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Mathematics LibreTexts

2.4E: Exercises

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    30480
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    Practice Makes Perfect

    Solve Equations Using the General Strategy for Solving Linear Equations

    In the following exercises, solve each linear equation.

    Exercise \(\PageIndex{1}\)

    \(15(y-9)=-60\)

    Exercise \(\PageIndex{2}\)

    \(21(y-5)=-42\)

    Answer

    \(y=3\)

    Exercise \(\PageIndex{3}\)

    \(-9(2 n+1)=36\)

    Exercise \(\PageIndex{4}\)

    \(-16(3 n+4)=32\)

    Answer

    \(n=-2\)

    Exercise \(\PageIndex{5}\)

    \(8(22+11 r)=0\)

    Exercise \(\PageIndex{6}\)

    \(5(8+6 p)=0\)

    Answer

    \(p=-\frac{4}{3}\)

    Exercise \(\PageIndex{7}\)

    \(-(w-12)=30\)

    Exercise \(\PageIndex{8}\)

    \(-(t-19)=28\)

    Answer

    \(t=-9\)

    Exercise \(\PageIndex{9}\)

    \(9(6 a+8)+9=81\)

    Exercise \(\PageIndex{10}\)

    \(8(9 b-4)-12=100\)

    Answer

    \(b=2\)

    Exercise \(\PageIndex{11}\)

    \(32+3(z+4)=41\)

    Exercise \(\PageIndex{12}\)

    \(21+2(m-4)=25\)

    Answer

    \(m=6\)

    Exercise \(\PageIndex{13}\)

    \(51+5(4-q)=56\)

    Exercise \(\PageIndex{14}\)

    \(-6+6(5-k)=15\)

    Answer

    \(k=\frac{3}{2}\)

    Exercise \(\PageIndex{15}\)

    \(2(9 s-6)-62=16\)

    Exercise \(\PageIndex{16}\)

    \(8(6 t-5)-35=-27\)

    Answer

    \(t=1\)

    Exercise \(\PageIndex{17}\)

    \(3(10-2 x)+54=0\)

    Exercise \(\PageIndex{18}\)

    \(-2(11-7 x)+54=4\)

    Answer

    \(x=-2\)

    Exercise \(\PageIndex{19}\)

    \(\frac{2}{3}(9 c-3)=22\)

    Exercise \(\PageIndex{20}\)

    \(\frac{3}{5}(10 x-5)=27\)

    Answer

    \(x=5\)

    Exercise \(\PageIndex{21}\)

    \(\frac{1}{5}(15 c+10)=c+7\)

    Exercise \(\PageIndex{22}\)

    \(\frac{1}{4}(20 d+12)=d+7\)

    Answer

    \(d=1\)

    Exercise \(\PageIndex{23}\)

    \(18-(9 r+7)=-16\)

    Exercise \(\PageIndex{24}\)

    \(15-(3 r+8)=28\)

    Answer

    \(r=-7\)

    Exercise \(\PageIndex{25}\)

    \(5-(n-1)=19\)

    Exercise \(\PageIndex{26}\)

    \(-3-(m-1)=13\)

    Answer

    \(m=-15\)

    Exercise \(\PageIndex{27}\)

    \(11-4(y-8)=43\)

    Exercise \(\PageIndex{28}\)

    \(18-2(y-3)=32\)

    Answer

    \(y=-4\)

    Exercise \(\PageIndex{29}\)

    \(24-8(3 v+6)=0\)

    Exercise \(\PageIndex{30}\)

    \(35-5(2 w+8)=-10\)

    Answer

    \(w=\frac{1}{2}\)

    Exercise \(\PageIndex{31}\)

    \(4(a-12)=3(a+5)\)

    Exercise \(\PageIndex{32}\)

    \(-2(a-6)=4(a-3)\)

    Answer

    \(a=4\)

    Exercise \(\PageIndex{33}\)

    \(2(5-u)=-3(2 u+6)\)

    Exercise \(\PageIndex{34}\)

    \(5(8-r)=-2(2 r-16)\)

    Answer

    \(r=8\)

    Exercise \(\PageIndex{35}\)

    \(3(4 n-1)-2=8 n+3\)

    Exercise \(\PageIndex{36}\)

    \(9(2 m-3)-8=4 m+7\)

    Answer

    \(m=3\)

    Exercise \(\PageIndex{37}\)

    \(12+2(5-3 y)=-9(y-1)-2\)

    Exercise \(\PageIndex{38}\)

    \(-15+4(2-5 y)=-7(y-4)+4\)

    Answer

    \(y=-3\)

    Exercise \(\PageIndex{39}\)

    \(8(x-4)-7 x=14\)

    Exercise \(\PageIndex{40}\)

    \(5(x-4)-4 x=14\)

    Answer

    \(x=34\)

    Exercise \(\PageIndex{41}\)

    \(5+6(3 s-5)=-3+2(8 s-1)\)

    Exercise \(\PageIndex{42}\)

    \(-12+8(x-5)=-4+3(5 x-2)\)

    Answer

    \(x=-6\)

    Exercise \(\PageIndex{43}\)

    \(4(u-1)-8=6(3 u-2)-7\)

    Exercise \(\PageIndex{44}\)

    \(7(2 n-5)=8(4 n-1)-9\)

    Answer

    \(n=-1\)

    Exercise \(\PageIndex{45}\)

    \(4(p-4)-(p+7)=5(p-3)\)

    Exercise \(\PageIndex{46}\)

    \(3(a-2)-(a+6)=4(a-1)\)

    Answer

    \(a=-4\)

    Exercise \(\PageIndex{47}\)

    \(\begin{array}{l}{-(9 y+5)-(3 y-7)} \\ {=16-(4 y-2)}\end{array}\)

    Exercise \(\PageIndex{48}\)

    \(\begin{array}{l}{-(7 m+4)-(2 m-5)} \\ {=14-(5 m-3)}\end{array}\)

    Answer

    \(m=-4\)

    Exercise \(\PageIndex{49}\)

    \(\begin{array}{l}{4[5-8(4 c-3)]} \\ {=12(1-13 c)-8}\end{array}\)

    Exercise \(\PageIndex{50}\)

    \(\begin{array}{l}{5[9-2(6 d-1)]} \\ {=11(4-10 d)-139}\end{array}\)

    Answer

    \(d=-3\)

    Exercise \(\PageIndex{51}\)

    \(\begin{array}{l}{3[-9+8(4 h-3)]} \\ {=2(5-12 h)-19}\end{array}\)

    Exercise \(\PageIndex{52}\)

    \(\begin{array}{l}{3[-14+2(15 k-6)]} \\ {=8(3-5 k)-24}\end{array}\)

    Answer

    \(k=\frac{3}{5}\)

    Exercise \(\PageIndex{53}\)

    \(\begin{array}{l}{5[2(m+4)+8(m-7)]} \\ {=2[3(5+m)-(21-3 m)]}\end{array}\)

    Exercise \(\PageIndex{54}\)

    \(\begin{array}{l}{10[5(n+1)+4(n-1)]} \\ {=11[7(5+n)-(25-3 n)]}\end{array}\)

    Answer

    \(n=-5\)

    Exercise \(\PageIndex{55}\)

    \(5(1.2 u-4.8)=-12\)

    Exercise \(\PageIndex{56}\)

    \(4(2.5 v-0.6)=7.6\)

    Answer

    \(v=1\)

    Exercise \(\PageIndex{57}\)

    \(0.25(q-6)=0.1(q+18)\)

    Exercise \(\PageIndex{58}\)

    \(0.2(p-6)=0.4(p+14)\)

    Answer

    \(p=-34\)

    Exercise \(\PageIndex{59}\)

    \(0.2(30 n+50)=28\)

    Exercise \(\PageIndex{60}\)

    \(0.5(16 m+34)=-15\)

    Answer

    \(m=-4\)

    Classify Equations

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

    Exercise \(\PageIndex{61}\)

    \(23 z+19=3(5 z-9)+8 z+46\)

    Exercise \(\PageIndex{62}\)

    \(15 y+32=2(10 y-7)-5 y+46\)

    Answer

    identity; all real numbers

    Exercise \(\PageIndex{63}\)

    \(5(b-9)+4(3 b+9)=6(4 b-5)-7 b+21\)

    Exercise \(\PageIndex{64}\)

    \(9(a-4)+3(2 a+5)=7(3 a-4)-6 a+7\)

    Answer

    identity; all real numbers

    Exercise \(\PageIndex{65}\)

    \(18(5 j-1)+29=47\)

    Exercise \(\PageIndex{66}\)

    \(24(3 d-4)+100=52\)

    Answer

    conditional equation; \(d=\frac{2}{3}\)

    Exercise \(\PageIndex{67}\)

    \(22(3 m-4)=8(2 m+9)\)

    Exercise \(\PageIndex{68}\)

    \(30(2 n-1)=5(10 n+8)\)

    Answer

    conditional equation; \(n=7\)

    Exercise \(\PageIndex{69}\)

    \(7 v+42=11(3 v+8)-2(13 v-1)\)

    Exercise \(\PageIndex{70}\)

    \(18 u-51=9(4 u+5)-6(3 u-10)\)

    Answer

    contradiction; no solution

    Exercise \(\PageIndex{71}\)

    \(3(6 q-9)+7(q+4)=5(6 q+8)-5(q+1)\)

    Exercise \(\PageIndex{72}\)

    \(5(p+4)+8(2 p-1)=9(3 p-5)-6(p-2)\)

    Answer

    contradiction; no solution

    Exercise \(\PageIndex{73}\)

    \(12(6 h-1)=8(8 h+5)-4\)

    Exercise \(\PageIndex{74}\)

    \(9(4 k-7)=11(3 k+1)+4\)

    Answer

    conditional equation; \(k=26\)

    Exercise \(\PageIndex{75}\)

    \(45(3 y-2)=9(15 y-6)\)

    Exercise \(\PageIndex{76}\)

    \(60(2 x-1)=15(8 x+5)\)

    Answer

    contradiction; no solution

    Exercise \(\PageIndex{77}\)

    \(16(6 n+15)=48(2 n+5)\)

    Exercise \(\PageIndex{78}\)

    \(36(4 m+5)=12(12 m+15)\)

    Answer

    identity; all real numbers

    Exercise \(\PageIndex{79}\)

    \(9(14 d+9)+4 d=13(10 d+6)+3\)

    Exercise \(\PageIndex{80}\)

    \(11(8 c+5)-8 c=2(40 c+25)+5\)

    Answer

    identity; all real numbers

    Everyday Math

    Exercise \(\PageIndex{81}\)

    Fencing Micah has 44 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)=44.

    Exercise \(\PageIndex{82}\)

    Coins Rhonda has \(\$ 1.90\) in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the
    number of nickels, \(n,\) by solving the equation \(0.05 n+0.10(2 n-1)=1.90 .\)

    Answer

    8 nickels

    Writing Exercises

    Exercise \(\PageIndex{83}\)

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

    Exercise \(\PageIndex{84}\)

    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.

    Answer

    Answers will vary.

    Exercise \(\PageIndex{85}\)

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

    Exercise \(\PageIndex{86}\)

    Solve the equation \(\frac{1}{4}(8 x+20)=3 x-4\) explaining all the steps of your solution as in the examples in this section.

    Answer

    Answers will vary.

    Self Check

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

    This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads: “solve equations using the general strategy for solving linear equations,” and “classify equations.” The rest of the cells are blank.

    ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

    Glossary

    conditional equation
    An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.
    contradiction
    An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.
    identity
    An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers.