# Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations

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## Go Math Grade 8 Chapter 7 Solving Linear Equations Answer Key

Improve your performance in exams with the help of Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations. You can find the review questions at the end of the chapter to test your knowledge. The topics covered in this Solving Linear Equations chapter are equations with the variable on both sides, equations with rational numbers, equations with the distributive property, equations with many solutions or no solution, etc. Just go through the online pdf and start practicing now. By looking into the questions and answers available on Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations.

Lesson 1: Equations with the Variable on Both Sides

Lesson 2: Equations with Rational Numbers

Lesson 3: Equations with the Distributive Property

Lesson 4: Equations with Many Solutions or No Solution

Lesson 5: Equations with the Variable on Both Sides

Reviews

### Guided Practice – Equations with the Variable on Both Sides – Page No. 200

Use algebra tiles to model and solve each equation.

Question 1.
x + 4 = -x – 4
x = ________

Answer:
x = -4

Explanation:
Model x + 4 on the left side of the mat and -x -4 on the right side.

Add one c-tile to both sides. This represents adding x to both sides of the equation. Remove zero pairs.

Place four -1-tiles on both sides. This represents subtracting -4 from both sides of the equation. Remove zero pairs.

Separate each side into 2 equal groups. One x-tile is equivalent to four -1-tiles.

x = -4

Question 2.
2 – 3x = -x – 8
x = ________

Answer:

Explanation:
Given 2 – 3x = -x – 8
Model 2-3x on the left side of the mat and -x-8 on the right side.

Place one x tile to both sides. This represents subtracting from both sides of the equation.

Remove 2 1 tiles from sides. This represents subtracting from both sides of the equation.

Separate each side into 2 equal groups. One -x tile is equivalent to 5 – 1 tile.

The solution is -x = -5 or x = 5

Question 3.
At Silver Gym, membership is $25 per month, and personal training sessions are$30 each. At Fit Factor, membership is $65 per month, and personal training sessions are$20 each. In one month, how many personal training sessions would Sarah have to buy to make the total cost at the two gyms equal?
________ sessions

Answer:
4 sessions

Explanation:
At Silver Gym, membership is $25 per month, and personal training sessions are$30 each.
Membership + Personal training session = 25 + 30x
At Fit Factor, membership is $65 per month, and personal training sessions are$20 each.
Membership + Personal training session = 65 + 20x
Membership at Silver Gym = Membership at Fit Factor
25 + 30x = 65 + 20x
30x – 20x = 65 – 25
10x = 40
x = 4
Sarah would have to buy 4 sessions for the total cost at the two gyms to be equal.

Question 4.
Write a real-world situation that could be modeled by the equation 120 + 25x = 45x.
Type below:
_______________

Answer:
120 + 25x = 45x
Sarah offers a plan to tutor a student at $25 per her plus a one-time registration fee of$ 120.
y = 2x + 45
y = 2(11) + 45 = $67 Since when x > 10, Company B is cheaper so it should be better to use when the length of the call is greater than 10 minutes. H.O.T. Focus on Higher Order Thinking Question 13. Draw Conclusions Liam is setting up folding chairs for a meeting. If he arranges the chairs in 9 rows of the same length, he has 3 chairs left over. If he arranges the chairs in 7 rows of that same length, he has 19 left over. How many chairs does Liam have? ______ chairs Answer: 75 chairs Explanation: Number of chairs in 9 row + left over chairs = 9x + 3 Number of chairs in 7 row + left over chairs = 7x + 19 9x + 3 = 7x + 19 9x – 7x = 19 – 3 2x = 16 x = 16/2 x = 8 Total number of chairs = 9(8) + 3 = 75 Question 14. Explain the Error Rent-A-Tent rents party tents for a flat fee of$365 plus $125 a day. Capital Rentals rents party tents for a flat fee of$250 plus $175 a day. Delia wrote the following equation to find the number of days for which the total cost charged by the two companies would be the same: 365x + 125 = 250x + 175 Find and explain the error in Delia’s work. Then write the correct equation. Type below: ____________ Answer: Delia’s equation 365x + 125 = 250x + 175 The error is that she attached the variable with the flat fee (which is constant) and put the daily rent as a constant (which is variable). Correct equation 125x + 365 = 175x + 250 Question 15. Persevere in Problem Solving Lilliana is training for a marathon. She runs the same distance every day for a week. On Monday, Wednesday, and Friday, she runs 3 laps on a running trail and then runs 6 more miles.On Tuesday and Sunday, she runs 5 laps on the trail and then runs 2 more miles. On Saturday, she just runs laps. How many laps does Lilliana run on Saturday? ________ laps Answer: 2 laps Explanation: Miles covered by lap + Addition number of miles = 3x + 6 Miles covered by lap + Addition number of miles = 5x + 2 3x + 6 = 5x + 2 5x – 3x = 6 – 2 2x = 4 x = 4/2 x = 2 ### Guided Practice – Equations with Rational Numbers – Page No. 206 Question 1. Sandy is upgrading her Internet service. Fast Internet charges$60 for installation and $50.45 per month. Quick Internet has free installation but charges$57.95 per month.
a. Write an equation that can be used to find the number of months at which the Internet service would cost the same.
Type below:
____________

Answer:
50.45x + 60 = 57.95x

Explanation:
Write an equation for Fast Internet, where x is the number of months.
Charge per Month × Number of Month + Installation Fee
50.45x + 60
Write an equation for Quick Internet, where x is the number of months.
Charge per Month × Number of Month + Installation Fee
57.95x
50.45x + 60 = 57.95x

Question 1.
b. Solve the equation.
_______ hours

Answer:
8

Explanation:
50.45x + 60 = 57.95x
57.95x – 50.45x = 60
7.5x = 60
x = 60/7.5
x = 8
The total cost will be the same for 8 months.

Solve.

Question 2.
$$\frac{3}{4}$$ n – 18 = $$\frac{1}{4}$$ n – 4
______

Answer:
n = 28

Explanation:
3/4 . n – 18 = 1/4 . n – 4
Determine the least common multiple of the denominators
LCM is 4
Multiply both sides of the equation by the LCM
4(3/4 . n – 18) = 4(1/4 . n – 4)
3n – 72 = n – 16
3n – n = -16 + 72
2n = 56
n = 56/2
n = 28

Question 3.
6 + $$\frac{4}{5}$$ b = $$\frac{9}{10}$$ b
_______

Answer:
b = 60

Explanation:
6 + $$\frac{4}{5}$$ b = $$\frac{9}{10}$$ b
LCM is 10
10(6 + $$\frac{4}{5}$$ b) = 10($$\frac{9}{10}$$ b)
60 + 8b = 9b
9b – 8b = 60
b = 60

Question 4.
$$\frac{2}{11}$$ m + 16 = 4 + $$\frac{6}{11}$$ m
_______

Answer:
m = 33

Explanation:
$$\frac{2}{11}$$ m + 16 = 4 + $$\frac{6}{11}$$ m
The LCM is 11
11($$\frac{2}{11}$$ m + 16) = 11(4 + $$\frac{6}{11}$$ m)
2m + 176 = 44 + 6m
6m – 2m = 176 – 44
4m = 132
m = 132/4
m = 33

Question 5.
2.25t + 5 = 13.5t + 14
_______

Answer:
t = -0.8

Explanation:
2.25t + 5 = 13.5t + 14
13.5t – 2.25t = 5 – 14
11.25t = -9
t = -9/11.25
t = -0.8

Question 6.
3.6w = 1.6w + 24
_______

Answer:
w = 12

Explanation:
3.6w = 1.6w + 24
3.6w – 1.6w = 24
2w = 24
w = 24/2
w = 12

Question 7.
-0.75p – 2 = 0.25p
_______

Answer:
p = -2

Explanation:
-0.75p – 2 = 0.25p
-2 = 0.25p + 0.75p
-2 = p
p = -2

Question 8.
Write a real-world problem that can be modeled by the equation 1.25x = 0.75x + 50.
Type below:
______________

Answer:
1.25x = 0.75x + 50.
Cell offer Plan A for no base fee and $1.25 per minute. Cell offer Plan B for a$50 base fee and $0.75 per minute. The equation shows when the total cost of the plan would be equal. Essential Question Check-In Question 9. How does the method for solving equations with fractional or decimal coefficients and constants compare with the method for solving equations with integer coefficients and constants? Type below: ______________ Answer: When solving equations with fractional or decimal coefficients, the equations need to be multiplied by the multiple of denominator such that the equations have integer coefficients and constants. ### Independent Practice – Equations with Rational Numbers – Page No. 207 Question 10. Members of the Wide Waters Club pay$105 per summer season, plus $9.50 each time they rent a boat. Nonmembers must pay$14.75 each time they rent a boat. How many times would a member and a non-member have to rent a boat in order to pay the same amount?
_______ times

Answer:
20 times

Explanation:
Members of the Wide Waters Club pay $105 per summer season, plus$9.50 each time they rent a boat.
9.5x + $105 Nonmembers must pay$14.75 each time they rent a boat.
9.5x + $105 = 14.75x 9.5x – 14.75x =$105
5.25x = 105
x = 105/5.25
x = 20
The cost for members and non-members will be the same for 8 visits.

Question 11.
Margo can purchase tile at a store for $0.79 per tile and rent a tile saw for$24. At another store she can borrow the tile saw for free if she buys tiles there for $1.19 per tile. How many tiles must she buy for the cost to be the same at both stores? _______ tiles Answer: 60 tiles Explanation: Margo can purchase tile at a store for$0.79 per tile and rent a tile saw for $24. 0.79x + 24 At another store she can borrow the tile saw for free if she buys tiles there for$1.19 per tile.
1.19x
0.79x + 24 = 1.19x
1.19x – 0.79x = 24
0.4x = 24
x = 24/0.4
x = 60
Margo should buy 60 tiles for the cost to be the same at both stores.

Question 12.
The charges for two shuttle services are shown in the table. Find the number of miles for which the cost of both shuttles is the same.

_______ miles

Answer:
40 miles

Explanation:
0.1x + 10
0.35x
0.1x + 10 = 0.35x
0.35x – 0.1x = 10
0.25x = 10
x = 10/0.25
x = 40
The cost of shuttles would be the same for 40 miles.

Question 13.
Multistep
Rapid Rental Car charges a $40 rental fee,$15 for gas, and $0.25 per mile driven. For the same car, Capital Cars charges$45 for rental and gas and $0.35 per mile. a. For how many miles is the rental cost at both companies the same? _______ miles Answer: 100 miles Explanation: 0.25x + 40 + 15 = 0.35x + 45 0.35x – 0.25x = 55 – 45 0.1x = 10 x = 10/0.1 x = 100 The cost of car rentals would be the same for 100 miles. Question 13. b. What is that cost?$ _______

Answer:
$80 Explanation: Let y be the total cost. Substitute 100 miles in any one of the two equations y = 0.35x + 45 y = 0.35(100) + 45 =$80
Total cost would be $80. Question 14. Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side. Type below: ______________ Answer: 4/3x + 10 = 50/3 + x Explanation: Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side. 1/3 . x = 1/3 . 20 1/3 . x + x = 1/3 . 20 + x 4/3x = 20/3 + x 4/3x + 10 = 20/3 + x + 10 4/3x + 10 = 50/3 + x Question 15. Write an equation with the solution x = 25. The equation should have the variable on both sides, a decimal coefficient on the left side, and a decimal anywhere on the right side. One of the decimals should be written in tenths, the other in hundredths. Type below: ______________ Answer: x=25 divide both sides by 25 x/25 = 1 convert 1/25 to decimal form 0.04 0.04x = 1 add x on both sides 1.04x = 1 + x add 0.1 on both sides 1.04x + 0.1 = x + 1.1 Question 16. Geometry The perimeters of the rectangles shown are equal. What is the perimeter of each rectangle? Perimeter = _______ Answer: Perimeter = 3.2 Explanation: Perimeter of the first rectangle P = 2(n + n + 0.6) = 2(2n + 0.6) = 4n + 1.2 Perimeter of the second rectangle P = 2(n + 0.1 + 2n) = 2(3n + 0.1) = 6n + 0.2 the perimeter is equal 4n + 1.2 = 6n + 0.2 6n – 4n = 1.2 – 0.2 2n = 1 n = 1/2 n = 0.5 P = 4n + 1.2 = 4(0.5) + 1.2 = 3.2 Question 17. Analyze Relationships The formula F = 1.8C + 32 gives the temperature in degrees Fahrenheit (F) for a given temperature in degrees Celsius (C). There is one temperature for which the number of degrees Fahrenheit is equal to the number of degrees Celsius. Write an equation you can solve to find that temperature and then use it to find the temperature Type below: ______________ Answer: x = 1.8x + 32 Explanation: F = 1.8C +32 let x be the temperature such that it is same in both celsius and in fahrenheit Then the required equation is x = 1.8x + 32 subtract 1.8x from both sides -0.8x = 32 divide by -0.8 on both sides x = -40 So -40 degree celsius ### Equations with Rational Numbers – Page No. 208 Question 18. Explain the Error Agustin solved an equation as shown. What error did Agustin make? What is the correct answer? x = _______ Answer: x = -12 Explanation: Agustin did not multiply by 12 on both sides in step 2. He only partially multiplied the variable and left the constants as such, which doesn’t make any sense. The correct solution is 12(x/3 – 4) = 12(3x/4 + 1) 4x – 48 = 9x + 12 subtract 12 on both sides 4x – 60 = 9x subtract 4x on both sides -60 = 5x x = -12 H.O.T. Focus on Higher Order Thinking Question 19. Draw Conclusions Solve the equation $$\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4$$. Explain your results. Type below: _____________ Answer: $$\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4$$ The least common multiple of the denominators: LCM(2, 3, 6) = 6 6($$\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4$$) 6.1/2x – 6.5 + 6.2/3x = 6.7/6x +6.4 3x – 30 + 4x = 7x + 24 7x – 30 = 7x + 24 -30 = 24 This is not true. The equation has no solution. Question 20. Look for a Pattern Describe the pattern in the equation. Then solve the equation. 0.3x + 0.03x + 0.003x + 0.0003x + .. = 3 x = ______ Answer: x = 9 Explanation: 0.3x + 0.03x + 0.003x + 0.0003x + .. = 3 0.3x = 3 0.9x = 9 x = 9 Question 21. Critique Reasoning Jared wanted to find three consecutive even integers whose sum was 4 times the first of those integers. He let k represent the first integer, then wrote and solved this equation : k + (k + 1) + (k + 2) = 4k. Did he get the correct answer? Explain. __________ Answer: No, it is wrong on two accounts. First, he has not specified if k is even or not. An easy way of doing so would assume x to be any integer and k=2a This ensures that k is an even integer. Nest the question asks for 3 consecutive even integers, Jared just took 3 consecutive integers, and thus at least 1 of them is odd. So correct representation would be k + (k+2) + (k + 4) = 4k which upon solving yields k=6 ### Guided Practice – Equations with the Distributive Property – Page No. 212 Solve each equation. Question 1. 4(x + 8) – 4 = 34 – 2x ________ Answer: x = 1 Explanation: 4(x + 8) – 4 = 34 – 2x 4x + 32 – 4 = 34 – 2x 4x + 2x = 34 – 28 6x = 6 x = 6/6 x = 1 Question 2. $$\frac{2}{3}$$(9 + x) = -5(4 – x) ________ Answer: x = 6 Explanation: $$\frac{2}{3}$$(9 + x) = -5(4 – x) 2/3(9 + x) = -5(4 – x) 3 (2/3(9 + x)) = 3(-5(4 – x)) 2(9 + x ) = -15 (4 – x) 18 + 2x = -60 + 15x 15x – 2x = 18 + 60 13x = 78 x = 78/13 x = 6 Question 3. -3(x + 4) + 15 = 6 – 4x ________ Answer: x = 3 Explanation: -3(x + 4) + 15 = 6 – 4x -3x – 12 + 15 = 6 – 4x -3x + 3 = 6 – 4x -3x + 4x = 6 – 3 x = 3 Question 4. 10 + 4x = 5(x – 6) + 33 ________ Answer: x = 7 Explanation: 10 + 4x = 5(x – 6) + 33 10 + 4x = 5x – 30 + 33 10 + 4x = 5x + 3 5x – 4x = 10 – 3 x = 7 Question 5. x – 9 = 8(2x + 3) – 18 ________ Answer: x = -1 Explanation: x – 9 = 8(2x + 3) – 18 x – 9 = 16x + 24 – 18 x – 9 = 16x + 6 16x – x = -9 – 6 15x = – 15 x = -15/15 x = -1 Question 6. -6(x – 1) – 7 = -7x + 2 ________ Answer: x = 3 Explanation: -6(x – 1) – 7 = -7x + 2 -6x + 6 – 7 = -7x + 2 -6x – 1 = -7x + 2 -7x + 6x = -1 -2 -x = -3 x = 3 Question 7. $$\frac{1}{10}$$(x + 11) = -2(8 – x) ________ Answer: x = 9 Explanation: $$\frac{1}{10}$$(x + 11) = -2(8 – x) 10($$\frac{1}{10}$$(x + 11)) = 10 (-2(8 – x)) x + 11 = -20(8 – x) x + 11 = -160 + 20x 20x – x = 11 + 160 19x = 171 x = 171/19 = 9 Question 8. -(4 – x) = $$\frac{3}{4}$$(x – 6) ________ Answer: x = -2 Explanation: -(4 – x) = $$\frac{3}{4}$$(x – 6) 4(-(4 – x)) = 4 (3/4(x – 6)) -16 + 4x = 3x – 18 4x – 3x = -18 + 16 x = -2 Question 9. -8(8 – x) = $$\frac{4}{5}$$(x + 10) ________ Answer: x = 10 Explanation: -8(8 – x) = $$\frac{4}{5}$$(x + 10) 5(-8(8 – x)) = 5($$\frac{4}{5}$$(x + 10)) -40(8 – x) = 4(x + 10) -320 + 40x = 4x + 40 40x – 4x = 40 + 320 36x = 360 x = 360/36 x = 10 Question 10. $$\frac{1}{2}$$(16 – x) = -12(x + 7) ________ Answer: x = 8 Explanation: $$\frac{1}{2}$$(16 – x) = -12(x + 7) 2 ($$\frac{1}{2}$$(16 – x)) = 2 (-12(x + 7)) 16 – x = -24 (x + 7) 16 – x = -24x – 168 24x – x = -168 – 16 23x = 184 x = 184/23 x = 8 Question 11. Sandra saves 12% of her salary for retirement. This year her salary was$3,000 more than in the previous year, and she saved $4,200.What was her salary in the previous year? Write an equation _____ Sandra’s salary in the previous year was _____ Salary =$ _____

Answer:
Write an equation 0.12x + 360 = 4200
Sandra’s salary in the previous year was $32000 Salary =$3000

Explanation:
0.12(x + 3000) = 4200
0.12x + 360 = 4200
0.12x = 4200 – 360
0.12x = 3840
x = 3840/0.12
x = 32000
Sandra’s salary in the previous year was $32000 Essential Question Check-In Question 12. When solving an equation using the Distributive Property, if the numbers being distributed are fractions, what is your first step? Why? Type below: ___________ Answer: Multiply both sides by the denominator of the fraction ### Independent Practice – Equations with the Distributive Property – Page No. 213 Question 13. Multistep Martina is currently 14 years older than her cousin Joey. In 5 years she will be 3 times as old as Joey. Use this information to answer the following questions. a. If you let x represent Joey’s current age, what expression can you use to represent Martina’s current age? Type below: ___________ Answer: y = x + 14 Explanation: y = x + 14 where x is Joey’s current age and t is Martna’s current age. Question 13. b. Based on your answer to part a, what expression represents Joey’s age in 5 years? What expression represents Martina’s age in 5 years? Type below: ___________ Answer: Ages in 5 years Joey’s age = x + 5 Martina’s age = x + 14 + 5 = x + 19 Question 13. c. What equation can you write based on the information given? Type below: ___________ Answer: 3(x + 5) = x + 19 Explanation: In 5 years, Martina will be three times as old as Joey 3(x + 5) = x + 19 Question 13. d. What is Joey’s current age? What is Martina’s current age? Joey’s current age ___________ Martina’s current age ___________ Answer: Joey’s current age 2 Martina’s current age 16 Explanation: 3(x + 5) = x + 19 3x + 15 = x + 19 3x – x = 19 – 15 2x = 4 x = 2 Question 14. As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.” What equation can Luis write based on Sarah’s clues and what is Sarah’s number? Type below: ___________ Answer: x = 12 Explanation: As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.” 4 (x – 5) + 7 = 35 4x – 20 + 7 = 35 4x – 13 = 35 4x = 35 + 13 4x = 48 x = 48/4 x = 12 Question 15. Critical Thinking When solving an equation using the Distributive Property that involves distributing fractions, usually the first step is to multiply by the LCD to eliminate the fractions in order to simplify computation. Is it necessary to do this to solve $$\frac{1}{2}$$(4x + 6) = 13(9x – 24)? Why or why not? ___________ Answer: It is not necessary. In this case, distributing the fractions directly results in whole-number coefficients and constants, however, if the results are not in whole-number coefficients and constants it is harder to solve fractions. Question 16. Solve the equation given in Exercise 15 with and without using the LCD of the fractions. Are your answers the same? ___________ Answer: x = 11 Explanation: $$\frac{1}{2}$$(4x + 6) = 13(9x – 24) 6($$\frac{1}{2}$$(4x + 6)) = 6(13(9x – 24)) 3(4x + 6) = 2(9x – 24) 12x + 18 = 18x – 48 18x – 12x = 18 + 48 6x = 66 x = 66/6 x = 11 ### Equations with the Distributive Property – Page No. 214 Question 17. Represent Real-World Problems A chemist mixed x milliliters of 25% acid solution with some 15% acid solution to produce 100 milliliters of a 19% acid solution. Use this information to fill in the missing information in the table and answer the questions that follow. a. What is the relationship between the milliliters of acid in the 25% solution, the milliliters of acid in the 15% solution, and the milliliters of acid in the mixture? Type below: _____________ Answer: The milliliters of acid in the 25% solution plus the milliliters of acid in the 15% solution equals the milliliters of acid in the mixture Explanation: Question 17. b. What equation can you use to solve for x based on your answer to part a? Type below: _____________ Answer: 0.25x + 0.15(100 – x) = 19 Question 17. c. How many milliliters of the 25% solution and the 15% solution did the chemist use in the mixture? Type below: _____________ Answer: 0.25x + 0.15(100 – x) = 19 0.25x + 15 – 0.15x = 19 0.1x + 15 = 19 0.1x = 4 x = 4/0.1 x = 40 The chemist used 40ml of the 25% solution and 100 – 40 = 60ml of the 15% solution. H.O.T. Focus on Higher Order Thinking Question 18. Explain the Error Anne solved 5(2x) – 3 = 20x + 15 for x by first distributing 5 on the left side of the equation. She got the answer x = -3. However, when she substituted -3 into the original equation for x, she saw that her answer was wrong. What did Anne do wrong, and what is the correct answer? x = ________ Answer: x = -1.8 Explanation: Dado que 5 solo se multiplica por 2x, no tiene sentido usar la distribución aquí. Básicamente, distribuir 5 fue el problema Solución correcta: 5 (2x) – 3 = 20x + 15 10x -3 = 20x + 15 restar 15 en ambos lados 10x – 18 = 20x restar 10x de ambos lados -18 = 10x x = -1.8 Question 19. Communicate Mathematical Ideas Explain a procedure that can be used to solve 5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45. Then solve the equation. x = ________ Answer: x = 1 Explanation: 5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45 5[3x + 12 – 2 + 2x] – x – 15 = 14x + 45 5[5x + 10] – x – 15 = 14x + 45 25x + 50 – x – 15 = 14x + 45 24x + 35 = 14x + 45 24x – 14x = 45 – 35 10x = 10 x = 1 ### Guided Practice – Equations with Many Solutions or No Solution – Page No. 218 Use the properties of equality to simplify each equation. Tell whether the final equation is a true statement. Question 1. The statement is: _______ Answer: The statement is: true Explanation: 3x – 2 = 25 – 6x 3x + 6x -2 = 25 -6x + 6x 9x – 2 = 25 9x -2 + 2 = 25 + 2 9x = 27 x = 27/9 x = 3 The statement is true. Question 2. ____________ Answer: The statement is false. Explanation: 2x – 4 = 2(x – 1) + 3 2x – 4 = 2x – 2 + 3 2x – 4 = 2x + 1 2x – 4 – 2x = 2x + 1 – 2x -4 not equal to 1 The statement is false. Question 3. How many solutions are there to the equation in Exercise 2? ____________ Answer: There is no solution to exercise 2. Question 4. After simplifying an equation, Juana gets 6 = 6. Explain what this means. ____________ Answer: When 6 = 6, there are infinite solutions. Write a linear equation in one variable that has infinitely many solutions. Question 5. Start with a _____ statement. Add the _____ to both sides. Add the _____ to both sides. Combine _____ terms. Type below: ____________ Answer: Start with a “true” statement Add the “same variable” to both sides Add the “same constant” to both sides Combine “like” terms Explanation: Start with a “true” statement 10 = 10 Add the “same variable” to both sides 10 + x = 10 + x Add the “same constant” to both sides 10 + x + 5 = 10 + x + 5 Combine “like” terms 15 + x = 15 + x Essential Question Check-In Question 6. Give an example of an equation with an infinite number of solutions. Then make one change to the equation so that it has no solution. Type below: ____________ Answer: An equation with infinitely many solutions x – 2x + 3 = 3 – x -x + 3 = 3 – x +x/3 = +x/3 An equation for no solution x – 2x + 3 = 3 – x + 4 -x + 3 = 7 – x -x/3 = -x/7 ### Independent Practice – Equations with Many Solutions or No Solution – Page No. 219 Tell whether each equation has one, zero, or infinitely many solutions. Question 7. -(2x + 2) – 1 = -x – (x + 3) ____________ Answer: The statement is true Explanation: -(2x + 2) – 1 = -x – (x + 3) -2x – 2 – 1 = -x – x + 3 -2x – 3 = -2x + 3 -3 = -3 The statement is true Question 8. -2(z + 3) – z = -z – 4(z + 2) ____________ Answer: The statement is false. Explanation: -2(z + 3) – z = -z – 4(z + 2) -3z – 6 = -3z -8 -3z -6 + 3z = -3z – 8 + 3z -6 not equal to -8 The statement is false. Create an equation with the indicated number of solutions. Question 9. No solution: 3(x – $$\frac{4}{3}$$) = 3x + _____ Type below: ______________ Answer: 3(x – $$\frac{4}{3}$$) = 3x + ? 3x – 4 = 3x + ? 3x – 4 = 3x + 2 When there is no solution, the statement should be false. Any number except -4 would make the equation have no solutions. Question 10. Infinitely many solutions: 2(x – 1) + 6x = 4( _____ – 1) + 2 Type below: ______________ Answer: 2(x – 1) + 6x = 4( _____ – 1) + 2 2(x – 1) + 6x = 4( ? – 1) + 2 2x – 2 + 6x = 4(? – 1) + 2 8x – 2 = 4(? – 1) + 2 8x – 2 = 4(2x – 1) + 2 8x – 2 = 8x – 4 + 2 8x – 2 = 8x – 2 When there are infinitely many solutions, the statement should be true Question 11. One solution of x = -1: 5x – (x – 2) = 2x – ( _____ ) Type below: ______________ Answer: Put x = -1 in the equation -5 – (-1 – 2) = -2 – blank simplifying -2 = -2 – blank add 2 on both sides 0 = blank Question 12. Infinitely many solutions: -(x – 8) + 4x = 2( _____ ) + x Type below: ______________ Answer: -(x – 8) + 4x = 2( ?) + x -x + 8 + 4x = 2(?) + x 3x + 8 = 2(?) + x 3x + 8 = 2 (x + 4) + x 3x + 8 = 2x + 8x + x 3x + 8 = 3x + 8 When there are infinitely many solutions, the statement should be true. Question 13. Persevere in Problem Solving The Dig It Project is designing two gardens that have the same perimeter. One garden is a trapezoid whose nonparallel sides are equal. The other is a quadrilateral. Two possible designs are shown at the right. a. Based on these designs, is there more than one value for x? Explain how you know this. ______________ Answer: There are more than one value of x Explanation: Perimeter of the trapezoid P = 2x – 2 + x + 1 + x + x + 1 = 5x Perimeter of the quadrilateral P = 2x – 9 + x + x + 8 + x + 1 = 5x 5x = 5x There are more than one value of x Question 13. b. Why does your answer to part a make sense in this context? Type below: ______________ Answer: The condition was that the two perimeters are to be equal. However, a specific number was not given, so there are an infinite number of possible perimeters Explanation: Interpretation of part a in this context The condition was that the two perimeters are to be equal. However, a specific number was not given, so there are an infinite number of possible perimeters Question 13. c. Suppose the Dig It Project wants the perimeter of each garden to be 60 meters. What is the value of x in this case? How did you find this? ______ meters Answer: 12 meters Explanation: 2x – 2 + x + 1 + x + x + 1 = 60 5x = 60 x = 60/5 x = 12 ### Equations with Many Solutions or No Solution – Page No. 220 Question 14. Critique Reasoning Lisa says that the indicated angles cannot have the same measure. Marita disagrees and says she can prove that they can have the same measure. Who do you agree with? Justify your answer. I agree with: ______________ Answer: I agree with: Marita Explanation: 9x – 25 + x = x + 50 + 2x – 12 10x – 25 = 3x + 38 10x – 3x = 38 + 25 7x = 63 x = 63/7 x = 9 When x = 9 the angles will be same and for any other value of x, the angles will not be the same. Question 15. Represent Real-World Problems Adele opens an account with$100 and deposits $35 a month. Kent opens an account with$50 and also deposits $35 a month. Will they have the same amount in their accounts at any point? If so, in how many months and how much will be in each account? Explain. ______________ Answer: Adele’s amount after x months A = 100 + 35x Kent’s amount after x months A = 50 + 35x 100 + 35x = 50 + 35x 100 is not equal to 50 The statement is false, the amounts in two accounts would never be equal. H.O.T. Focus on Higher Order Thinking Question 16. Communicate Mathematical Ideas Frank solved an equation and got the result x = x. Sarah solved the same equation and got 12 = 12. Frank says that one of them is incorrect because you cannot get different results for the same equation. What would you say to Frank? If both results are indeed correct, explain how this happened. Frank is: ____________ Answer: Both of them can be correct as both equations give the same result i.e. there are infinitely many solutions. Frank eliminated the constant from both sides while Sarah eliminated the variable from both sides. Question 17. Critique Reasoning Matt said 2x – 7 = 2(x – 7) has infinitely many solutions. Is he correct? Justify Matt’s answer or show how he is incorrect. Matt is: ____________ Answer: Explanation: 2x – 7 = 2(x – 7) 2x – 7 = 2x – 14 -7 not equal to -14 The statement is false, there is no solution. Matt is incorrect. ### 7.1 Equations with the Variable on Both Sides – Model Quiz – Page No. 221 Solve. Question 1. 4a – 4 = 8 + a _______ Answer: a = 4 Explanation: 4a – 4 = 8 + a 4a – a = 8 + 4 3a = 12 a = 12/3 a = 4 Question 2. 4x + 5 = x + 8 _______ Answer: x = 1 Explanation: 4x + 5 = x + 8 4x – x = 8 – 5 3x = 3 x = 3/3 x = 1 Question 3. Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs. Write and solve an equation to find how many chairs are in that row length. _______ chairs Answer: 7 chairs Explanation: Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs. 6x + 3 = 8x – 11 8x – 6x = 3 + 11 2x = 14 x = 14/2 x = 7 There are 7 chairs in each row. 7.2 Equations with Rational Numbers Solve. Question 4. $$\frac{2}{3} n-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}$$ _______ Answer: n = 4 Explanation: $$\frac{2}{3} n-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}$$ The LCM is 6. 6(2/3n – 2/3) = 6(n/6 + 4/3) 6(2/3n) -6(2/3) = 6(n/6) + 6(4/3) 4n – 4 = n + 8 4n – n = 8 + 4 3n = 12 n = 12/3 n = 4 Question 5. 1.5d + 3.25 = 1 + 2.25d _______ Answer: d = 3 Explanation: 1.5d + 3.25 = 1 + 2.25d 2.25d – 1.5d = 3.25 – 1 0.75d = 2.25 d = 2.25/0.75 d = 3 Question 6. Happy Paws charges$19.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges$14.00 plus $2.75 per hour. Write and solve an equation to find for how many hours the total cost of the services is equal. _______ hours Answer: 3.2 hours Explanation: Happy Paws charges$19.00 plus $1.50 per hour to keep a dog during the day. 1.5x + 19 Woof Watchers charges$14.00 plus $2.75 per hour. 2.75x + 15 1.5x + 19 = 2.75x + 15 2.75x – 1.5x = 19 – 15 1.25x = 4 x = 4/1.25 x = 3.2 The total cost of the services is equal after 3.2 hrs. 7.3 Equations with the Distributive Property Solve. Question 7. 14 + 5x = 3(-x + 3) – 11 _______ Answer: x = -2 Explanation: 14 + 5x = 3(-x + 3) – 11 14 + 5x = -3x + 9 – 11 14 + 5x = -3x – 2 5x + 3x = -2 – 14 8x = – 16 x = -16/8 x = -2 Question 8. $$\frac{1}{4}$$(x – 7) = 1 + 3x _______ Answer: x = -1 Explanation: $$\frac{1}{4}$$(x – 7) = 1 + 3x 4($$\frac{1}{4}$$(x – 7)) = 4(1 + 3x) (x – 7) = 4 + 12x 12x – x = -7 – 4 11x = -11 x = -11/11 x = -1 Question 9. -5(2x – 9) = 2(x – 8) – 11 _______ Answer: x = 6 Explanation: -5(2x – 9) = 2(x – 8) – 11 -10x + 45 = 2x – 16 – 11 -10x + 45 = 2x – 27 2x + 10x = 45 + 27 12x = 72 x = 72/12 x = 6 Question 10. 3(x + 5) = 2(3x + 12) _______ Answer: x = -3 Explanation: 3(x + 5) = 2(3x + 12) 3x + 15 = 6x + 24 6x – 3x = 15 – 24 3x = -9 x = -9/3 x = -3 7.4 Equations with Many Solutions or No Solution Tell whether each equation has one, zero, or infinitely many solutions. Question 11. 5(x – 3) + 6 = 5x – 9 ____________ Answer: There are infinitely many solutions Explanation: 5(x – 3) + 6 = 5x – 9 5x – 15 + 6 = 5x – 9 5x – 9 = 5x – 9 The statement is true. There are infinitely many solutions. Question 12. 5(x – 3) + 6 = 5x – 10 ____________ Answer: There are no solutions Explanation: 5(x – 3) + 6 = 5x – 10 5x – 15 + 6 = 5x – 10 5x – 9 = 5x – 10 -9 not equal to -10 The statement is false. There are no solutions. Question 13. 5(x – 3) + 6 = 4x + 3 ____________ Answer: There is one solution Explanation: 5(x – 3) + 6 = 4x + 3 5x – 15 + 6 = 4x + 3 5x – 9 = 4x + 3 5x – 4x = 3 + 9 x = 12 There is one solution ### Selected Response – Mixed Review – Page No. 222 Question 1. Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car. Which equation could you solve to find how many hours it will take for the second car to catch up to the first car? Options: a. 55t + 3 = 40t b. 55t + 165 = 40t c. 40t + 3 = 55t d. 40t + 120 = 55t Answer: d. 40t + 120 = 55t Explanation: Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car. 3 × 40 + 40t = 120 + 40t 55t 40t + 120 = 55t Question 2. Which linear equation is represented by the table? Options: a. y = -x + 5 b. y = 2x – 1 c. y = x + 3 d. y = -3x + 11 Answer: a. y = -x + 5 Explanation: Find the slope using m = (y2 – y1)/(x2 – x1) where (x1, y1) = (3, 2), (x2, y2) = (1, 4) Slope = (4 – 2)/(1 – 3) = -2/2 = -1 Question 3. Shawn’s Rentals charges$27.50 per hour to rent a surfboard and a wetsuit. Darla’s Surf Shop charges $23.25 per hour to rent a surfboard plus$17 extra for a wetsuit. For what total number of hours are the charges for Shawn’s Rentals the same as the charges for Darla’s Surf Shop?
Options:
a. 3
b. 4
c. 5
d. 6

Answer:
b. 4

Explanation:
Shawn’s Rentals charges $27.50 per hour to rent a surfboard and a wetsuit. 27.5x Darla’s Surf Shop charges$23.25 per hour to rent a surfboard plus $17 extra for a wetsuit. 23.25x + 17 23.25x + 17 = 27.5x 27.5x – 23.25x = 17 4.25x = 17 x = 17/4.25 x = 4 The charge would be equal after 4 hrs Question 4. Which of the following is irrational? Options: a. -8 b. 4.63 c. $$\sqrt { x }$$ d. $$\frac{1}{3}$$ Answer: c. $$\sqrt { x }$$ Explanation: $$\sqrt { x }$$ is irrational Question 5. Greg and Jane left a 15% tip after dinner. The amount of the tip was$9. Greg’s dinner cost \$24. Which equation can you use to find x, the cost of Jane’s dinner?
Options:
a. 0.15x + 24 = 9
b. 0.15(x + 24) = 9
c. 15(x + 24) = 9
d. 0.15x = 24 + 9

Answer:
b. 0.15(x + 24) = 9

Explanation:
Let x be the cost of Jane’s dinner. The amount of tip is the 15% of the total cost of dinner.
0.15(x + 24) = 9

Question 6.
For the equation 3(2x − 5) = 6x + k, which value of k will create an equation with infinitely many solutions?
Options:
a. 15
b. -5
c. 5
d. -15

Answer:
d. -15

Explanation:
3(2x – 5) = 6x + k
6x – 15 = 6x + k
6x – 15 = 6x – 15
The statement is true. k = -15

Question 7.
Which of the following is equivalent to 2−4?
Options:
a. $$\frac{1}{16}$$
b. $$\frac{1}{8}$$
c. -2
d. -16

Answer:
a. $$\frac{1}{16}$$

Explanation:
2−4
1/24
1/16

Mini-Task

Question 8.
Use the figures below for parts a and b.

a. Both figures have the same perimeter. Solve for x.
_______

Answer:
x=12

Explanation:
4x+10=3x+22
4x – 3x = 22 – 10
x = 12
Answer: x=12

Question 8.
b. What is the perimeter of each figure?
_______

Answer:
Both are 58

Explanation:
x + x + 5 + x + x + 5
12 + 12 + 5 + 12 + 12 + 5
58
x + 7 + x + 4 + x + 11
12 + 7 + 12 + 4 + 12 + 11
58

### Conclusion:

Go Math Grade 8 Answer Key is the best source to enhance your math skills. Learn all the solutions from Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations and complete your homework. Hope the information provided in Go Math Grade 8 Chapter 7 Solving Linear Equations Answer Key is satisfactory for all the 8th standard students. Stay with our CCSS Math Answers website to get the solutions of all grade 8 chapters.