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Mathematics LibreTexts

8.E: Solving Linear Equations (Exercises)

  • Page ID
    5024
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    8.1 - Solve Equations using the Subtraction and Addition Properties of Equality

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

    1. x + 16 = 31, x = 15
    2. w − 8 = 5, w = 3
    3. −9n = 45, n = 54
    4. 4a = 72, a = 18

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

    1. x + 7 = 19
    2. y + 2 = −6
    3. a + \(\frac{1}{3} = \frac{5}{3}\) 
    4. n + 3.6 = 5.1

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

    1. u − 7 = 10
    2. x − 9 = −4
    3. c − \(\frac{3}{11} = \frac{9}{11}\) 
    4. p − 4.8 = 14

    In the following exercises, solve the equation.

    1. n − 12 = 32
    2. y + 16 = −9
    3. f + \(\frac{2}{3}\) = 4
    4. d − 3.9 = 8.2
    5. y + 8 − 15 = −3
    6. 7x + 10 − 6x + 3 = 5
    7. 6(n − 1) − 5n = −14
    8. 8(3p + 5) − 23(p − 1) = 35

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

    1. The sum of −6 and m is 25.
    2. Four less than n is 13.

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

    1. Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?
    2. Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?
    3. 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?
    4. Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

    8.2 - Solve Equations using the Division and Multiplication Properties of Equality

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

    1. 8x = 72
    2. 13a = −65
    3. 0.25p = 5.25
    4. −y = 4

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

    1. \(\frac{n}{6}\) = 18
    2. y −10 = 30
    3. 36 = \(\frac{3}{4}\)x
    4. \(\frac{5}{8} u = \frac{15}{16}\)

    In the following exercises, solve each equation.

    1. −18m = −72
    2. \(\frac{c}{9}\) = 36
    3. 0.45x = 6.75
    4. \(\frac{11}{12} = \frac{2}{3} y\) 
    5. 5r − 3r + 9r = 35 − 2
    6. 24x + 8x − 11x = −7−14

    8.3 - Solve Equations with Variables and Constants on Both Sides

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

    1. 8p + 7 = 47
    2. 10w − 5 = 65
    3. 3x + 19 = −47
    4. 32 = −4 − 9n

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

    1. 7y = 6y − 13
    2. 5a + 21 = 2a
    3. k = −6k − 35
    4. 4x − \(\frac{3}{8}\) = 3x

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

    1. 12x − 9 = 3x + 45
    2. 5n − 20 = −7n − 80
    3. 4u + 16 = −19 − u
    4. \(\frac{5}{8} c\) − 4 = \(\frac{3}{8} c\) + 4

    In the following exercises, solve each linear equation using the general strategy.

    1. 6(x + 6) = 24
    2. 9(2p − 5) = 72
    3. −(s + 4) = 18
    4. 8 + 3(n − 9) = 17
    5. 23 − 3(y − 7) = 8
    6. \(\frac{1}{3}\)(6m + 21) = m − 7
    7. 8(r − 2) = 6(r + 10)
    8. 5 + 7(2 − 5x) = 2(9x + 1) − (13x − 57)
    9. 4(3.5y + 0.25) = 365
    10. 0.25(q − 8) = 0.1(q + 7)

    8.4 - Solve Equations with Fraction or Decimal Coefficients

    In the following exercises, solve each equation by clearing the fractions.

    1. \(\frac{2}{5} n − \frac{1}{10} = \frac{7}{10}\) 
    2. \(\frac{1}{3} x + \frac{1}{5} x = 8\)
    3. \(\frac{3}{4} a − \frac{1}{3} = \frac{1}{2} a + \frac{5}{6}\) 
    4. \(\frac{1}{2}\)(k + 3) = \(\frac{1}{3}\)(k + 16)

    In the following exercises, solve each equation by clearing the decimals.

    1. 0.8x − 0.3 = 0.7x + 0.2
    2. 0.36u + 2.55 = 0.41u + 6.8
    3. 0.6p − 1.9 = 0.78p + 1.7
    4. 0.10d + 0.05(d − 4) = 2.05

    PRACTICE TEST

    1. Determine whether each number is a solution to the equation. 3x + 5 = 23.
      1. 6
      2. \(\frac{23}{5}\)

    In the following exercises, solve each equation.

    1. n − 18 = 31
    2. 9c = 144
    3. 4y − 8 = 16
    4. −8x − 15 + 9x − 1 = −21
    5. −15a = 120
    6. \(\frac{2}{3}\)x = 6
    7. x + 3.8 = 8.2
    8. 10y = −5y + 60
    9. 8n + 2 = 6n + 12
    10. 9m − 2 − 4m + m = 42 − 8
    11. −5(2x + 1) = 45
    12. −(d + 9) = 23
    13. \(\frac{1}{3}\)(6m + 21) = m − 7
    14. 2(6x + 5) − 8 = −22
    15. 8(3a + 5) − 7(4a − 3) = 20 − 3a
    16. \(\frac{1}{4} p + \frac{1}{3} = \frac{1}{2}\) 
    17. 0.1d + 0.25(d + 8) = 4.1
    18. Translate and solve: The difference of twice x and 4 is 16.
    19. Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much did he pay last week?

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