/9z – 8.
\n1\/9z + 5 – 5\/9z – 8.
\n-4\/9z + 5 – 8.
\n-4\/9z – 3.
\n4\/9z – 3\/9z + 5 – 5\/9z – 8 = -4\/9z – 3.<\/p>\n

Question 17.
\nConstruct Arguments<\/strong> Explain whether 11t – 4t is equivalent to 4t – 11t. Support your answer by evaluating the expression for t = 2.<\/p>\n

Answer:
\nThe values are the same but 11t – 4t is positive and 4t – 11t is negative.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n11t – 4t is equivalent to 4t – 11t.
\nt = 2.
\n11(2) – 4(2).
\n22 – 8.
\n14.
\n4t – 11t.
\n4(2) – 11(2).
\n8 – 22.
\n-14.<\/p>\n

Question 18.
\nThe signs show the costs of different games at a math festival. How much would it cost n people to play Decimal Decisions and Ratio Rage?
\n\"Envision<\/p>\n

Answer:
\nThe cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n\/4 – 3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe cost of 1 Game is 5.5n – 3.
\nthe cost of 1 Game is n\/4.
\n5.5n – 3 + n\/4.
\n6.6n\/4 – 3.
\nthe cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n\/4 – 3.<\/p>\n

Question 19.
\nHigher Order Thinking<\/strong> in the expression ax + bx, a is a decimal and b is a fraction. How do you decide whether to write a as a fraction or b as a decimal?<\/p>\n

Answer:
\nYes, we can write an as a fraction and b as a decimal.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nin the expression ax + bx, a is a decimal and b is a fraction.
\nfor example:
\na = 1.1.
\nb = 1\/2.
\nax + bx.
\n1.1x + 1\/2x.
\nso 1.1 is a decimal and 1\/2 is a fraction.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 20.
\nSelect all expressions equivalent to -6z + (-5.5) + 3.5z + 5y – 2.5.
\n\u2610 -8 + 5y + 2.52
\n\u2610 -2.5z + 5y – 8
\n\u2610 -8 + 5y +(-2.5z)
\n\u2610 2.5y + (-2.5z) \u2013 5.5
\n\u2610 5y – 8 – 2.5z<\/p>\n

Answer:
\nOption B and C are correct.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-6z + (-5.5) + 3.5z + 5y – 2.5.
\n-6z + 3.5z – 5.5 – 2.5 + 5y.
\n-2.5z -8 + 5y.
\nso options B and C are correct.<\/p>\n

Lesson 4.4 Expand Expressions<\/h3>\n

Solve & Discuss It!<\/strong>
\nThe school is planning to add a weight room to the gym. If the total area of the gym and weight room should stay under 5,500 square feet, what is one possible length for the new weight room? Show your work. Are there other lengths that would work? Why or why not? -90 ft
\nI can… expand expressions using the Distributive Property.
\n\"Envision<\/p>\n

Look for Relationships
\nWhat is the relationship between the areas of the gym and weight room?<\/p>\n

Answer:
\nThe relationship between the areas of the gym and weight room = 550 ft.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe school is planning to add a weight room to the gym.
\nIf the total area of the gym and weight room should stay under 5,500 square feet.
\nthe area of the school is l x b.
\nwhere l = 90 ft and b = 55 ft.
\narea = l x b.
\narea = 90 x 55.
\narea = 4950.
\n5500 – 4950 = 550.
\nso the relationship between the areas of the gym and weight room = 550 ft.<\/p>\n

Focus on math practices
\nModel with Math What is an expression using x that represents the total area of the gym and the weight room?
\n\"Envision<\/p>\n

Answer:
\nThe relationship between the areas of the gym and weight room = 550 ft.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe school is planning to add a weight room to the gym.
\nIf the total area of the gym and weight room should stay under 5,500 square feet.
\nthe area of the school is l x b.
\nwhere l = 90 ft and b = 55 ft.
\narea = l x b.
\narea = 90 x 55.
\narea = 4950.
\n5500 – 4950 = 550.
\nso the relationship between the areas of the gym and weight room = 550 ft.<\/p>\n

Essential Question<\/strong>
\nHow does the value of an expression change when it is expanded?<\/p>\n

Try It!<\/strong><\/p>\n

What is the expanded form of the expression 3.6(t + 5)?
\n\"Envision
\n3.6(t + 5)
\n= ________t + _______ \u2022 5
\n= _______ + _______
\nThe expanded expression is _______.<\/p>\n

Answer:
\n3.6t + 18.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3.6(t + 5).
\n3.6 x t = 3.6t.
\n3.6t + 3.6 x 5.
\n3.6t + 18.<\/p>\n

Convince Me! If you know the value of t, would the evaluated expression be different if you added the known value of t and 5 and then multiplied by 3.6? Explain.<\/p>\n

Try It!<\/strong><\/p>\n

Expand the expression t(-1.2w + 3).<\/p>\n

Answer:
\nThe expression is -1.2tw + 3t.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is t(-1.2w + 3).
\n-1.2tw + 3t.
\nso the expanded expression is -1.2tw + 3t.<\/p>\n

Try It!<\/strong><\/p>\n

Simplify the expression –\\(\\frac{2}{5}\\)(10 + 15m \u2013 20n).<\/p>\n

Answer:
\nThe expression is -4 -6m – 8n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n–\\(\\frac{2}{5}\\)(10 + 15m \u2013 20n).
\n-2\/5 (10 + 15m – 20n).
\n-2\/5(10 + 15m – 20n).
\n-20\/5 – 30\/5m – 40\/5n.
\n-4 – 6m – 8n.<\/p>\n

KEY CONCEPT<\/strong>
\nYou can expand an expression using the Distributive Property.
\nMultiply, or distribute, the factor outside the parentheses with each term inside the parentheses.
\n-7(3y – 1)
\n= (-7)(3y) + (-7)(-1)
\n= -21y + 7
\nThe sign of each term is included in all calculations.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How does the value of an expression change when it is expanded?<\/p>\n

Answer:
\nThe value of an expression change when it is expanded.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe value of an expression change when it is expanded.
\nfor example:
\n-8(2y – 2).
\n-8(-2y) + (-8) (-2).
\n-16y + -16.<\/p>\n

Question 2.
\nUse Structure How does the subtraction part of the expression change when a(b – c) is expanded?<\/p>\n

Answer:
\nThe subtraction part of the expression change.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\na(b – c).
\nax b – a x c.
\nab – ac.
\nthe product of in terms is multiplied with outterms.<\/p>\n

Question 3.
\nMake Sense and Persevere When does expanding and simplifying a(b + c) result in a positive value for ac?<\/p>\n

Answer:
\nab + ac.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is a(b + c).
\na x b + a x c.
\nab + ac.
\nthe sign is positive.
\nso the value for ac is also positive.<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nShoes and hats are on sale. The expression \\(\\frac{1}{4}\\)(s + 24.80) can be used to determine the discount when you buy shoes with a retail price of s dollars and a hat with a retail price of $24.80. Write another expression that can be used to determine the discount.
\n\"Envision<\/p>\n

Answer:
\nAnother expression is $1.55.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nShoes and hats are on sale.
\nThe expression \\(\\frac{1}{4}\\)(s + 24.80).
\nwhen you buy shoes with a retail price of s dollars and a hat with a retail price of $24.80.
\n1\/4 (s + 24.80).
\ns\/4 + 24.80\/4.
\ns\/4 + 6.2.
\ns\/4 = – 6.2.
\ns = -6.2\/4.
\ns = 1.55.
\nso the retail price of the shoes = $1.55.<\/p>\n

Question 5.
\nExpand x(4 – 3.4y).<\/p>\n

Answer:
\nThe expression is 4x – 3.4xy.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nx(4 – 3.4y).
\n4x X – 3.4 x X x Y.
\n4x – 3.4xy.
\nso the expanded expression is 4x – 3.4xy.<\/p>\n

Question 6.
\nExpand –\\(\\frac{2}{10}\\)(1 – 2x + 2).<\/p>\n

Answer:
\nThe expanded expression is -3\/5 – 2\/5x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n–\\(\\frac{2}{10}\\)(1 – 2x + 2).
\n-2\/10 (1 – 2x + 2).
\n-1\/5 (1 – 2x + 2).
\n-1\/5 – 2\/5x – 2\/5.
\n-3\/5 – 2\/5x.<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice For 7-8, fill in the boxes to expand each expression.
\nQuestion 7.
\n3(n + 7)
\n= (3) (_____) + (3) (_____)
\n= ____ + _____<\/p>\n

Answer:
\n3n + 21.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3(n + 7).
\n3 x n + 3 x 7.
\n3n + 21.<\/p>\n

Question 8.
\n4(x – 3)
\n= ______ x – ______ (3)
\n= ______ – ______<\/p>\n

Answer:
\n4x – 12.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4(x – 3).
\n4 x X – 4 x 3.
\n4x – 12.<\/p>\n

For 9-14, write the expanded form of the expression.
\nQuestion 9.
\ny(0.5 + 8)<\/p>\n

Answer:
\ny(0.5 + 8)\u00a0= 8.5y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\ny(0.5 + 8).
\n0.5y + 8y.
\n8.5y.
\ny(0.5 + 8) = 8.5y.<\/p>\n

Question 10.
\n4(3 + 4x – 2)<\/p>\n

Answer:
\n4(3 + 4x – 2) = 4 + 16x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4(3 + 4x – 2).
\n4 x 3 + 4x x 4 – 2 x 4.
\n12 + 16x – 8.
\n4 + 16x.
\n4(3 + 4x – 2) = 4 + 16x.<\/p>\n

Question 11.
\n6(y + x)<\/p>\n

Answer:
\n6(y + x) = 6y + 6x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n6(y + x).
\n6 x y + 6 x x.
\n6y + 6x.
\n6(y + x) = 6y + 6x.<\/p>\n

Question 12.
\n-2.5(-3 + 4n + 8)<\/p>\n

Answer:
\n-2.5 (-3 + 4n + 8) = -14.5 – 10n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-2.5 (-3 + 4n + 8).
\n-2.5 x -3 – 2.5 x 4n – 2.5 x 8.
\n-5.5 – 10n – 20.
\n-14.5 – 10n.
\n-2.5 (-3 + 4n + 8) = -14.5 – 10n.<\/p>\n

Question 13.
\n–\\(\\frac{1}{3}\\)(y – x)<\/p>\n

Answer:
\n–\\(\\frac{1}{3}\\)(y – x) = -1\/3y + x\/3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n–\\(\\frac{1}{3}\\)(y – x).
\n-1\/3(y – x).
\n-1\/3y + x\/3.
\n–\\(\\frac{1}{3}\\)(y – x) = -1\/3y + x\/3.<\/p>\n

Question 14.
\n8(6x – 4)<\/p>\n

Answer:
\n8(6x – 4) = 48x – 32.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n8(6x – 4).
\n8 x 6x – 4 x 8.
\n48x – 32.
\n8(6x – 4) = 48x – 32.<\/p>\n

Question 15.
\nHigher Order Thinking<\/strong> A grocery store has a 13%-off sale on all bread. You decide to purchase 6 loaves of bread. Let b be the original price of a loaf of bread. Expand the expression 6(b – 0.13b). Once the expression is expanded, what do the terms represent?<\/p>\n

Answer:
\n6(b – 0.13b) = -5.22b.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA grocery store has a 13%-off sale on all bread.
\nYou decide to purchase 6 loaves of bread.
\nLet b be the original price of a loaf of bread.
\n6(b – 0.13b).
\n6 x b – 0.13b x 6.
\n6b – 0.78b.
\n-5.22b.
\n6(b – 0.13b) = -5.22b.<\/p>\n

Question 16.
\nA gardener plans to extend the length of a rectangular garden. Let x represent the garden’s original length. The expression 4(x + 7) represents the area of the extended garden. When asked for the area of the extended portion, the gardener incorrectly said it was 11 square feet. Describe the error the gardener made.
\n\"Envision<\/p>\n

Answer:
\nThe error the gardener made = 4x + 28.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nLet x represent the garden’s original length.
\nThe expression 4(x + 7) represents the area of the extended garden.
\n4(x + 7).
\n4 x x + 4 x 7.
\n4x + 28.<\/p>\n

Question 17.
\nFind a difference equivalent to the product 11(x – y).<\/p>\n

Answer:
\n11(x – y) = 11x – 11y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n11(x – y).
\n11 x x – 11 x y.
\n11x – 11y.
\n11(x – y) = 11x – 11y.<\/p>\n

Question 18.
\nUse the Distributive Property to write an expression equivalent to 0.4(-5 \u2013 7y – 13.8).<\/p>\n

Answer:
\n0.4(-5 \u2013 7y – 13.8) = -10.12 – 2.8y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 0.4(-5 \u2013 7y – 13.8).
\n0.4 x (-5) – 0.4 (7y) – 0.4 (-13.8).
\n-4.6 – 2.8y – 5.52.
\n-10.12 – 2.8y.<\/p>\n

Question 19.
\nMake Sense and Persevere Use the Distributive Property to expand 7(7x – 3y) – 6.<\/p>\n

Answer:
\n7(7x – 3y) – 6 = 49x – 21y – 6.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n7(7x – 3y) – 6.
\n7 x 7x – 7 (3y) – 6.
\n49x – 21y – 6.
\n7(7x – 3y) – 6 = 49x – 21y – 6.<\/p>\n

Question 20.
\nUse the Distributive Property to write an expression equivalent to y(-3 – 8x).<\/p>\n

Answer:
\ny(-3 – 8x) = -3y – 8xy.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\ny(-3 – 8x).
\n-3 x y – 8x (y).
\n-3y -8xy.<\/p>\n

Question 21.
\nAn architect plans to build an extension to Meiling’s rectangular deck. Let x represent the increase, in meters, of her deck’s length. The expression 5(X + 8) represents the area of the deck, where 5 is the width, in meters, and (x + 8) represents the extended length, in meters. Use the Distributive Property to write an expression that represents the total area of Meiling’s new deck.
\n\"Envision<\/p>\n

Answer:
\nThe total area of Meiling’s new deck = 5x + 40.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAn architect plans to build an extension to Meiling’s rectangular deck.
\nLet x represent the increase, in meters, of her deck’s length.
\nThe expression 5(X + 8) represents the area of the deck, where 5 is the width, in meters, and (x + 8) represents the extended length in meters.
\n5(x + 8).
\n5 x x + 5 x 8.
\n5x + 40.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 22.
\nSelect all expressions equivalent to –\\(\\frac{1}{2}\\)(4 \u2013 2 + 8x).
\n\u2610 -4x – 1
\n\u2610 4x – 1
\n\u2610 3x
\n\u2610 -2 + 1 – 4x
\n\u2610 2 + 1 – 4x
\n\u2610 4x + 1<\/p>\n

Answer:
\n-4x -1 and -2 + 1 – 4x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n–\\(\\frac{1}{2}\\)(4 \u2013 2 + 8x).
\n-1\/2(4 – 2 + 8x).
\n-4\/2 – 2\/2 + 8\/2x.
\n-2 + 1 – 4x.
\n-1 – 4x.<\/p>\n

Question 23.
\nAn expression is shown.
\n\\(\\frac{1}{5}\\)(5 \u2013 7y + 10) Create an equivalent expression without parentheses.<\/p>\n

Answer:
\n1\/5(5 – 7y + 10) = 1 – 7\/5y + 2.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n1\/5(5 – 7y + 10).
\n5\/5 – 7\/5y + 10\/5.
\n1 – 7\/5y + 2.<\/p>\n

Lesson 4.5 Factor Expressions<\/h3>\n

Explain It!<\/strong>
\nTasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. Tasha believes that she can pack no more than 6 bags using all of her supplies.<\/p>\n

Answer:
\nYes, she can pack 44 bags.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTasha is packing gift bags that include the same items.
\nShe has 72 glow sticks, 36 markers, and 24 bottles of bubbles.
\n72 + 36 + 24 = 132.
\n132\/3 = 44.
\nshe can pack 44 bags.<\/p>\n

I can… use common factors and the Distributive Property to factor expressions.
\n\"Envision<\/p>\n

Make Sense and Persevere
\nHow can you use what you know about common factors to solve the problem?<\/p>\n

A. Critique Reasoning Do you agree with Tasha? Explain.<\/p>\n

Answer:
\nNo, Tasha is wrong.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTasha is packing gift bags that include the same items.
\nShe has 72 glow sticks, 36 markers, and 24 bottles of bubbles.
\n72 + 36 + 24 = 132.
\n132\/3 = 44.
\nshe can pack 44 bags.<\/p>\n

B. If Tasha creates the greatest number of gift bags, how many of each item is in each bag? Explain how you know.<\/p>\n

Answer:
\nTasha packs 44 bags.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTasha is packing gift bags that include the same items.
\nShe has 72 glow sticks, 36 markers, and 24 bottles of bubbles.
\n72 + 36 + 24 = 132.
\n132\/3 = 44.
\nshe can pack 44 bags.<\/p>\n

Focus on math practices
\nReasoning Tasha added more markers and now has a total of 48 markers. Does this change the possible number of gift bags? Explain.<\/p>\n

Essential Question<\/strong>
\nHow does the Distributive Property relate to factoring expressions?<\/p>\n

Try It!<\/strong><\/p>\n

Use factoring to write an expression for the length of the pool with the given width.
\n\"Envision
\n4x + 20 = _____ (x + _____)
\nSo, the length of the pool is ______ meters.<\/p>\n

Answer:
\nThe length of the pool is 4(x + 5) meters.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4x + 20.
\n4(x + 5).
\nthe length of the pool is 4(x + 5) meters.<\/p>\n

Convince Me! How can you use the Distributive Property to check the factored expression? Use the factored expression for Example 1 in your explanation.<\/p>\n

Try It!<\/strong><\/p>\n

Show two different ways to factor -(4x – 28).<\/p>\n

Answer:
\n-(4x – 28) = -4(x – 7).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-(4x – 28).
\n-4x + 28.
\n-4(x – 7).<\/p>\n

Try It!<\/strong><\/p>\n

Write an equivalent expression for the expression above using a negative factor.<\/p>\n

Answer:
\nThe equivalent expression for the expression is -4(x – 7).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe equivalent expression for the expression is -4(x – 7).
\n-(4x – 28).
\n-4x + 28.
\n-4(x – 7).<\/p>\n

KEY CONCEPT<\/strong>
\nThe greatest common factor (GCF) can be used to factor expressions.
\nThe Distributive Property can be applied to factor an expression. Factoring an expression creates an equivalent expression.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How does the Distributive Property relate to factoring expressions?<\/p>\n

Answer:
\n2x + 8 = 2(x + 4).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe distributive property can be applied to factor an expression.
\nfactoring an expression creates an equivalent expression.
\n2x + 8.
\n2(x + 4).<\/p>\n

Question 2.
\nSusan incorrectly factored the expression below.
\n12a – 156 + 6
\n3(4a + 5b + 3)<\/p>\n

a. Explain any errors Susan may have made when factoring.<\/p>\n

Answer:
\n12a + 15b + 9.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3(4a + 5b + 3).
\n(3 x 4a) + (3 x 5b) + (3 x 3).
\n12a + 15b + 9.<\/p>\n

b. Factor the expression correctly.<\/p>\n

Answer:
\n12a + 15b + 9.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n3(4a + 5b + 3).
\n(3 x 4a) + (3 x 5b) + (3 x 3).
\n12a + 15b + 9.<\/p>\n

Do You Know How?<\/strong>
\nQuestion 3.
\nSahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
\na. Write an expression to show the total number of items.<\/p>\n

Answer:
\nThe expression to show the total number of items = 36x + 12y + 24.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nSahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
\n36 x X + 12 x Y + 24,
\n36x + 12y + 24.
\nso the expression to show the total number of items = 36x + 12y + 24.<\/p>\n

b. Use factoring to show many kits Sahil can make while putting every type of item in each kit.<\/p>\n

Answer:
\nThe expression to show the total number of items = 36x + 12y + 24.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nSahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
\n36 x X + 12 x Y + 24,
\n36x + 12y + 24.
\nso the expression to show the total number of items = 36x + 12y + 24.<\/p>\n

c. Use the factored expression to find the number of each item in each kit.<\/p>\n

Answer:
\nThe expression to show the total number of items = 36x + 12y + 24.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nSahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
\n36 x X + 12 x Y + 24,
\n36x + 12y + 24.
\nso the expression to show the total number of items = 36x + 12y + 24.<\/p>\n

Question 4.
\nShow two different ways to factor – 12x + 24 – 18y.<\/p>\n

Answer:
\n-12x + 24 – 18y and -2(6x – 12 + 9y).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is – 12x + 24 – 18y.
\n-2(6x – 12 + 9y).<\/p>\n

Question 5.
\nHow can you use the Distributive Property to factor the expression 6x + 15?<\/p>\n

Answer:
\n3(2x + 5).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe given expression is 6x + 15.
\n3x + 3x +15.
\n3(2x + 5).<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice In 6-9, factor the expression.
\nQuestion 6.
\n16a + 10.
\nThe GCF of 16a and 10 is 2.
\n2 \u00d7 ______ = 16a
\n2 \u00d7 _______ = 10
\nThe factored expression is ________<\/p>\n

Answer:
\n2 x 8a = 16a.
\n2 x 5 = 10.
\nThe factored expression is 16a + 10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 16a + 10.
\n2(8a + 5).
\n2 x 8a = 16a.
\n2 x 5 = 10.<\/p>\n

Question 7.
\n-9y – 3.
\nThe positive GCF of -9y and -3 is 3.
\n3 \u00d7 ______ = -9y
\n3 \u00d7 ______ = -3
\nThe factored expression is ________<\/p>\n

Answer:
\n3 x -3y = -9y.
\n3 x 1 = -3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is -9y – 3.
\n-3(3y + 1).
\n-3 x 3y = -9y.
\n-3 x 1 = -3.<\/p>\n

Question 8.
\n14x + 49<\/p>\n

Answer:
\n7 x 2x = 14x.
\n7 x 7 = 49.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 14x + 49.
\n7(2x + 7).
\n7 x 2x = 14x.
\n7 x 7 = 49.<\/p>\n

Question 9.
\n12y – 16<\/p>\n

Answer:
\n2 x 6y = 12y.
\n2 x 8 = 16.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 12y – 16.
\n2(6y – 8).
\n2 x 6y = 12y.
\n2 x 8 = 16.<\/p>\n

Question 10.
\nThis model shows the area of a garden. Write two expressions that represent the area.
\n\"Envision<\/p>\n

Answer:
\nThe area of the garden = 5x + 10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 5x + 10.
\n5(x + 2).
\n5 x x = 5x.
\n5 x 2 = 10.
\nso the area of the garden = 5x + 10.<\/p>\n

Question 11.
\nUse the GCF to write the factored form of the expression 18x + 24y.<\/p>\n

Answer:
\nThe factored form of the expression is 2(9x + 12y).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 18x + 24y.
\n2(9x + 12y).
\n6(3x + 4y).
\nso the factored form of the expression is 2(9x + 12y).<\/p>\n

Question 12.
\nFind the dimensions of the sports field at the right if the width is at least 60 yards.
\n\"Envision<\/p>\n

Answer:
\nThe dimensions of the sports field = -160 and 60.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 240 – 400x.
\n-160x.
\n-160(60).
\n-9600.
\nso the dimensions of the sports field = -160 and 60.<\/p>\n

Question 13.
\nYour friend incorrectly factors the expression 15x \u2013 20xy as 5x( 3 – 4xy).
\na. Factor the expression correctly.<\/p>\n

Answer:
\nThe expression correctly = 5(3x – 4xy).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 15x – 20xy.
\n5(3x – 4xy).
\n5 x 3x = 15x.
\n5 x 4xy = 20xy.
\nso the expression correctly is 5(3x – 4xy).<\/p>\n

b. What error did your friend likely make?<\/p>\n

Answer:
\nThe expression correctly = 5(3x – 4xy).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 15x – 20xy.
\n5(3x – 4xy).
\n5 x 3x = 15x.
\n5 x 4xy = 20xy.
\nso the expression correctly is 5(3x – 4xy).<\/p>\n

Question 14.
\nYou are given the expression 12x + 18y + 26.
\na. Make Sense and Persevere What is the first step in factoring the expression?<\/p>\n

Answer:
\nThe first step in factoring is 2(6x + 9y + 18).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 12x + 18y + 26.
\n2(6x + 9y + 13).
\n2 x 6x = 12x.
\n2 x 9y = 18y.
\n2 x 13 = 26.<\/p>\n

b. Factor the expression.<\/p>\n

Answer:
\nThe expression is 2(6x + 9y + 18).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 12x + 18y + 26.
\n2(6x + 9y + 13).
\n2 x 6x = 12x.
\n2 x 9y = 18y.
\n2 x 13 = 26.<\/p>\n

Question 15.
\nA hotel manager is adding a tile border around the hotel’s rectangular pool. Let x represent the width of the pool, in feet. The length is 3 more than 2 times the width, as shown. Write two expressions that give the perimeter of the pool.
\n\"Envision<\/p>\n

Answer:
\nThe perimeter of the pool = 2XxX + 3x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nLet x represent the width of the pool, in feet.
\nThe length is 3 more than 2 times the width, as shown.
\nthe perimeter of the rectangle = length x width.
\nperimeter = 2x + 3 x X.
\nperimeter = 2×2 x 3x.<\/p>\n

Question 16.
\nHigher Order Thinking<\/strong> Use the expressions below.
\n14m + mn
\n2y + 2x + 4
\n–\\(\\frac{3}{4}\\)m + 8m + m
\n4 – 3p
\n5.75t + 7.75t – t
\n8xy – 6xy
\na. Circle the expressions that have like terms.<\/p>\n

Answer:
\nThe expressions that have the like terms = 14m + mn, 5.75t + 7.75t -t, and 8xy – 6xy.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expressions are 14m + mn
\n2y + 2x + 4.
\n–\\(\\frac{3}{4}\\)m + 8m + m.
\n4 – 3p.
\n5.75t + 7.75t – t.
\n8xy – 6xy.
\nthe like terms are 8xy – 6xy = 2xy.
\n5.75t + 7.75t – t = 13.5t – t.
\n12.5 t.<\/p>\n

b. Explain why the other expressions do not have like terms.<\/p>\n

Answer:
\nThe other expressions that do not have like terms are 4 – 3p, 14m + mn.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expressions are 14m + mn
\n2y + 2x + 4.
\n–\\(\\frac{3}{4}\\)m + 8m + m.
\n4 – 3p.
\n5.75t + 7.75t – t.
\n8xy – 6xy.
\nthe unlike terms are 14m + mn.
\n2y + 2x + 4.<\/p>\n

Question 17.
\nConstruct Arguments<\/strong> Ryan says the expression 3 + 5y cannot be factored using a GCF. Is he correct? Explain why or why not.<\/p>\n

Answer:
\nYes, Ryan was correct.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 3 + 5y.
\nwe cannot be factored in using a GCF.
\nso Ryan was correct.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 18.
\nSelect all the expressions equivalent to 12 + 30y.
\n\u2610 3(4 + 10y)
\n\u2610 4(3 + 10y)
\n\u2610 6(2 + 5y)
\n\u2610 2(6 + 30y)
\n\u2610 6(3 + 10y)<\/p>\n

Answer:
\nThe expressions equivaent to 12 + 30y is 3(4 + 10y), 6(2 + 5y).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 12 + 30y.
\n3(4 + 10y).
\n6(2 + 5y).
\n3 x 4 = 12 + 3 x 10y = 30y.
\n6 x 2 = 12 + 6 x 5y = 30y.<\/p>\n

Question 19.
\nWrite an expression that is the product of two factors and is equivalent to -2x \u2013 10.<\/p>\n

Answer:
\nThe expression is -2(x + 5).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is -2x – 10.
\n-2(x + 5).
\n-2x + 10.<\/p>\n

Topic 4 Mid-Topic Checkpoint<\/h3>\n

Question 1.
\nVocabulary If you write an expression to represent the following situation, how can you determine which is the constant and which is the coefficient of the variable? Lesson 4-1
\nThe zoo charges the Garcia family an admission fee of $5.25 per person and a one-time fee of $3.50 to rent a wagon for their young children.<\/p>\n

Answer:
\nThe admission fee is the coefficient and rents a wagon is constant.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe zoo charges the Garcia family an admission fee of $5.25 per person.
\na one-time fee of $3.50 to rent a wagon for their young children.
\n$5.25x + $3.50.
\n$5.25 is the coefficient of the variable.
\n$3.50 is the constant.<\/p>\n

Question 2.
\nAn online photo service charges $20 to make a photo book with 16 pages. Each extra page costs $1.75. The cost to ship the completed photo book is $5. Write an expression to determine the total cost in dollars to make and ship a photo book with x extra pages. Lesson 4-1<\/p>\n

Answer:
\nThe total cost in dollars to make and ship a photo book with X extra pages is $20x + $5 = $1.75.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAn online photo service charges $20 to make a photo book with 16 pages.
\nEach extra page costs $1.75.
\nThe cost to ship the completed photo book is $5.
\n$20x + $5 = $1.75.
\n$25x -$1.75 = $5.<\/p>\n

Question 3.
\nWrite an expression equivalent to 2a + (\\(\\frac{3}{4}\\)a + \\(\\frac{1}{5}\\)b) by combining like terms. Lesson 4-3<\/p>\n

Answer:
\nThe expression equivalent to 2a + 3\/4a + 1\/5b is 5\/4a + 1\/5b.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe given expression is 2a + (\\(\\frac{3}{4}\\)a + \\(\\frac{1}{5}\\)b).
\n2a + 3\/4a + 1\/5b.
\n5\/4a + 1\/5b.<\/p>\n

Question 4.
\nWhich expression is equivalent to 3.2y – \\(\\frac{1}{3}\\) + (-7y) + \\(\\frac{2}{3}\\)? Lesson 4-2
\nA. -10.2y + \\(\\frac{1}{3}\\)
\nB. -3.8y + \\(\\frac{1}{3}\\)
\nC. -3\\(\\frac{7}{15}\\)y
\nD. -3y<\/p>\n

Answer:
\nOption B is the correct.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 3.2y – \\(\\frac{1}{3}\\) + (-7y) + \\(\\frac{2}{3}\\).
\n3.2y – 1\/3 -7y + 2\/3.
\n-3.8y + 1\/3.
\nso option B is the correct.<\/p>\n

Question 5.
\nRay wants to buy a hat that costs $10 and some shirts that cost $12 each. The sales tax rate is 6.5%. Write an expression to determine the amount of sales tax that Ray will pay on his entire purchase. Expand to simplify the expression. Lesson 4-4<\/p>\n

Answer:
\nThe expression to determine the amount of sales tax that Ray will pay on his entire purchase = $22 + 6.5%.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nRay wants to buy a hat that costs $10 and some shirts that cost $12 each.
\nThe sales tax rate is 6.5%.
\n$10 + $12 + 6.5%.
\n$22 + 6.5%.<\/p>\n

Question 6.
\nFactor the expression 28r + 425 – 35. Lesson 4-5<\/p>\n

Answer:
\nThe expression is 7(4r + 61 – 5).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 28r + 425 – 35.
\n7(4r + 61 – 5).
\n7 x 4r = 28r.
\n61 x 7 = 425.
\n7 x 5 = 35.<\/p>\n

Question 7.
\nDescribe two ways the Distributive Property can be used to write equivalent expressions. Lessons 4-4 and 4-5<\/p>\n

Answer:
\nThe two ways the distributive property can be used to write equivalent expressions<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-1\/2(x + 8), -1\/2x + (-4) and -4 +(-1\/2x) are equivalent.
\n-1\/2(x + 8).
\n-1\/2x + (-1\/2) . 8.
\n-1\/2x + (-4).
\n-4 + (-1\/2x).
\nthe three expressions are true.<\/p>\n

Topic 4 Mid-Topic Performance Task<\/h3>\n

Alison is a buyer for a chain of 6 flower shops. This means that she buys flowers in bulk from a supplier and then distributes them to the 6 flower shops in the chain.
\nPART A
\nThis week Alison bought 108 bunches of carnations and 96 bunches of roses from the supplier. Let c represent the number of carnations in each bunch, and let r represent the number of roses in each bunch. Write an expression to show the total number of carnations and roses that Alison bought.<\/p>\n

Answer:
\nThe total number of carnations and roses that Alison bought = 12(9c + 8r).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThis week Alison bought 108 bunches of carnations and 96 bunches of roses from the supplier.
\nLet c represent the number of carnations in each bunch, and let r represent the number of roses in each bunch.
\n108c + 96r.
\n12(9c + 8r).
\n12 x 9c = 108c.
\n12 x 8r = 96r.<\/p>\n

PART B
\nAlison wants to distribute the carnations and roses equally among the 6 flower shops. Factor the expression from Part A using 6 as the common factor. How does the factored expression help Alison determine how many carnations and how many roses each flower shop should get?<\/p>\n

Answer:
\nThe common factor is 3(c + r).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAlison wants to distribute the carnations and roses equally among the 6 flower shops.
\n3c + 3r.
\n3(c + r).
\n1(3c + 3r).
\nso the common factor is 3(c + r).<\/p>\n

PART C
\nThere are 24 carnations in each bunch and 12 roses in each bunch. Use your answer to Part B to determine the total number of carnations and the total number of roses Alison will distribute to each flower shop this week.<\/p>\n

Answer:
\nThe total number of carnations and roses = 6(4c + 2r).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThere are 24 carnations in each bunch and 12 roses in each bunch.
\n24 c + 12 r.
\n2(12c + 6r).
\n6(4c + 2r).
\n6 x 4c = 24c.
\n6 x 2r = 12r.<\/p>\n

PART D
\nJake manages one of the flower shops. He wants to use the carnations and roses to make bouquets. He wants each bouquet to have the same combination of carnations and roses, with no flowers left over. Determine a way that Jake can divide the flowers to make the bouquets. How many bouquets will there be?<\/p>\n

Answer:
\nThe number of bouquets will there be = 6(c + r).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nJake manages one of the flower shops.
\nHe wants to use the carnations and roses to make bouquets.
\nHe wants each bouquet to have the same combination of carnations and roses, with no flowers left over.
\n6 and 6.
\n6 x 6 = 36.
\n6c + 6r.
\n6(c + r).
\nso the number of bouquets will there be = 6(c + 1).<\/p>\n

3-Act Mathematical Modeling:<\/strong>
\nI’ve Got You Covered
\n\"Envision<\/p>\n

ACT 1<\/strong>
\nQuestion 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nConstruct Arguments Predict an answer to this Main Question. Explain your prediction.
\n\"Envision
\nAnswer:<\/p>\n

Question 4.
\nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
\n\"Envision<\/p>\n

Answer:
\nThe two numbers are 1 and 10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe number line is 10 cm long.
\nthe short is 1 cm.
\nthe large is 10 cm.
\nso the two numbers that are too small and too large is 1 and 10.<\/p>\n

Question 5.
\nPlot your prediction on the same number line.<\/p>\n

Answer:
\nThe two numbers are 1 and 10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe number line is 10 cm long.
\nthe short is 1 cm.
\nthe large is 10 cm.
\nso the two numbers that are too small and too large is 1 and 10.<\/p>\n

ACT 2<\/strong>
\nQuestion 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision<\/p>\n

Answer:
\nThe figure contain 7 objects.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe figure contains 7 objects.
\n3 objects on the left side.
\n4 objects on the right side.
\nso the figure contain 7 objects.<\/p>\n

Question 7.
\nUse Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.<\/p>\n

Answer:
\nThe figure contain 7 objects.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe figure contains 7 objects.
\n3 objects on the left side.
\n4 objects on the right side.
\nso the figure contain 7 objects.<\/p>\n

Question 8.
\nModel with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong>
\nQuestion 10.
\nWrite the answer you saw in the video.
\n\"Envision
\nAnswer:<\/p>\n

Question 11.
\nReasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Extension
\nReflect
\nQuestion 13.
\nModel with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\nAnswer:<\/p>\n

Question 14.
\nGeneralize What pattern did you notice in your calculations? How did that pattern help you solve the problem?
\nAnswer:<\/p>\n

SEQUEL
\nQuestion 15.
\nReasoning A classmate says that another object needs 512 tiles. What do you know about the dimensions of the object?
\n\"Envision<\/p>\n

Answer:
\nThe dimensions of the object = 22 and 26.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA classmate says that another object needs 512 tiles.
\n22 x 26 = 512.
\nthe length = 22.
\nthe width = 26.
\narea = l x b.
\n22 x 26.
\n512.<\/p>\n

Lesson 4.6 Add Expressions<\/h3>\n

Solve & Discuss It!<\/strong>
\nThe Smith family took a 2-day road trip. On the second day, they drove the distance they traveled on the first day. What is a possible distance they could have traveled over the 2 days? Is there more than one possible distance? Justify your response.
\nI can… add expressions that represent real-world problems.
\n\"Envision<\/p>\n

Make Sense and Persevere
\nHow are the quantities in the problem related?<\/p>\n

Answer:
\nThe quantities in the problem are the smith family took a 2-day road trip.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe Smith family took a 2-day road trip.
\nOn the second day, they drove the distance they traveled on the first day.
\n2 + 2 = 4.<\/p>\n

Focus on math practices
\nUse Structure How can two different expressions be used to represent the total distance?<\/p>\n

Essential Question<\/strong>
\nHow can properties of operations be used to add expressions?<\/p>\n

Try It!<\/strong><\/p>\n

Sophia and Ollie each deposit $120 to open a joint account. They each make monthly deposits as shown. What expression represents the amount in the account after m months?
\n\"Envision
\nThe amount of money in the joint account after m months is ______ + _______.<\/p>\n

Answer:
\nThe amount of money in the joint account after m months is 120 + 120 = $525..<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nSophia and Ollie each deposit $120 to open a joint account.
\n120 + 150 = 270.
\n120 + 135 = 255.
\n270 + 255 = 525.
\nso the amount of money in the joint account after m months is $525.<\/p>\n

Convince Me! Explain why the initial deposits and monthly deposits are not combined into one term?<\/p>\n

Try It!<\/strong><\/p>\n

Find each sum.
\na. (9.740 – 250.50) + (-5.48p + 185.70)<\/p>\n

Answer:
\nTe sum is 60.54.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(9.740 – 250.50) + (-5.48p + 185.70).
\n(-240.76) + (180.22).
\n60.54.<\/p>\n

b. (\\(\\frac{2}{11}\\)x – 3 – 5y) + (-\\(\\frac{3}{11}\\) + 5y + 5.5)<\/p>\n

Answer:
\n-1\/11 x + 2.5 + 5y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(\\(\\frac{2}{11}\\)x – 3 – 5y) + (-\\(\\frac{3}{11}\\) + 5y + 5.5)
\n(2\/11)x – 3 – 5y + (-3\/11) + 5y + 5.5.
\n-1\/11x + 2.5 + 5y.
\n-1\/11 x + 2.5 + 5y.<\/p>\n

C. (-14.2b – 97.35) + (6.76d – 118.7 – 3.4d)<\/p>\n

Answer:
\n-14.2b – 216.05 + 3.36d.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(-14.2b – 97.35) + (6.76d – 118.7 – 3.4d).
\n-14.2b – 97.35 + 3.36d – 118.7.
\n-14.2b – 216.05 + 3.36d.<\/p>\n

d. (\\(\\frac{3}{8}\\) – \\(\\frac{1}{6}\\)m + 5t) + (\\(\\frac{7}{10}\\)m + 9t + \\(\\frac{1}{4}\\))<\/p>\n

Answer:
\n1\/3 + 3\/2m + 14t.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\\(\\frac{3}{8}\\) – \\(\\frac{1}{6}\\)m + 5t) + (\\(\\frac{7}{10}\\)m + 9t + \\(\\frac{1}{4}\\))
\n3\/8 – 1\/6 m + 5t + 7\/10 m + 9t + 1\/4.
\n3\/8 + 1\/4 – 1\/6 m + 7\/10 m + 5t + 9t.
\n3\/8 + 1\/4 + 3\/2 m + 14 t.
\n1\/3 + 3\/2m + 14t.<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Adding expressions may require combining like terms.
\nTerms with the same variables are added together and constants are added together.
\nWhen adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.
\n\"Envision
\n(3.6 + 22.4t) + (2 + 18.9t) = 5.6 + 41.3t<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How can properties of operations be used to add expressions?<\/p>\n

Answer:
\nThe properties of operations be used to add expressions with the like terms.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAdding expressions may require combining like terms.
\nTerms with the same variables are added together and constants are added together.
\nWhen adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.
\n(3.6 + 22.4t) + (2 + 18.9t).
\n5.6 + 41.3t.<\/p>\n

Question 2.
\nReasoning Explain whether the coefficients of two terms with different variables can be added to make one new term.<\/p>\n

Answer:
\nYes the coefficients of two terms with different variables can be added to make the new term.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAdding expressions may require combining like terms.
\nTerms with the same variables are added together and constants are added together.
\nWhen adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.
\n(3.6 + 22.4t) + (2 + 18.9t).
\n5.6 + 41.3t.<\/p>\n

Question 3.
\nBe Precise which properties of operations could be used to show that (-5p + 9) + (-2 + p) is equivalent to (-5p) + p + 9 – 2?<\/p>\n

Answer:
\nThe equivalent expression is -4p + 7.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is (-5p + 9) + (-2 + p).
\n-5p + 9 -2 + p.
\n-4p + 7.<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nDillon says that 4b and -2b are not like terms because 4b is positive and -2b is negative. Is he correct? Explain.<\/p>\n

Answer:
\nNo, they are like terms.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nDillon says that 4b and -2b are not like terms because 4b is positive and -2b is negative.
\n4b and 2b are same.
\n4b – 2b = 2b.
\nso they are like terms.<\/p>\n

Question 5.
\nJoel spent $28 for an Internet data service and pays $14.50 per month. He spent $24.50 to join an online movie streaming site and pays $13.25 per month. Write an expression to represent Joel’s total cost for both memberships after m months.
\n\"Envision<\/p>\n

Answer:
\nThe expression to represent Joel’s total cost for both memberships after m months = $27.75m + $52.50.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nJoel spent $28 for an Internet data service and pays $14.50 per month.
\nHe spent $24.50 to join an online movie streaming site and pays $13.25 per month.
\n$14.50m + $13.25m + $28 + $24.50.
\n$27.75m + $52.50.<\/p>\n

Question 6.
\nAdd \\(\\frac{1}{3}\\)n + \\(\\frac{2}{3}\\) and –\\(\\frac{1}{6}\\)n + \\(\\frac{1}{6}\\)m.<\/p>\n

Answer:
\n-1\/3n + 2\/3 + 1\/6m.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\\(\\frac{1}{3}\\)n + \\(\\frac{2}{3}\\) and –\\(\\frac{1}{6}\\)n + \\(\\frac{1}{6}\\)m.
\n1\/3 n + 2\/3 – 1\/6 n + 1\/6m.
\n-1\/3n + 2\/3 + 1\/6m.<\/p>\n

Question 7.
\nFind the sum.
\n(-3.5t – 4s +4.5) + (-7.1 – 0.3s + 4.1t)<\/p>\n

Answer:
\n2.6 + 0.6t – 4.3s.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(-3.5t – 4s +4.5) + (-7.1 – 0.3s + 4.1t).
\n-3.5t – 4s + 4.5 -7.1 – 0.3s + 4.1t.
\n0.6 t – 4.3s – 2.6.
\n2.6 + 0.6t – 4.3s.<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice For 8-9, fill in the boxes to add the expressions.
\nQuestion 8.
\n(2a + 8) + (4a + 5)
\n= (2a + ______) + (8 + _____)
\n= _______ + 13<\/p>\n

Answer:
\n6a + 13.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is (2a + 8) + (4a + 5).
\n(2a + 4a) + ( 8 + 5).
\n6a + 13.<\/p>\n

Question 9.
\n(\\(\\frac{2}{7}\\)x – 7) + (\\(\\frac{1}{7}\\)x + 8)
\n= (_______ + _______) + (-7 + ______)
\n= ______x + ______<\/p>\n

Answer:
\n3\/7 + 1.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(\\(\\frac{2}{7}\\)x – 7) + (\\(\\frac{1}{7}\\)x + 8).
\n(2\/7 + 1\/7)x – 7 + 8.
\n3\/7x + 1.<\/p>\n

Question 10.
\nFind the sum.
\n(8b + 7) + (6x – 4) + (5c + 8)<\/p>\n

Answer:
\n8b + 6x + 11.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(8b + 7) + (6x – 4) + (5c + 8).
\n8b + 7 + 6x – 4 + 5c + 8.
\n8b + 3 + 6x + 8.
\n8b + 6x + 11.<\/p>\n

Question 11.
\nCombine like terms.
\n(-3y – 5) + (5m + 7y) + (6 + 9m)<\/p>\n

Answer:
\n14m + 4y + 1.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(-3y – 5) + (5m + 7y) + (6 + 9m)
\n-3y – 5 + 5m + 7y + 6 + 9m.
\n4y + 1 + 14m.
\n14m + 4y + 1.<\/p>\n

Question 12.
\nFelipe is going to plant b sunflower seeds in one garden and 5b + 10 sunflower seeds in another. How many seeds is Felipe going to plant altogether?
\n\"Envision<\/p>\n

Answer:
\nThe number of seeds Felipe going to plant altogether = 5bsquare + 10b.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nFelipe is going to plant b sunflower seeds in one garden and 5b + 10 sunflower seeds in another.
\n(5b + 10) b.
\n5b x b + 10b.
\nso the number of seeds Felipe goind to plant altogether = 5bsquare + 10b.<\/p>\n

Question 13.
\nAn art class is making a mural for the school that has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 1 more than three times the length of the bottom of the triangle. The last side is 2 more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.<\/p>\n

Answer:
\nThe perimeter of the triangle =<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAn art class is making a mural for the school that has a triangle drawn in the middle.
\nThe length of the bottom of the triangle is x.
\nAnother side is 1 more than three times the length of the bottom of the triangle.
\nThe last side is 2 more than the bottom of the triangle.
\nx + 3x + 2x.
\np = x + 2x + 3x.
\np = 6x.
\nso the perimeter of the triangle = 6x.<\/p>\n

Question 14.
\nOn a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m. Sarah identifies the only coefficient as 7 and the only constant as 4.
\na. Identify all the coefficients of the expression.<\/p>\n

Answer:
\nThe coefficients of the expression is 7.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nOn a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
\nSarah identifies the only coefficient as 7.
\nso the coefficient is 7.<\/p>\n

b. Identify all the constants of the expression.<\/p>\n

Answer:
\nThe constants of the expression is n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nOn a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
\nSarah identifies the only constant as n.
\nso the constant is n.<\/p>\n

c. What error did Sarah likely make?<\/p>\n

Answer:
\nSarah make the mistake was 4 is the constant.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nOn a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
\nSarah identifies the only constant as 4.
\nso the constant is 4.<\/p>\n

Question 15.
\nThe width of a rectangle is 5x – 2.5 feet and the length is 2.5x + 8 feet. Find the perimeter of the rectangle.<\/p>\n

Answer:
\nThe perimeter of the rectangle = 6.25x – 20.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe width of a rectangle is 5x – 2.5 feet and the length is 2.5x + 8 feet.
\n(5x – 2.5) x (2.5x + 8).
\n6.25x – 20.
\nso the perimeter of the rectangle = 6.25x – 20.<\/p>\n

Question 16.
\nNina has x coins. Clayton has 5 fewer coins than six times the number of coins Nina has. Write an expression for the total number of coins Nina and Clayton have altogether. Then simplify the expression.<\/p>\n

Answer:
\nThe expression for the total number of coins Nina and Clayton have altogether = 6x\u00a0 – 30.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nNina has x coins.
\nClayton has 5 fewer coins than six times the number of coins Nina has.
\nx – 5 x 6.
\n6x – 30.
\nso the expression for the total number of coins Nina and Clayton have altogether = 6x – 30.<\/p>\n

Question 17.
\nHigher Order Thinking<\/strong> Use the expression (8x + 2) + (-9x + 7).
\na. Find the sum.<\/p>\n

Answer:
\n9 – x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that
\n(8x + 2) + (-9x + 7).
\n8x + 2 -9x + 7.
\n-x + 9.
\n9 – x.<\/p>\n

b. Reasoning Explain how you know when to combine terms with variables.<\/p>\n

Answer:
\nWe have to combine the variables when they have like terms.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that
\n(8x + 2) + (-9x + 7).
\n8x + 2 -9x + 7.
\n-x + 9.
\n9 – x.<\/p>\n

Question 18.
\nGabe went to the Florida Mall. He bought k model planes and spent $24 on books. Then he spent another $25 at another store.
\n\"Envision
\na. Write an expression that represents the amount Gabe spent at the mall.<\/p>\n

Answer:
\nThe expression that represents the amount Gabe = $49k + $14.99.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nGabe went to the Florida Mall.
\nHe bought k model planes and spent $24k on books.
\nThen he spent another $25k at another store.
\neach model panel cost $14.99.
\n$24k + $25k + $14.99.
\n$49k + $14.99.<\/p>\n

b. How much did Gabe spend in all if he bought 3 model planes?<\/p>\n

Answer:
\nThe amount did Gabe spend in all if he bought 3 models planes = $93.97.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nGabe went to the Florida Mall.
\nHe bought k model planes and spent $24k on books.
\nThen he spent another $25k at another store.
\neach model panel cost $14.99.
\n3 x $14.99 = 93.97.
\nso the amount did Gabe spend in all if he bought 3 models planes = $93.97.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 19.
\nA middle school with x students conducted a survey to determine students’ Tuesday afternoon activities.
\nPART A
\nWrite an expression for each activity.
\n25 more than one-tenth of the students dance.<\/p>\n

20 fewer than three-tenths of the students play soccer.<\/p>\n

21 more than one-tenth of the students play baseball<\/p>\n

Answer:
\n25 + 1\/10, 20 – 3\/10, and 21 + 1\/10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n25 more than one-tenth of the students dance.
\n20 fewer than three-tenths of the students play soccer.
\n21 more than one-tenth of the students play baseball.
\n25 + 1\/10, 20 – 3\/10, and 21\/10.<\/p>\n

PART B
\nWrite a simplified expression to represent the number of students who either dance or play baseball on Tuesday afternoons.<\/p>\n

Answer:
\nThe number of students who either dance or play baseball on\u00a0 Tuesday afternoons = 25 + 1\/10 and 21\/10.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n25 more than one-tenth of the students dance.
\n21 more than one-tenth of the students play baseball.
\n25 + 1\/10 and 21\/10.<\/p>\n

Lesson 4.7 Subtract Expressions<\/h3>\n

Explore It!<\/strong>
\nThe East Side Bulldogs and the West Side Bears are playing a football game. A fan is keeping score using T for a touchdown plus extra point, worth 7 points total, and F for a field goal, worth 3 points.<\/p>\n

I can… subtract expressions using properties of operations.
\n\"Envision<\/p>\n

A. How can you represent the score of each team using expressions?<\/p>\n

Answer:
\nThe score of the each team using expressions = 10 points.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nEast Side Bulldogs and the West Side Bears are playing a football game.
\nfan is keeping score using T for a touchdown.
\nplus extra point, worth 7 points total, and F for a field goal, worth 3 points.
\n7 + 5 = 12.<\/p>\n

B. How can you represent the difference of the teams’ scores using an expression?<\/p>\n

Answer:
\nThe difference of the teams scores using an expression is 4.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nEast Side Bulldogs and the West Side Bears are playing a football game.
\nfan is keeping score using T for a touchdown.
\nplus extra point, worth 7 points total, and F for a field goal, worth 3 points.
\n12 – 1 = 11.<\/p>\n

C. How can you determine how many more points the winning team had than the losing team?<\/p>\n

Answer:
\nThe number of points the winning team had than the losing team = 1.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nEast Side Bulldogs and the West Side Bears are playing a football game.
\nfan is keeping score using T for a touchdown.
\nplus extra point, worth 7 points total, and F for a field goal, worth 3 points.
\n12 – 1 = 11.
\nso the number of points the winning team had than the losing team = 1.<\/p>\n

Focus on math practices
\nLook for Relationships How can looking at the coefficients help you determine which team scored the greater number of points?<\/p>\n

Essential Question<\/strong>
\nHow can properties of operations be used to subtract expressions?<\/p>\n

Try It!<\/strong><\/p>\n

A frame holds a picture that is 15 inches long and x inches wide. The frame border is 3 inches wide around the picture. What expression represents the area of the frame border?
\nArea of frame border = Area of entire frame \u2013 Area of photo = ________ – ________
\nThe area of the frame is _________ in2<\/sup><\/p>\n

Answer:
\nThe expression represents the area of the frame border =<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA frame holds a picture that is 15 inches long and x inches wide.
\nThe frame border is 3 inches wide around the picture.
\n3 = 15 – x.
\nx = 15\/3.
\nx = 5.
\nso the expression represents the area of the frame border is 5 square inches.<\/p>\n

Convince Me! Why can you choose to add or subtract when subtracting an expression?<\/p>\n

Try It!<\/strong><\/p>\n

Subtract (0.95x – 0.04) – (0.99x – 0.13).<\/p>\n

Answer:
\n0.04x + 0.09.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(0.95x – 0.04) – (0.99x – 0.13).
\n0.95x – 0.04 – 0.99x + 0.13.
\n0.04x + 0.09.
\n0.04x + 0.09.<\/p>\n

Try It!<\/strong><\/p>\n

Subtract (17 + 4.5m + 8k) – (7.5m – 9 + 4k).<\/p>\n

Answer:
\n4k – 3m + 26.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(17 + 4.5m + 8k) – (7.5m – 9 + 4k).
\n17 + 4.5m + 8k – 7.5m + 9 – 4k.
\n26 – 3m + 4k.
\n4k – 3m + 26.<\/p>\n

KEY CONCEPT<\/strong>
\nTo subtract expressions, you can use properties of operations.
\nWrite the subtraction as addition and use the Distributive Property to multiply – 1 to the terms in the expression being subtracted.
\n5 – (2x – 7)
\n= 5 – (-2x \u2013 7)
\n= 5 +(-1)(-2x \u2013 7)
\n= 5+ (-1)(-2)x + (-1)(-7)
\n= 5 + 2x + 7
\nYou can use the Distributive Property to distribute the minus sign to the second expression, which changes the signs of the terms.
\n5 – (-2x – 7)
\n= 5 + 2x + 7<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question How can properties of operations be used to subtract expressions?<\/p>\n

Answer:
\nThe properties be used to subtract expressions are Distributive property.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n5 – (2x – 7)
\n= 5 – (-2x \u2013 7)
\n= 5 +(-1)(-2x \u2013 7)
\n= 5+ (-1)(-2)x + (-1)(-7)
\n= 5 + 2x + 7.<\/p>\n

Question 2.
\nUse Structure How is subtracting – 4x from 9x similar to subtracting -4 from 9?<\/p>\n

Answer:
\nThe difference is 5x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-4x and 9x.
\n9x – 4x.
\n5x.
\nso the difference is 5x.<\/p>\n

Question 3.
\nIs adding the quantity – 12 + 8r to an expression the same as subtracting -8r+ 12 from the same expression? Explain your reasoning.<\/p>\n

Answer:
\nNo, the expressions are not equal.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-12 + 8r.
\n8r – 12.
\n-8r + 12.
\n12 – 8r.
\nso both the expressions are not equal.<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nSubtract.
\na. (21x) \u2013 (-16 + 7x)<\/p>\n

Answer:
\n28x + 16.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nsubtract the expressions.
\n21x – (-16 + 7x).
\n21x + (-1) (-16 + 7x).
\n21x + (-1)(-16) + (-1)(-7x).
\n21x + 16 + 7x.
\n28x + 16.<\/p>\n

b. (-13n) \u2013 (17 \u2013 5n)<\/p>\n

Answer:
\n18n – 17.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nsubtract the expressions.
\n(-13n) – (17 – 5n).
\n13n – 17 + 5n.
\n18n – 17.<\/p>\n

c. (4y \u2013 7) \u2013 (y \u2013 7)<\/p>\n

Answer:
\n3y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nsubtract the expressions.
\n(4y \u2013 7) \u2013 (y \u2013 7).
\n4y – 7 -y + 7.
\n3y.<\/p>\n

d. (-w + 0.4) – (-w – 0.4)<\/p>\n

Answer:
\n0.8.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nsubtract the expressions.
\n(-w + 0.4) – (-w – 0.4).
\n-w + 0.4 + w + 0.4.
\n0.8.<\/p>\n

Question 5.
\nJude has 5 pairs of sunglasses that cost the same in his online shopping cart but then decides to get only 2. Each pair of sunglasses is the same price. Let p represent the cost of each pair. Write an expression for the original cost, the updated cost, and the difference in cost.
\n\"Envision<\/p>\n

Answer:
\nThe expression for the original cost, the updated cost, and the difference in cost = $2.02.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nJude has 5 pairs of sunglasses that cost the same in his online shopping cart but then decides to get only 2.
\nEach pair of sunglasses is the same price. Let p represent the cost of each pair.
\n5 – 2 = 3p.
\n3 x $1.49 = 4.47.
\n$6.49 – $4.47 = $2.02.
\nso the difference in cost = $2.02.<\/p>\n

Question 6.
\nSubtract and simplify.
\n\\(\\frac{1}{6}\\)m – (-\\(\\frac{5}{8}\\)m + \\(\\frac{1}{3}\\))<\/p>\n

Answer:
\n53m + 1\/3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\\(\\frac{1}{6}\\)m – (-\\(\\frac{5}{8}\\)m + \\(\\frac{1}{3}\\)).
\n1\/6m + 5\/8m + 1\/3.
\n6 x 8 = 48.
\n48 + 5 = 53.
\n53m + 1\/3.<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Multimedia Leveled Practice In 7-9, fill in the missing signs or numbers.
\nQuestion 7.
\nRewrite the expression 14m – (5 + 8m)
\n14m \"Envision 5 \"Envision 8m<\/p>\n

Answer:
\nThe missing signs are minus.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n14m – (5 + 8m).
\n14m – 5 – 8m.
\n6m – 5.<\/p>\n

Question 8.
\nRewrite the expression 13d – (-9d – 4) without parentheses. without parentheses.
\n13d \"Envision 9d \"Envision 4<\/p>\n

Answer:
\nThe expression is 22d + 4.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 13d – (-9d – 4).
\n13d + 9d + 4.
\n22d + 4.
\nso the expression without parentheses is 22d + 4.<\/p>\n

Question 9.
\nWrite an equivalent expression to 8k – (5 + 2k) without parentheses. Then simplify.
\n8k – (5 + 2k) = 8k \"Envision 5 \"Envision 2k
\n= 8k \"Envision 2k \"Envision 5
\n= \"Envision k \"Envision 5<\/p>\n

Answer:
\nThe equivalent expression is 6k – 5.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n8k – (5 + 2k).
\n8k – 5 – 2k.
\n6k – 5.<\/p>\n

Question 10.
\nA company has two manufacturing plants with daily production levels of 5x + 11 items and 2x – 3 items, respectively, where x represents a minimum quantity. The first plant produces how many more items daily than the second plant?<\/p>\n

Answer:
\nThe first plant produces 3 more items daily than the second plant.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA company has two manufacturing plants with daily production levels of 5x + 11 items and 2x – 3 items.
\n5x + 11 – 2x – 3.
\n3x + 8.
\nso the first plant produces 3 more items daily than the second plant.<\/p>\n

Question 11.
\nTwo communications companies offer calling plans. With Company X, it costs 35\u00a2 to connect and then 5\u00a2 for each minute. With Company Y, it costs 15\u00a2 to connect and then 4\u00a2 for each minute.
\nWrite and simplify an expression that represents how much more Company X charges than Company Y, in cents, for n minutes.<\/p>\n

Answer:
\nThe expression that represents company X charges than company Y, in cents, for n minutes =<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTwo communications companies offer calling plans. With Company X, it costs 35\u00a2 to connect and then 5\u00a2 for each minute.
\nWith Company Y, it costs 15\u00a2 to connect and then 4\u00a2 for each minute.
\n35 + 5 = 40\u00a2.
\n15 + 4 = 19\u00a2.
\n40 – 19 = 21\u00a2.
\nso the expression that represents company x charges than company y, in cents, for n minutes = 21\u00a2.<\/p>\n

Question 12.
\nMake Sense and Persevere The base and height of a triangle are each extended 2 cm. What is the area of the shaded region? How do you know?
\n\"Envision<\/p>\n

Answer:
\nThe area of the shaded region = x cm.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe base and height of a triangle are each extended 2 cm.
\narea = 1\/2 x b x h.
\narea = 1\/2 x 2 x x.
\narea = 2\/2x.
\narea = x cm.
\nso the area of the shaded region is x cm.<\/p>\n

Question 13.
\nTwo friends shop for fresh fruit. Jackson buys a watermelon for $7.65 and 5 pounds of cherries. Tim buys a pineapple for $2.45 and 4 pounds of cherries. Use the variable p to represent the price, in dollars, per pound of cherries. Write and simplify an expression to represent how much more Jackson spent.<\/p>\n

Answer:
\nThe expression to represent how much more jackson spent = $6.2p.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTwo friends shop for fresh fruit.
\nJackson buys a watermelon for $7.65 and 5 pounds of cherries.
\nTim buys a pineapple for $2.45 and 4 pounds of cherries.
\n$2.45 + 4 = $6.45.
\n$7.65 + 5 = $12.65.
\n$12.65 – $6.45 = $6.2.
\nso the expression to represent how much more jacson spent = $6.2p.<\/p>\n

Question 14.
\nYu’s family wants to rent a car to go on vacation. Envocar charges $50.50 and 8\u00a2 per mile. Freedomride charges $70.50 and 12\u00a2 per mile. How much more does Freedomride charge for driving d miles than EnvoCar?<\/p>\n

Answer:
\nThe much does Freedomride charge for driving d miles than Envo Car = $20 and 4\u00a2 .<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nYu’s family wants to rent a car to go on vacation.
\nEnvocar charges $50.50 and 8\u00a2 per mile.
\nFreedomride charges $70.50 and 12\u00a2 per mile.
\n$50.50 and 8\u00a2 .
\n$70.50 and 12\u00a2 .
\n70.50 – 50.50 and 12 – 8.
\n$20 and 4\u00a2 .<\/p>\n

Question 15.
\nA rectangular garden has a walkway around it. Find the area of the walkway.
\n\"Envision<\/p>\n

Answer:
\nThe area of the walkway = 168x + 136.5 sq ft.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe area of the walkway = l x b.
\narea = (8x + 6.5 ft ) x 21 ft.
\narea = 168x + 136.5 square ft.<\/p>\n

Question 16.
\nCritique Reasoning<\/strong> Tim incorrectly rewrote the expression \\(\\frac{1}{2}\\)p – (\\(\\frac{1}{4}\\)p + 4) as \\(\\frac{1}{2}\\)p + \\(\\frac{1}{4}\\)p – 4. Rewrite the expression without parentheses. What was Tim’s error?<\/p>\n

Answer:
\nThe Tim’s error = 1\/2p – 4.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nTim incorrectly rewrote the expression \\(\\frac{1}{2}\\)p – (\\(\\frac{1}{4}\\)p + 4).
\n\\(\\frac{1}{2}\\)p + \\(\\frac{1}{4}\\)p – 4.
\n1\/2p – 1\/4p – 4.
\n1\/2p – 4.<\/p>\n

Question 17.
\nHigher Order Thinking<\/strong> Find the difference.
\n(7x – 6\\(\\frac{2}{3}\\)) – (-3x +4\\(\\frac{3}{4}\\))<\/p>\n

Answer:
\nThe difference is 10x – 7.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(7x – 6\\(\\frac{2}{3}\\)) – (-3x +4\\(\\frac{3}{4}\\))
\n7x – 6 x 2\/3 – (-3x + 4 (3\/4).
\n7x – 6 (2\/3) + 3x – 4 (-3\/4).
\n10x -4 – 3.
\n10x – 7.<\/p>\n

Question 18.
\nEach month, a shopkeeper spends 5x + 11 dollars on rent and electricity. If he spends 2x – 3 dollars on rent, how much does he spend on electricity?<\/p>\n

Answer:
\nThe much he spend on electricity = 3x + 8.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nEach month, a shopkeeper spends 5x + 11 dollars on rent and electricity.
\nIf he spends 2x – 3 dollars on rent.
\n5x + 11 – 2x – 3.
\n3x + 8.<\/p>\n

Question 19.
\nUse the expression \\(\\frac{1}{4}\\)p – (1 – \\(\\frac{1}{3}\\)p).
\na. Rewrite the expression without parentheses. Simplify. Show your work.<\/p>\n

Answer:
\nThe expression is 1\/7p – 1.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\\(\\frac{1}{4}\\)p – (1 – \\(\\frac{1}{3}\\)p).
\n1\/4 p – (1 – 1\/3p).
\n1\/4p – 1 + 1\/3p.
\n1\/7p – 1.<\/p>\n

b. Use a different method to write the expression without parentheses. Do not simplify.<\/p>\n

Answer:
\nThe expression is 1\/7p – 1.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\\(\\frac{1}{4}\\)p – (1 – \\(\\frac{1}{3}\\)p).
\n1\/4 p – (1 – 1\/3p).
\n1\/4p – 1 + 1\/3p.
\n1\/7p – 1.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 20.
\nAn expression is shown.
\n(0.25n – 0.3) \u2013 (0.8n – 0.25)
\nCreate an equivalent expression without parentheses.<\/p>\n

Answer:
\nThe equivalent expression without parantheses = 0.05 – 0.55n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(0.25n – 0.3) \u2013 (0.8n – 0.25).
\n0.25n – 0.3 – 0.8n + 0.25.
\n-0.55n + 0.05.<\/p>\n

Lesson 4.8 Analyze Equivalent Expressions<\/h3>\n

Solve & Discuss It!<\/strong>
\nHow many toothpicks make a triangle? Two triangles? Write an expression that represents the number of toothpicks needed to make x triangles that appear side-by-side in a single row, as shown. Explain your reasoning.
\nI can… use an equivalent expression to find new information.
\n\"Envision<\/p>\n

Look for Relationships
\nWhat do you notice about the number of toothpicks needed for more than 1 triangle?<\/p>\n

Focus on math practices
\nReasoning Can there be more than one expression that represents the total number of toothpicks needed to make x triangles in the arrangement shown? Explain.<\/p>\n

Essential Question<\/strong>
\nHow can writing equivalent expressions show how quantities are related?<\/p>\n

Try It!<\/strong><\/p>\n

Joe is buying gift cards that are on sale for 15% off. He uses c – 0.15c to determine the sale price of gift cards. What is an equivalent expression that Joe could also use to determine the sale price of a gift card?
\n\"Envision<\/p>\n

Answer:
\nThe equivalent expression that Joe could use to determine the sale price = 0.15 – 0.15c.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nJoe is buying gift cards that are on sale for 15% off.
\nHe uses c – 0.15c to determine the sale price of gift cards.
\n15\/100 – 0.15c.
\n0.15 – 0.15c.<\/p>\n

Convince Me! How do you know if an expression is describing a percent increase or a percent decrease?<\/p>\n

Try It!<\/strong><\/p>\n

The total area, in square feet, of a rectangular stage that has been widened by x feet is represented by 1,900 + 76x. Use the Distributive Property to factor the expression. What does each factor in the equivalent expression tell you about the stage?
\n\"Envision<\/p>\n

Answer:
\nThe equivalent expression about the stage = 1900x + 76 x square.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe total area, in square feet, of a rectangular stage that has been widened by x feet is represented by 1,900 + 76x.
\n1900 + 76x x x.
\n1900x + 76xsquare.<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Rewriting expressions can clarify relationships among quantities or variables. When you rewrite an expression, you are writing an equivalent expression.
\n4x + 12 is equivalent to 4(x + 3) is equivalent to x + x + x + x + 3 + 3 + 3 + 3<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question How can writing equivalent expressions show how quantities are related?<\/p>\n

Answer:
\nThe equivalent expressions are 4x + 12.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n4x + 12.
\n4(x + 3).
\nx + x + x + x + 3 + 3 + 3 + 3.<\/p>\n

Question 2.
\nUse Structure The total area, in square feet, of a rectangular mural that has been extended by x feet is represented by 5.5(7.5 + x). Expand the expression using the Distributive Property. What do each of the terms in the equivalent expression tell you about the mural?
\n\"Envision<\/p>\n

Answer:
\nThe expressions tell you about the mural = 41.25 + 5.5x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 5.5(7.5 + x).
\n5.5 x 7.5 + 5.5 x x.
\n41.25 + 5.5x.
\nso the equivalent expression is 41.25 + 5.5x.<\/p>\n

Question 3.
\nThe expression (2x + 6) + x represents the perimeter of an isosceles triangle. If x represents the length of one side of the triangle, explain how you can use the Distributive Property to find the length of each of the two equivalent sides?<\/p>\n

Answer:
\nThe Distributive property to find the length of each of the two equivalent sides = 2x square + 6x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe expression (2x + 6) + x represents the perimeter of an isosceles triangle.
\n(2x + 6) + x.
\n2x X x + 6x.
\nso the distributive property to find the length of each of the two equivalent sides = 2x x x + 6x.<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nRewrite the expression 12x + 8 to find an equivalent expression. Show three possible expressions. What do the rewritten expressions tell you about the relationships among the quantities?<\/p>\n

Answer:
\nThe expression are 2(6x + 4) and 4(3x + 2).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nRewrite the expression 12x + 8 to find an equivalent expression.
\n2 (6x + 4).
\n4 (3x + 2).<\/p>\n

Question 5.
\nA rope is used to make a fence in the shape of an equilateral triangle around a newly planted tree. The length of the rope is represented with the expression 9x + 15
\n\"Envision
\na. Rewrite the expression to represent the three side lengths of the rope fence.<\/p>\n

Answer:
\nThe length of the rope is represented with the expression =<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n9x + 15.
\n3(3x + 5).
\n3 x 3x = 9x.
\n3 x 5 = 15.
\nso the length of the rope is 3(3x + 5).<\/p>\n

b. What is the length of one side?<\/p>\n

Answer:
\nThe length of one side = 3x.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n9x + 15.
\n3(3x + 5).
\nso the length of one side = 3x.<\/p>\n

Question 6.
\nThe expression (x – 0.35x) represents 35% off the cost of an item x. How is this equivalent to multiplying x by 0.65?<\/p>\n

Answer:
\nThe expression equivalent to multiplying = 0.4225.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe expression (x – 0.35x) represents 35% off the cost of an item x.
\nx = 0.65.
\n0.65 – 0.35(0.65).
\n0.65 – 0.2275.
\n0.4225.<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nReasoning Eric is planning an event at a hotel. Let g stand for the number of Eric’s guests. The two expressions represent the difference between the cost of the rooms. Expression 1: (326 + 37g) – (287 + 23g). Expression 2: 39 + 14g. What can you tell about Expression 2 and Expression 1?
\n\"Envision<\/p>\n

Answer:
\nThe two expressions are same.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nEric is planning an event at a hotel. Let g stand for the number of Eric’s guests.
\nThe two expressions represent the difference between the cost of the rooms.
\nExpression 1: (326 + 37g) – (287 + 23g). Expression 2: 39 + 14g.
\n326 + 37g – 287 – 23g.
\n39 + 14g.
\nso the expressions 1 and 2 are same.<\/p>\n

Question 8.
\nA student received a coupon for 17% off the total purchase price at a clothing store. Let b be the original price of the purchase. Use the expression b-0.17b for the new price of the purchase. Write an equivalent expression by combining like terms.<\/p>\n

Answer:
\nThe equivalent expression is 0.16b.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA student received a coupon for 17% off the total purchase price at a clothing store.
\nLet b be the original price of the purchase.
\nUse the expression b-0.17b for the new price of the purchase.
\n0.17b – b.
\n0.16b.
\nso the equivaent expression is 0.16b.<\/p>\n

Question 9.
\nKirana buys boxes of crackers that each have the same cost, c. She represents the cost of 3 boxes of cheese crackers, 2 boxes of poppy seed crackers, and 2 boxes of plain crackers using the expression 3c + 2c + 2c. What equivalent expression can represent the cost?<\/p>\n

Answer:
\nThe equivalent epression that represent the cost = 7c.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nKirana buys boxes of crackers that each have the same cost c.
\nShe represents the cost of 3 boxes of cheese crackers.
\n2 boxes of poppy seed crackers, and 2 boxes of plain crackers using the expression 3c + 2c + 2c.
\n3c + 2c + 2c.
\n5c + 2c.
\n7c.
\nso the equivalent expression that represent the cost = 7c.<\/p>\n

Question 10.
\nA student received a coupon for 14% off the total purchase price at a clothing store. Let c be the original price of the purchase. The expression c – 0.14c represents the new price of the purchase. Write an equivalent expression to show another way to represent the new price.<\/p>\n

Answer:
\nThe equivalent expression to show another way to represent the new price = 0.13c.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA student received a coupon for 14% off the total purchase price at a clothing store.
\nLet c be the original price of the purchase.
\nThe expression c – 0.14c represents the new price of the purchase.
\nc – 0.14c.
\n0.13c.
\nso the equivalent expression to show another way to represent the new price = 0.13c.<\/p>\n

Question 11.
\nA farmer recently sold a large plot of land. The sale decreased his total acreage by 8%. Let v be the original acreage.
\na. Find two equivalent expressions that will give the new acreage.<\/p>\n

Answer:
\nThe two equivalent expressions that will give the new acreage = v – 0.08v and 0.07v.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA farmer recently sold a large plot of land.
\nThe sale decreased his total acreage by 8%.
\nv – 0.08v.
\n0.07v.<\/p>\n

b. Use the expressions to describe two ways to find the new acreage.<\/p>\n

Answer:
\nThe two equivalent expressions that will give the new acreage = v – 0.08v and 0.07v.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA farmer recently sold a large plot of land.
\nThe sale decreased his total acreage by 8%.
\nv – 0.08v.
\n0.07v.<\/p>\n

Question 12.
\nAn art teacher enlarged the area of a copy of a painting by 49%. Let d represent the area of the original painting. The expression d + 0.49d is one way to represent the area of the new painting. Write two additional expressions that will give the area of the new painting.<\/p>\n

Answer:
\nThe two additional expressions that will give the area of the new painting = d + 0.49d and 0.50d.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAn art teacher enlarged the area of a copy of a painting by 49%.
\nLet d represent the area of the original painting.
\nThe expression d + 0.49d is one way to represent the area of the new painting.
\nso the expressions are d nd d + 0.49d.
\n0.50d.<\/p>\n

Question 13.
\nUse Structure The area of a rectangular playground has been extended on one side. The total area of the playground, in square meters, can be written as 352 + 22x. Rewrite the expression to give a possible set of dimensions for the playground.<\/p>\n

Answer:
\nThe expression to give a possible set of dimensions for the playground = 22(16 + x).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe area of a rectangular playground has been extended on one side.
\nThe total area of the playground, in square meters, can be written as 352 + 22x.
\n352 + 22x.
\n22(16 + x).
\n22 x 16 = 352.
\n22 x x = 22x.<\/p>\n

Question 14.
\nThe manager of a store increases the price of the bathing suits by 7%. Let t be the original price of a bathing suit. The new price is t + 0.07t. a. Find an expression equivalent to t + 0.07t.<\/p>\n

Answer:
\nThe expression equivalent to t + 0.07t = 0.08t.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe manager of a store increases the price of the bathing suits by 7%.
\nLet t be the original price of a bathing suit.
\nThe new price is t + 0.07t.
\n0.08t.<\/p>\n

b. If the original price of a bathing suit was $19.99, estimate the new price by first rounding the original price to the nearest dollar.<\/p>\n

Answer:
\nThe original price to the nearest dollar = $20 + $1.4t.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nIf the original price of a bathing suit was $19.99.
\n19.99 = 20.
\n20 + 0.07 x 20.
\n20 + 1.4t.<\/p>\n

Question 15.
\nHigher Order Thinking<\/strong> A customer at a clothing store is buying a pair of pants and a shirt. The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase. Let n represent the original price of the pants and s represent the price of the shirt
\n\"Envision
\na. Write two expressions that represent the “15% off sale on all pants” option.<\/p>\n

Answer:
\nThe expressions that represents is n – 0.015.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA customer at a clothing store is buying a pair of pants and a shirt.
\nThe customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase.
\nn – 0.015.<\/p>\n

b. Write two expressions that represent the “10% off her entire purchase” option.<\/p>\n

Answer:
\nThe expressions that represents is n – 0.010.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA customer at a clothing store is buying a pair of pants and a shirt.
\nThe customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase.
\nn – 0.010.<\/p>\n

c. If the original cost of the pants is $25 and the shirt is $10, which option should the customer choose? Explain.<\/p>\n

Answer:
\nThe customer choose the both pants and shirts = $35.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nIf the original cost of the pants is $25 and the shirt is $10.
\n$25 + $10.
\n$35.
\nso the customer choose the both pants and shirts = $35.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 16.
\nAt a college, the cost of tuition increased by 10%. Let b represent the former cost of tuition. Use the expression b + 0.10b for the new cost of tuition.
\nPART A
\nWrite an equivalent expression for the new cost of tuition.<\/p>\n

Answer:
\nThe new cost of tuition is 0.11b.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nAt a college, the cost of tuition increased by 10%.
\nLet b represent the former cost of tuition.
\nb + 0.10b.
\n0.11b.
\nso the new cost of tution is 0.11b.<\/p>\n

PART B
\nWhat does your equivalent expression tell you about how to find the new cost of tuition?
\nAnswer:<\/p>\n

250
\n4-8 Analyze Equivalent Expressions<\/p>\n

Topic 4 Review<\/h3>\n

Topic Essential Question<\/strong>
\nHow can properties of operations help to generate equivalent expressions that can be used in solving problems?<\/p>\n

Vocabulary Review<\/strong>
\nComplete each definition and then provide an example of each vocabulary word.<\/p>\n

Vocabulary
\ncoefficient
\nconstant
\nvariable
\nfactor
\nexpression
\n\"Envision<\/p>\n

Answer:
\nA term that contains only a number is constant.
\nThe number part of the term that contains a variable is coefficient.
\nA variable is a letter that represents an unknown value.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA term that contains only a number is constant.
\nThe number part of the term that contains a variable is coefficient.
\nA variable is a letter that represents an unknown value.
\nfor example:
\n2x + 4y – 9.
\nwhere 2 and 4 are coefficients.
\nx and y are variables.
\n9 is the constant.<\/p>\n

Use Vocabulary in Writing<\/strong>
\nMembership in a digital library has a $5 startup fee and then costs $9.95 per month. Membership in a video streaming service costs $7.99 per month with no startup fee. Use vocabulary words to explain how this information could be used to write an expression for the total cost of both memberships after m months.<\/p>\n

Answer:
\nThe total cost of memberships after m months = $49.75 + $7.99m.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nMembership in a digital library has a $5 startup fee and then costs $9.95 per month.
\nMembership in a video streaming service costs $7.99 per month with no startup fee.
\n5 x $9.95 + $7.99m.
\n$49.75 + $7.99m.
\nso the total cost of memberships after m months = $49.75 + $7.99m.<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

Lesson 4.1 Write and Evaluate Algebraic Expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can use an algebraic expression to represent and solve a problem with unknown values. The expression can consist of coefficients, constants, and variables. You can substitute values for variables to evaluate expressions.<\/p>\n

Example<\/strong>
\nA farm charges $1.75 for each pound of strawberries picked and $2 for a basket to hold the strawberries. What is the total cost to pick 5 pounds of strawberries?<\/p>\n

Answer:
\nThe total cost to pick 5 pounds of strawberries = $10.75.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nA farm charges $1.75 for each pound of strawberries picked and $2 for a basket to hold the strawberries.
\n($1.75 x $5) + 2.
\n($8.75) + 2.
\n$10.75.<\/p>\n

Write an expression to represent the total cost in dollars to pick p pounds of strawberries.
\n1.75p + 2
\nSubstitute 5 for p.
\n1.75(5) + 2 = 8.75 + 2 = 10.75
\nIt costs $10.75 to pick 5 pounds of strawberries.<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nHaddie makes and sells knit scarves. Next week she will pay a $25 fee for the use of a booth at a craft fair. She will charge $12 for each scarf she sells at the fair. Write an expression to determine Haddie’s profit for selling s scarves after paying the fee for the use of the booth.<\/p>\n

Answer:
\nThe expression to determine Haddie’s profit for selling scarves = $37.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nHaddie makes and sells knit scarves.
\nNext week she will pay a $25 fee for the use of a booth at a craft fair.
\nShe will charge $12 for each scarf she sells at the fair.
\n$25 + $12.
\n$37.
\nso the expression to determine Haddie’s profit for selling scarves = $37.<\/p>\n

Question 2.
\nThe cost to buy p pounds of potatoes at $0.32 per pound and n pounds of onions at $0.48 per pound can be determined by using the expression 0.32p + 0.48n. How much will it cost to buy 4.5 pounds of potatoes and 2.5 pounds of onions?<\/p>\n

Answer:
\nThe cost to buy 4.5 pounds of potatoes and 2.5 pounds of onions = 0.4608 + 1.2n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nThe cost to buy p pounds of potatoes at $0.32 per pound and
\nn pounds of onions at $0.48 per pound can be determined by using the expression 0.32p + 0.48n.
\n0.32 x 4.5p = 0.4608p.
\n0.48 x 2.5n = 1.2.
\n0.4608 + 1.2n.<\/p>\n

Lessons 4-2 AND 4-3 Generate Equivalent Expressions and Simplify Expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can use properties of operations and combine like terms to simplify expressions. Like terms are terms that have the same variable part.<\/p>\n

Example<\/strong>
\nSimplify the expression below.
\n-7 + \\(\\frac{1}{3}\\)n – \\(\\frac{4}{3}\\) + 2n
\nUse the Commutative Property to put like terms together,
\n\\(\\frac{1}{3}\\)n + 2n \u2013 7 – \\(\\frac{4}{3}\\)
\nCombine like terms.
\n2\\(\\frac{1}{3}\\)n – 8\\(\\frac{1}{3}\\)<\/p>\n

Practice<\/strong>
\nSimplify each expression below.
\nQuestion 1.
\n\\(\\frac{5}{8}\\)m + 9 – \\(\\frac{3}{8}\\)m – 15<\/p>\n

Answer:
\n3\/8 – 6.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n\\(\\frac{5}{8}\\)m + 9 – \\(\\frac{3}{8}\\)m – 15.
\nUse the Commutative Property to put like terms together,
\n5\/8 – 3\/8 + 9 – 15.
\ncombine like terms.
\n3\/8 – 6.<\/p>\n

Question 2.
\n-8w + (-4z) + 2 + 6w + 9z – 7<\/p>\n

Answer:
\n5z – 2w – 5.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-8w + (-4z) + 2 + 6w + 9z – 7.
\nUse the Commutative Property to put like terms together,
\n6w – 8w -4z + 9z -7 + 2.
\ncombine like terms.
\n-2w + 5z -5.
\n5z – 2w – 5.<\/p>\n

Question 3.
\n-6 + (-2d) + (-4d) + 3d<\/p>\n

Answer:
\n-3(d + 2).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n-6 + (-2d) + (-4d) + 3d.
\nUse the Commutative Property to put like terms together,
\n-6 – 2d – 4d + 3d.
\ncombine like terms.
\n3d – 6d -6.
\n-3d – 6.
\n-3(d + 2).<\/p>\n

Lesson 4.4 Expand Expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nThe Distributive Property allows you to multiply each term inside parentheses by a factor that is outside the parentheses. This means that you can use the Distributive Property to expand expressions.<\/p>\n

Example<\/strong>
\nExpand the expression (6 + 7).
\n(\\(\\frac{1}{4}\\) \u00d7 h) + (\\(\\frac{1}{4}\\) \u00d7 7) = \\(\\frac{1}{4}\\)h + 1.75<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nExpand the expression 3.5(-3n + 4).<\/p>\n

Answer:
\nThe expression is -10.5n + 14.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 3.5(-3n + 4).
\n3.5 x -3n + 3.5 x 4.
\n-10.5n + 14.
\nso the expression is -10.5n + 14.<\/p>\n

Question 2.
\nSimplify the expression –\\(\\frac{3}{5}\\)(-8 + \\(\\frac{5}{9}\\)x – 3).<\/p>\n

Answer:
\n5\/9 + 24\/5 – 3.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression –\\(\\frac{3}{5}\\)(-8 + \\(\\frac{5}{9}\\)x – 3).
\n-3\/5 -8 + 5\/9 x – 3.
\n24\/5 + 5\/9x – 3.
\n5\/9x + 24\/5 – 3.<\/p>\n

Lesson 4.5 Factor Expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nWhen you factor an expression, you write it as a product of two expressions. The new expression is equivalent to the original expression. The greatest common factor (GCF) and the Distributive Property are tools that you use when you need to factor an expression.<\/p>\n

Example<\/strong>
\nFactor the expression 12x – 9y + 15.
\nThe GCF of 12x, 15, and -9y is 3.
\nRewrite each term using 3 as a factor.
\n12x = 3 \u2022 4x
\n-9y = 3 \u2022 (-3y)
\n15= 3 \u2022 5
\nUse the Distributive Property to factor the expression.
\n3(4x – 3y + 5)<\/p>\n

Practice<\/strong>
\nFactor each expression.
\nQuestion 1.
\n63a – 42b<\/p>\n

Answer:
\n3(21a – 14b).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 63a – 42b.
\n3(21a – 14b).
\n3 x 21a = 63a.
\n3 x 14b = 42b.<\/p>\n

Question 2.
\n81y + 54<\/p>\n

Answer:
\n9(9y + 6).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 81y + 54.
\n9(9y + 6).
\n9 x 9y = 81y.
\n9 x 6 = 54.<\/p>\n

Question 3.
\nWhich show a way to factor the expression 32t – 48? Select all that apply.
\n\u2610 2(16t – 24)
\n\u2610 4(12t – 48)
\n\u2610 6(26 – 42)
\n\u2610 8(4t – 6)
\n\u2610 16(2t – 3)<\/p>\n

Answer:
\n2(16t – 24) and 8(4t – 6).<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nthe expression is 32t – 48.
\n2(16t – 24).
\n2 x 16t = 32t.
\n2 x 24 = 48.
\n8(4t – 6).
\n8 x 4t = 32t.
\n8 x 6 = 48.<\/p>\n

Lessons 4.6 AND 4.7 Add and Subtract Expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nAdding and subtracting expressions may require combining like terms. This means that you must use the Commutative and Associative Properties to reorder and group terms as needed.<\/p>\n

Example<\/strong>
\nKerry has n markers. Rachel has 1 marker fewer than twice the number of markers Kerry has. Write and simplify an expression for the total number of markers they have.
\nNumber of markers Kerry has: n
\nNumber of markers Rachel has: 2n – 1
\nTotal number of markers:
\nn + (2n – 1)
\n(n + 2n) – 1
\n3n – 1<\/p>\n

Practice<\/strong>
\nAdd the expressions.
\nQuestion 1.
\n5.2C \u2013 7.35) + (-3.9C + 2.65)<\/p>\n

Answer:
\n1.3c – 4.7.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n5.2C \u2013 7.35) + (-3.9C + 2.65).
\n5.2c – 7.35 – 3.9c + 2.65.
\n5.2c – 3.9c + 2.65 – 7.35.
\n1.3c – 4.7.<\/p>\n

Question 2.
\n(6x \u2013 2y \u2013 5) – (-5 + 9y \u2013 8x)<\/p>\n

Answer:
\n14x – 11y.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\n(6x \u2013 2y \u2013 5) – (-5 + 9y \u2013 8x).
\n6x – 2y – 5 + 5 – 9y + 8x.
\n14x – 11y.<\/p>\n

Question 3.
\nLast week Jean ran 2 fewer than 4m miles. This week she ran 0.5 miles more than last week. Write and simplify an expression for the total number of miles Jean ran in the two weeks.<\/p>\n

Answer:
\nThe total number of miles Jean ran in the two weeks = 2.5m.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nLast week Jean ran 2 fewer than 4m miles.
\nThis week she ran 0.5 miles more than last week.
\n2m + 0.5m.
\n2.5m.
\nso the total number of miles Jean ran in the two weeks = 2.5m.<\/p>\n

Lesson 4.8 Analyze Equivalent Expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nEquivalent expressions can help to show new information about a problem. Sometimes the equivalent expression will be an expanded expression. In other cases, it will be a factored expression.<\/p>\n

Example<\/strong>
\nThe perimeter of a square is represented with the expression 84 + 44s. What is the length of one side of the square?
\nA square has 4 sides, so factor 4 out of each term in the expression for the perimeter.
\n84 + 445 = 4 \u2022 21 + 4 \u2022 11s = 4(21 + 115)
\nThe factor within the parentheses represents the length of one side of the square.
\nThe length of one side is 21 + 11s.<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nHal earns n dollars per hour. Next month he will receive a 2% raise in pay per hour. The expression n + 0.02n is one way to represent Hal’s pay per hour after the raise. Write an equivalent simplified expression that will represent his pay per hour after the raise.<\/p>\n

Answer:
\nThe equivalent expression that will represent his pay per hour after the raise = 0.03n.<\/p>\n

Explanation:
\nIn the above-given question,
\ngiven that,
\nHal earns n dollars per hour.
\nNext month he will receive a 2% raise in pay per hour.
\nThe expression n + 0.02n is one way to represent Hal’s pay per hour after the raise.
\nn + 0.02n = 0.03n.<\/p>\n

Question 2.
\nThe area of a garden plot can be represented by the expression 84z – 54. The garden will be divided into six sections for planting six different vegetables. The sections will be equal in area. Write an expression that represents the area of each section.<\/p>\n

Answer:
\nThe expression that represents the area of each section =<\/p>\n

Explanati<\/p>\n

Topic 4 Fluency Practice<\/h3>\n

Hidden Clue
\nFor each ordered pair, solve the percent problems to find the coordinates. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help you answer the riddle below.<\/p>\n

I can… represent and solve percent problems.
\n\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

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