Skip to main content
Mathematics LibreTexts

8.E: Solving Linear Equations (Exercises)

  1. Last updated
  2. Save as PDF
  • Page ID
    5024

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    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 + \(\dfrac{1}{3} = \dfrac{5}{3}\)
    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 − \(\dfrac{3}{11} = \dfrac{9}{11}\)
    4. p − 4.8 = 14

    In the following exercises, solve the equation.

    1. n − 12 = 32
    2. y + 16 = −9
    3. f + \(\dfrac{2}{3}\) = 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. \(\dfrac{n}{6}\) = 18
    2. y −10 = 30
    3. 36 = \(\dfrac{3}{4}\)x
    4. \(\dfrac{5}{8} u = \dfrac{15}{16}\)

    In the following exercises, solve each equation.

    1. −18m = −72
    2. \(\dfrac{c}{9}\) = 36
    3. 0.45x = 6.75
    4. \(\dfrac{11}{12} = \dfrac{2}{3} y\)
    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 − \(\dfrac{3}{8}\) = 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. \(\dfrac{5}{8} c\) − 4 = \(\dfrac{3}{8} c\) + 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. \(\dfrac{1}{3}\)(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. \(\dfrac{2}{5} n − \dfrac{1}{10} = \dfrac{7}{10}\)
    2. \(\dfrac{1}{3} x + \dfrac{1}{5} x = 8\)
    3. \(\dfrac{3}{4} a − \dfrac{1}{3} = \dfrac{1}{2} a + \dfrac{5}{6}\)
    4. \(\dfrac{1}{2}\)(k + 3) = \(\dfrac{1}{3}\)(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. \(\dfrac{23}{5}\)

    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. \(\dfrac{2}{3}\)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. \(\dfrac{1}{3}\)(6m + 21) = m − 7
    14. 2(6x + 5) − 8 = −22
    15. 8(3a + 5) − 7(4a − 3) = 20 − 3a
    16. \(\dfrac{1}{4} p + \dfrac{1}{3} = \dfrac{1}{2}\)
    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?

    Contributors and Attributions

    This page titled 8.E: Solving Linear Equations (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Author
      OpenStax
      License
      CC BY
      License Version
      4.0
      Show Page TOC
      no
    2. Tags
      This page has no tags.