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CHAPTER 5 Solving First Degree Equations in One Variable

5.8 Chapter Review

Review Exercises

Verify a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.

1. w-8=5,\phantom{\rule{0.2em}{0ex}}w=3 2. x+16=31,\phantom{\rule{0.2em}{0ex}}x=15
3. 4a=72,\phantom{\rule{0.2em}{0ex}}a=18 4.-9n=45,\phantom{\rule{0.2em}{0ex}}n=54

Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, solve each equation using the Subtraction Property of Equality.

5. y+2=-6 6. x+7=19
7. n+3.6=5.1 8. a+\frac{1}{3}=\frac{5}{3}

In the following exercises, solve each equation using the Addition Property of Equality.

9. x-9=-4 10. u-7=10
11. p-4.8=14 12. c-\frac{3}{11}=\frac{9}{11}

In the following exercises, solve each equation.

13. y+16=-9 14. n-12=32
15. d-3.9=8.2 16. f+\frac{2}{3}=4

Solve Equations That Require Simplification

In the following exercises, solve each equation.

17. 7x+10-6x+3=5 18. y+8-15=-3
19. 8\left(3p+5\right)-23\left(p-1\right)=35 20. 6\left(n-1\right)-5n=-14

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

21. Four less than n is 13. 22. The sum of -6 and m is 25.

Translate and Solve Applications

In the following exercises, translate into an algebraic equation and solve.

23. Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh? 24. Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?
25. Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week? 26. Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the division and multiplication properties of equality and check the solution.

27. 13a=-65 28. 8x=72
29. -y=4 30. 0.25p=5.25
31. \frac{y}{-10}=30 32. \frac{n}{6}=18
33. \frac{5}{8}u=\frac{15}{16} 34. 36=\frac{3}{4}x
35. \frac{c}{9}=36 36. -18m=-72
37. \frac{11}{12}=\frac{2}{3}y 38. 0.45x=6.75

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

39. 24x+8x-11x=-7-14 40. 5r-3r+9r=35-2
41. -9\left(d-2\right)-15=-24 42. \frac{11}{12}n-\frac{5}{6}n=9-5

Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve.

43. The quotient of b and and 9 is -27. 44. 143 is the product of -11 and y.
45. The difference of s and one-twelfth is one fourth. 46. The sum of q and one-fourth is one.

Translate and Solve Applications

In the following exercises, translate into an equation and solve.

47. Janet gets paid $24 per hour. She heard that this is \frac{3}{4} of what Adam is paid. How much is Adam paid per hour? 48. Ray paid $21 for 12 tickets at the county fair. What was the price of each ticket?

Solve an Equation with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

49. 10w-5=65 50. 8p+7=47
51. 32=-4-9n 52. 3x+19=-47

Solve an Equation with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

53. 5a+21=2a 54. 7y=6y-13
55. 4x-\frac{3}{8}=3x 56. k=-6k-35

Solve an Equation with Variables and Constants on Both Sides

In the following exercises, solve the following equations with variables and constants on both sides.

57. 5n-20=-7n-80 58. 12x-9=3x+45
59. \frac{5}{8}c-4=\frac{3}{8}c+4 60. 4u+16=-19-u

Solve Equations Using the General Strategy for Solving Linear Equations

In the following exercises, solve each linear equation.

61. 9\left(2p-5\right)=72 62. 6\left(x+6\right)=24
63. 8+3\left(n-9\right)=17 64. \text{-}\left(s+4\right)=18
65. \frac{1}{3}\left(6m+21\right)=m-7 66. 23-3\left(y-7\right)=8
67. 0.25\left(q-8\right)=0.1\left(q+7\right) 68. 4\left(3.5y+0.25\right)=365
69. 5+7\left(2-5x\right)=2\left(9x+1\right) -\left(13x-57\right) 70. 8\left(r-2\right)=6\left(r+10\right)
71. 2\left[-16+5\left(8k-6\right)\right] =8\left(3-4k\right)-32 72. \left(9n+5\right)-\left(3n-7\right) =20-\left(4n-2\right)

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

73. 9u+32=15\left(u-4\right) -3\left(2u+21\right) 74. 17y-3\left(4-2y\right)=11\left(y-1\right) +12y-1
75. 21\left(c-1\right)-19\left(c+1\right) =2\left(c-20\right) 76. -8\left(7m+4\right)=-6\left(8m+9\right)

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

77. \frac{1}{3}x+\frac{1}{5}x=8 78. \frac{2}{5}n-\frac{1}{10}=\frac{7}{10}
79. \frac{1}{2}\left(k-3\right)=\frac{1}{3}\left(k+16\right) 80. \frac{3}{4}a-\frac{1}{3}=\frac{1}{2}a-\frac{5}{6}
81. \frac{5y-1}{3}+4=\frac{-8y+4}{6} 82. \frac{3x-2}{5}=\frac{3x+4}{8}

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

83. 0.36u+2.55=0.41u+6.8 84. 0.8x-0.3=0.7x+0.2

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

85. Mallory is taking the bus from Edmonton to North Battleford. The distance is 300 miles and the bus travels at a steady rate of 60 miles per hour. How long will the bus ride be? 86. Natalie drove for 7\frac{1}{2} hours at 60 miles per hour. How much distance did she travel?
87. Link rode his bike at a steady rate of 15 miles per hour for 2\frac{1}{2} hours. How much distance did he travel? 88. Aaron’s friend drove him from Williams Lake to Kamloops. The distance is 187 miles and the trip took 2.75 hours. How fast was Aaron’s friend driving?

Solve a Formula for a Specific Variable

In the following exercises, solve.

89. Use the formula. d=rt to solve for r
a) when when d=451 and t=5.5
b) in general		 90. Use the formula. d=rt to solve for t
a) when d=510 and r=60
b) in general
91. Use the formula A=\frac{1}{2}bh to solve for h
a) when A=153 and b=18
b) in general		 92. Use the formula A=\frac{1}{2}bh to solve for b
a) when A=390 and h=26
b) in general
93. Solve the formula 4x+3y=6 for y
a) when x=-2
b) in general		 94. Use the formula I=Prt to solve for the principal, P for
a) I=\$2,501,r=4.1\%,t=5\phantom{\rule{0.2em}{0ex}}\text{years}
b) in general
95. Solve the formula V=LWH for H. 96. Solve 180=a+b+c for c.

Everyday Math

97. Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

Review Exercises Answers

1. no 3. yes
5. y=-8 7. n=1.5
9. x=5 11. p=18.8
13. y=-25 15. d=12.1
17. x=-8 19. p=-28
21. n-4=13;n=17 23. 161 pounds
25. $131.19 27. a=-5
29. y=-4 31. y=-300
33. u=\frac{3}{2} 35. c=324
37. y=\frac{11}{8} 39. x=-1
41. d=3 43. \frac{b}{9}=-27;b=-243
45. s-\frac{1}{12}=\frac{1}{4};s=\frac{1}{3} 47. $32
49. w=7 51. n=-4
53. a=-7 55. x=\frac{3}{8}
57. n=-5 59. c=32
61. p=\frac{13}{2} 63. n=12
65. m=-14 67. q=18
69. x=-1 71. k=\frac{3}{4}
73. contradiction; no solution 75. identity; all real numbers
77. x=15 79. k=41
81. y=-1 83. u=-85
85. 5 hours 87. 37.5 miles
89. a)r=82\phantom{\rule{0.2em}{0ex}}\text{mph}; b)r=\frac{D}{t} 91. a)h=17b)h=\frac{2A}{b}
93. a)y=\frac{14}{3}b)y=\frac{6-4x}{3} 95. H=\frac{V}{LW}

Practice Test

Determine whether each number is a solution to the equation 3x+5=20.

1.
a) 5
b) \frac{23}{5}

In the following exercises, solve each equation.

2. n-18=31 3. 9c=144
4. 4y-8=16 5. -8x-15+9x-1=-21
6. -15a=120 7. \frac{2}{3}x=6
8. x-3.8=8.2 9. 10y=-5y-60
10. 8n-2=6n-12 11. 9m-2-4m-m=42-8
12. -5\left(2x-1\right)=45 13. \text{-}\left(d-9\right)=23
14. \frac{1}{4}\left(12m-28\right)=6-2\left(3m-1\right) 15. 2\left(6x-5\right)-8=-22
16.8\left(3a-5\right)-7\left(4a-3\right)=20-3a 17. \frac{1}{4}p-\frac{1}{3}=\frac{1}{2}
18. 0.1d+0.25\left(d+8\right)=4.1 19. 14n-3\left(4n+5\right)=-9+2\left(n-8\right)
20. 9\left(3u-2\right)-4\left[6-8\left(u-1\right)\right]=3\left(u-2\right) 21. Solve the formula x-2y=5 for y
a) when x=-3
b) in general
22. Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much had he paid last week?

Practice Test Answers

1. a) yes b) no 2. n=49
3. c=16 4. y=6
5. x=-5 6. a=8
7. x=9 8. x=12
9. y=-4 10. n=-5
11. m=9 12. x=-4
13. d=-14 14. m=\frac{5}{3}
15. x=-\frac{1}{3} 16. a=-39
17. p=\frac{10}{3} 18. d=6
19. contradiction; no solution 20. u=\frac{17}{14}
21. a) y=4 b) y=\frac{5-x}{2} 22. \$29.29

Attributions

This chapter has been adapted from “Review Exercises” and “Practice Test” in Chapter 2 of Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

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Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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