Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

2.2E: Use a General Strategy to Solve Linear Equations (Exercises)

  • Page ID
    29618
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Practice Makes Perfect

    Solve Equations Using the General Strategy

    In the following exercises, determine whether the given values are solutions to the equation.

    1.  \(6y+10=12y\)

    a. \(y=\frac{5}{3}\)

    b. \(y=−\frac{1}{2}\)

    Answer

    a. yes
    b. no

    2.  \(4x+9=8x\)

    a. \(x=−\frac{7}{8}\)

    b. \(x=\frac{9}{4}\)

    3.  \(8u−1=6u\)

    a. \(u=−\frac{1}{2}\)

    b. \(u=\frac{1}{2}\)

    Answer

    a. no
    b. yes

    4.  \(9v−2=3v\)

    a. \(v=−\frac{1}{3}\)

    b. \(v=\frac{1}{3}\)

    In the following exercises, solve each linear equation.

    5.  \(15(y−9)=−60\)

    Answer

    \(y=5\)

    6.  \(−16(3n+4)=32\)

    7.  \(−(w−12)=30\)

    Answer

    \(w=−18\)

    8.  \(−(t−19)=28\)

    9.  \(51+5(4−q)=56\)

    Answer

    \(q=3\)

    10.  \(−6+6(5−k)=15\)

    11.  \(3(10−2x)+54=0\)

    Answer

    \(x=14\)

    12.  \(−2(11−7x)+54=4\)

    13\(\frac{2}{3}(9c−3)=22\)

    Answer

    \(c=4\)

    14.  \(\frac{3}{5}(10x−5)=27\)

    15.  \(\frac{1}{5}(15c+10)=c+7\)

    Answer

    \(c=\frac{5}{2}\)

    16.  \(\frac{1}{4}(20d+12)=d+7\)

    17.  \(3(4n−1)−2=8n+3\)

    Answer

    \(n=2\)

    18.  \(9(2m−3)−8=4m+7\)

    19.  \(12+2(5−3y)=−9(y−1)−2\)

    Answer

    \(y=−5\)

    20.  \(−15+4(2−5y)=−7(y−4)+4\)

    21.  \(5+6(3s−5)=−3+2(8s−1)\)

    Answer

    \(s=10\)

    22.  \(−12+8(x−5)=−4+3(5x−2)\)

    23\(4(p−4)−(p+7)=5(p−3)\)

    Answer

    \(p=−4\)

    24.  \(3(a−2)−(a+6)=4(a−1)\)

    25.  \(4[5−8(4c−3)]=12(1−13c)−8\)

    Answer

    \(c=−4\)

    26.  \(5[9−2(6d−1)]=11(4−10d)−139\)

    27.  \(3[−9+8(4h−3)]=2(5−12h)−19\)

    Answer

    \(h=\frac{3}{4}\)

    28.  \(3[−14+2(15k−6)]=8(3−5k)−24\)

    29.  \(5[2(m+4)+8(m−7)]=2[3(5+m)−(21−3m)]\)

    Answer

    \(m=6\)

    30.  \(10[5(n+1)+4(n−1)]=11[7(5+n)−(25−3n)]\)

    Classify Equations

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

    31.  \(23z+19=3(5z−9)+8z+46\)

    Answer

    identity; all real numbers

    32.  \(15y+32=2(10y−7)−5y+46\)

    33.  \(18(5j−1)+29=47\)

    Answer

    conditional equation;  \(j=\frac{2}{5}\)

    34.  \(24(3d−4)+100=52\)

    35.  \(22(3m−4)=8(2m+9)\)

    Answer

    conditional equation; \(m=165\)

    36.  \(30(2n−1)=5(10n+8)\)

    37.  \(7v+42=11(3v+8)−2(13v−1)\)

    Answer

    contradiction; no solution

    38.  \(18u−51=9(4u+5)−6(3u−10)\)

    39.  \(45(3y−2)=9(15y−6)\)

    Answer

    contradiction; no solution

    40.  \(60(2x−1)=15(8x+5)\)

    41.  \(9(14d+9)+4d=13(10d+6)+3\)

    Answer

    identity; all real numbers

    42.  \(11(8c+5)−8c=2(40c+25)+5\)

    Solve Equations with Fraction or Decimal Coefficients

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

    43.  \(\frac{1}{4}x−\frac{1}{2}=−\frac{3}{4}\)

    Answer

    \(x=−1\)

    44.  \(\frac{3}{4}x−\frac{1}{2}=\frac{1}{4}\)

    45.  \(\frac{5}{6}y−\frac{2}{3}=−\frac{3}{2}\)

    Answer

    \(y=−1\)

    46.  \(\frac{5}{6}y−\frac{1}{3}=−\frac{7}{6}\)

    47.  \(\frac{1}{2}a+\frac{3}{8}=\frac{3}{4}\)

    Answer

    \(a=\frac{3}{4}\)

    48.  \(\frac{5}{8}b+\frac{1}{2}=−\frac{3}{4}\)

    49.  \(2=\frac{1}{3}x−\frac{1}{2}x+\frac{2}{3}x\)

    Answer

    \(x=4\)

    50.  \(2=\frac{3}{5}x−\frac{1}{3}x+\frac{2}{5}x\)

    51.  \(\frac{1}{3}w+\frac{5}{4}=w−\frac{1}{4}\)

    Answer

    \(w=\frac{9}{4}\)

    52.  \(\frac{1}{2}a−\frac{1}{4}=\frac{1}{6}a+\frac{1}{12}\)

    53.  \(\frac{1}{3}b+\frac{1}{5}=\frac{2}{5}b−\frac{3}{5}\)

    Answer

    \(b=12\)

    54.  \(\frac{1}{3}x+\frac{2}{5}=\frac{1}{5}x−\frac{2}{5}\)

    55.  \(\frac{1}{4}(p−7)=\frac{1}{3}(p+5)\)

    Answer

    \(p=−41\)

    56.  \(\frac{1}{5}(q+3)=\frac{1}{2}(q−3)\)

    57.  \(\frac{1}{2}(x+4)=\frac{3}{4}\)

    Answer

    \(x=−\frac{5}{2}\)

    58.  \(\frac{1}{3}(x+5)=\frac{5}{6}\)

    59.  \(\dfrac{4n+8}{4}=\dfrac{n}{3}\)

    Answer

    \(n=−3\)

    60.  \(\dfrac{3p+6}{3}=\dfrac{p}{2}\)

    61.  \(\dfrac{3x+4}{2}+1=\dfrac{5x+10}{8}\)

    Answer

    \(x=−2\)

    62.  \(\dfrac{10y−2}{3}+3=\dfrac{10y+1}{9}\)

    63.  \(\dfrac{7u−1}{4}−1=\dfrac{4u+8}{5}\)

    Answer

    \(u=3\)

    64.  \(\dfrac{3v−6}{2}+5=\dfrac{11v−4}{5}\)

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

    65.  \(0.4x+0.6=0.5x−1.2\)

    Answer

    \(x=18\)

    66.  \(0.7x+0.4=0.6x+2.4\)

    67.  \(0.9x−1.25=0.75x+1.75\)

    Answer

    \(x=20\)

    68.  \(1.2x−0.91=0.8x+2.29\)

    69.  \(0.05n+0.10(n+8)=2.15\)

    Answer

    \(n=9\)

    70.  \(0.05n+0.10(n+7)=3.55\)

    71.  \(0.10d+0.25(d+5)=4.05\)

    Answer

    \(d=8\)

    72.  \(0.10d+0.25(d+7)=5.25\)

    Everyday Math

    73Fencing Micah has 74 feet of fencing to make a dog run in his yard. He wants the length to be 2.5 feet more than the width. Find the length, L, by solving the equation \(2L+2(L−2.5)=74\).

    Answer

    \(L=19.75\) feet

    74Stamps Paula bought $22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was eight less than the number of 49-cent stamps. Solve the equation \(0.49s+0.21 (s−8) =22.82\) for s, to find the number of 49-cent stamps Paula bought.

    Writing Exercises

    75.  Using your own words, list the steps in the general strategy for solving linear equations.

    Answer

    Answers will vary.

    76.  Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.

    77.  What is the first step you take when solving the equation \(3−7(y−4)=38?\) Why is this your first step?

    Answer

    Answers will vary.

    78.  If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

    79.  If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

    Answer

    Answers will vary.

    80.  For the equation \(0.35x+2.1=3.85\), how do you clear the decimal?

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has four columns and four rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was solve linear equations using a general strategy. In row 3, the I can was classify equations. In row 4, the I can was solve equations with fraction or decimal coefficients.

    b. If most of your checks were:

    …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

    …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

    …no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.