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Mathematics LibreTexts

8.E: Solving Linear Equations (Exercises)

 

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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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