# 2.4E: Exercises

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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 $$\PageIndex{2}$$

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

$$y=3$$

##### Exercise $$\PageIndex{3}$$

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

##### Exercise $$\PageIndex{4}$$

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

$$n=-2$$

##### Exercise $$\PageIndex{5}$$

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

##### Exercise $$\PageIndex{6}$$

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

$$p=-\frac{4}{3}$$

##### Exercise $$\PageIndex{7}$$

$$-(w-12)=30$$

##### Exercise $$\PageIndex{8}$$

$$-(t-19)=28$$

$$t=-9$$

##### Exercise $$\PageIndex{9}$$

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

##### Exercise $$\PageIndex{10}$$

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

$$b=2$$

##### Exercise $$\PageIndex{11}$$

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

##### Exercise $$\PageIndex{12}$$

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

$$m=6$$

##### Exercise $$\PageIndex{13}$$

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

##### Exercise $$\PageIndex{14}$$

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

$$k=\frac{3}{2}$$

##### Exercise $$\PageIndex{15}$$

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

##### Exercise $$\PageIndex{16}$$

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

$$t=1$$

##### Exercise $$\PageIndex{17}$$

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

##### Exercise $$\PageIndex{18}$$

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

$$x=-2$$

##### Exercise $$\PageIndex{19}$$

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

##### Exercise $$\PageIndex{20}$$

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

$$x=5$$

##### Exercise $$\PageIndex{21}$$

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

##### Exercise $$\PageIndex{22}$$

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

$$d=1$$

##### Exercise $$\PageIndex{23}$$

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

##### Exercise $$\PageIndex{24}$$

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

$$r=-7$$

##### Exercise $$\PageIndex{25}$$

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

##### Exercise $$\PageIndex{26}$$

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

$$m=-15$$

##### Exercise $$\PageIndex{27}$$

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

##### Exercise $$\PageIndex{28}$$

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

$$y=-4$$

##### Exercise $$\PageIndex{29}$$

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

##### Exercise $$\PageIndex{30}$$

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

$$w=\frac{1}{2}$$

##### Exercise $$\PageIndex{31}$$

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

##### Exercise $$\PageIndex{32}$$

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

$$a=4$$

##### Exercise $$\PageIndex{33}$$

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

##### Exercise $$\PageIndex{34}$$

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

$$r=8$$

##### Exercise $$\PageIndex{35}$$

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

##### Exercise $$\PageIndex{36}$$

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

$$m=3$$

##### Exercise $$\PageIndex{37}$$

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

##### Exercise $$\PageIndex{38}$$

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

$$y=-3$$

##### Exercise $$\PageIndex{39}$$

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

##### Exercise $$\PageIndex{40}$$

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

$$x=34$$

##### Exercise $$\PageIndex{41}$$

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

##### Exercise $$\PageIndex{42}$$

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

$$x=-6$$

##### Exercise $$\PageIndex{43}$$

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

##### Exercise $$\PageIndex{44}$$

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

$$n=-1$$

##### Exercise $$\PageIndex{45}$$

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

##### Exercise $$\PageIndex{46}$$

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

$$a=-4$$

##### Exercise $$\PageIndex{47}$$

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

##### Exercise $$\PageIndex{48}$$

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

$$m=-4$$

##### Exercise $$\PageIndex{49}$$

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

##### Exercise $$\PageIndex{50}$$

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

$$d=-3$$

##### Exercise $$\PageIndex{51}$$

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

##### Exercise $$\PageIndex{52}$$

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

$$k=\frac{3}{5}$$

##### Exercise $$\PageIndex{53}$$

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

##### Exercise $$\PageIndex{54}$$

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

$$n=-5$$

##### Exercise $$\PageIndex{55}$$

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

##### Exercise $$\PageIndex{56}$$

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

$$v=1$$

##### Exercise $$\PageIndex{57}$$

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

##### Exercise $$\PageIndex{58}$$

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

$$p=-34$$

##### Exercise $$\PageIndex{59}$$

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

##### Exercise $$\PageIndex{60}$$

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

$$m=-4$$

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 $$\PageIndex{62}$$

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

identity; all real numbers

##### Exercise $$\PageIndex{63}$$

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

##### Exercise $$\PageIndex{64}$$

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

identity; all real numbers

##### Exercise $$\PageIndex{65}$$

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

##### Exercise $$\PageIndex{66}$$

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

conditional equation; $$d=\frac{2}{3}$$

##### Exercise $$\PageIndex{67}$$

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

##### Exercise $$\PageIndex{68}$$

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

conditional equation; $$n=7$$

##### Exercise $$\PageIndex{69}$$

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

##### Exercise $$\PageIndex{70}$$

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

##### Exercise $$\PageIndex{71}$$

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

##### Exercise $$\PageIndex{72}$$

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

##### Exercise $$\PageIndex{73}$$

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

##### Exercise $$\PageIndex{74}$$

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

conditional equation; $$k=26$$

##### Exercise $$\PageIndex{75}$$

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

##### Exercise $$\PageIndex{76}$$

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

##### Exercise $$\PageIndex{77}$$

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

##### Exercise $$\PageIndex{78}$$

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

identity; all real numbers

##### Exercise $$\PageIndex{79}$$

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

##### Exercise $$\PageIndex{80}$$

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

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 $$0.05 n+0.10(2 n-1)=1.90 .$$

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.

##### 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.

### Self Check

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

ⓑ 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.
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.

2.4E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.