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2.1 Use a General Strategy to Solve Linear Equations

A repeatable procedure for isolating a variable, clearing fraction and decimal coefficients, and classifying equations as conditional, identity, or contradiction.

Intermediate Algebra 2e, 2.1 Use a General Strategy to Solve Linear Equations

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69 exercises

2.1 Exercise 1

Before you get started, take this readiness quiz.

Simplify: 32(12x+20).\frac{3}{2} \left(12 x + 20\right) .
If you missed this problem, review the reference.

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2.1 Exercise 2

Simplify: 5−2(n+1).5 - 2 \left(n + 1\right) .
If you missed this problem, review the reference.

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2.1 Exercise 3

Find the LCD of 56\frac{5}{6} and 14.\frac{1}{4} .
If you missed this problem, review the reference.

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2.1 Exercise 5

Determine whether the values are solutions to the equation: 9y+2=6y+3.9 y + 2 = 6 y + 3 .

ⓐ y=43y = \frac{4}{3} ⓑ y=13y = \frac{1}{3}

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2.1 Exercise 6

Determine whether the values are solutions to the equation: 4x−2=2x+1.4 x - 2 = 2 x + 1 .

ⓐ x=32x = \frac{3}{2} ⓑ x=−12x = - \frac{1}{2}

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2.1 Exercise 8

Solve: 2(m−4)+3=−1.2 \left(m - 4\right) + 3 = −1 .

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2.1 Exercise 9

Solve: 5(a−3)+5=−10.5 \left(a - 3\right) + 5 = −10 .

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2.1 Exercise 11

Solve: 13(6u+3)=7−u.\frac{1}{3} \left(6 u + 3\right) = 7 - u .

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2.1 Exercise 12

Solve: 23(9x−12)=8+2x.\frac{2}{3} \left(9 x - 12\right) = 8 + 2 x .

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2.1 Exercise 14

Solve: 6(p−3)−7=5(4p+3)−12.6 \left(p - 3\right) - 7 = 5 \left(4 p + 3\right) - 12 .

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2.1 Exercise 15

Solve: 8(q+1)−5=3(2q−4)−1.8 \left(q + 1\right) - 5 = 3 \left(2 q - 4\right) - 1 .

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2.1 Exercise 17

Solve: 6[4−2(7y−1)]=8(13−8y).6 \left[4 - 2 \left(7 y - 1\right)\right] = 8 \left(13 - 8 y\right) .

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2.1 Exercise 18

Solve: 12[1−5(4z−1)]=3(24+11z).12 \left[1 - 5 \left(4 z - 1\right)\right] = 3 \left(24 + 11 z\right) .

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2.1 Exercise 20

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 4+9(3x−7)=−42x−13+23(3x−2).4 + 9 \left(3 x - 7\right) = −42 x - 13 + 23 \left(3 x - 2\right) .

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2.1 Exercise 21

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 8(1−3x)+15(2x+7)=2(x+50)+4(x+3)+1.8 \left(1 - 3 x\right) + 15 \left(2 x + 7\right) = 2 \left(x + 50\right) + 4 \left(x + 3\right) + 1 .

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2.1 Exercise 23

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 11(q+3)−5=19.11 \left(q + 3\right) - 5 = 19 .

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2.1 Exercise 24

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 6+14(k−8)=95.6+14 \left(k - 8\right) = 95 .

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2.1 Exercise 26

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 12c+5(5+3c)=3(9c−4).12 c + 5 \left(5 + 3 c\right) = 3 \left(9 c - 4\right) .

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2.1 Exercise 27

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 4(7d+18)=13(3d−2)−11d.4 \left(7 d + 18\right) = 13 \left(3 d - 2\right) - 11 d .

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2.1 Exercise 29

Solve: 14x+12=58.\frac{1}{4} x + \frac{1}{2} = \frac{5}{8} .

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2.1 Exercise 30

Solve: 18x+12=14.\frac{1}{8} x + \frac{1}{2} = \frac{1}{4} .

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2.1 Exercise 32

Solve: 7=12x+34x−23x.7 = \frac{1}{2} x + \frac{3}{4} x - \frac{2}{3} x .

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2.1 Exercise 33

Solve: −1=12u+14u−23u.−1 = \frac{1}{2} u + \frac{1}{4} u - \frac{2}{3} u .

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2.1 Exercise 35

Solve: 15(n+3)=14(n+2).\frac{1}{5} \left(n + 3\right) = \frac{1}{4} \left(n + 2\right) .

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2.1 Exercise 36

Solve: 12(m−3)=14(m−7).\frac{1}{2} \left(m - 3\right) = \frac{1}{4} \left(m - 7\right) .

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2.1 Exercise 38

Solve: 3r+56+1=4r+33.\frac{3 r + 5}{6} + 1 = \frac{4 r + 3}{3} .

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2.1 Exercise 39

Solve: 2s+32+1=3s+24.\frac{2 s + 3}{2} + 1 = \frac{3 s + 2}{4} .

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2.1 Exercise 41

Solve: 0.25n+0.05(n+5)=2.95.0.25 n + 0.05 \left(n + 5\right) = 2.95 .

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2.1 Exercise 42

Solve: 0.10d+0.05(d−5)=2.15.0.10 d + 0.05 \left(d - 5\right) = 2.15 .

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2.1 Exercise 43

Determine whether the given values are solutions to the equation.

6y+10=12y6 y + 10 = 12 y

ⓐ y=53y = \frac{5}{3} ⓑ y=−12y = - \frac{1}{2}

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2.1 Exercise 45

Determine whether the given values are solutions to the equation.

8u−1=6u8 u - 1 = 6 u

ⓐ u=−12u = - \frac{1}{2} ⓑ u=12u = \frac{1}{2}

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2.1 Exercise 47

Solve each linear equation.

15(y−9)=−6015 \left(y - 9\right) = −60

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2.1 Exercise 49

Solve each linear equation.

−(w−12)=30− \left(w - 12\right) = 30

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2.1 Exercise 51

Solve each linear equation.

51+5(4−q)=5651 + 5 \left(4 - q\right) = 56

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2.1 Exercise 53

Solve each linear equation.

3(10−2x)+54=03 \left(10 - 2 x\right) + 54 = 0

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2.1 Exercise 55

Solve each linear equation.

23(9c−3)=22\frac{2}{3} \left(9 c - 3\right) = 22

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2.1 Exercise 57

Solve each linear equation.

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

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2.1 Exercise 59

Solve each linear equation.

3(4n−1)−2=8n+33 \left(4 n - 1\right) - 2 = 8 n + 3

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2.1 Exercise 61

Solve each linear equation.

12+2(5−3y)=−9(y−1)−212 + 2 \left(5 - 3 y\right) = −9 \left(y - 1\right) - 2

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2.1 Exercise 63

Solve each linear equation.

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

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2.1 Exercise 65

Solve each linear equation.

4(p−4)−(p+7)=5(p−3)4 \left(p - 4\right) - \left(p + 7\right) = 5 \left(p - 3\right)

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2.1 Exercise 67

Solve each linear equation.

4[5−8(4c−3)]=12(1−13c)−84 \left[5 - 8 \left(4 c - 3\right)\right] = 12 \left(1 - 13 c\right) - 8

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2.1 Exercise 69

Solve each linear equation.

3[−9+8(4h−3)]=2(5−12h)−193 \left[−9 + 8 \left(4 h - 3\right)\right] = 2 \left(5 - 12 h\right) - 19

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2.1 Exercise 71

Solve each linear equation.

5[2(m+4)+8(m−7)]=2[3(5+m)−(21−3m)]5 \left[2 \left(m + 4\right) + 8 \left(m - 7\right)\right] = 2 \left[3 \left(5 + m\right) - \left(21 - 3 m\right)\right]

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2.1 Exercise 73

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

23z+19=3(5z−9)+8z+4623 z + 19 = 3 \left(5 z - 9\right) + 8 z + 46

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2.1 Exercise 75

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

18(5j−1)+29=4718 \left(5 j - 1\right) + 29 = 47

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2.1 Exercise 77

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

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

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2.1 Exercise 79

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

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

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2.1 Exercise 81

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

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

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2.1 Exercise 83

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

9(14d+9)+4d=13(10d+6)+39 \left(14 d + 9\right) + 4 d = 13 \left(10 d + 6\right) + 3

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2.1 Exercise 85

Solve each equation with fraction coefficients.

14x−12=−34\frac{1}{4} x - \frac{1}{2} = - \frac{3}{4}

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2.1 Exercise 87

Solve each equation with fraction coefficients.

56y−23=−32\frac{5}{6} y - \frac{2}{3} = - \frac{3}{2}

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2.1 Exercise 89

Solve each equation with fraction coefficients.

12a+38=34\frac{1}{2} a + \frac{3}{8} = \frac{3}{4}

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2.1 Exercise 91

Solve each equation with fraction coefficients.

2=13x−12x+23x2 = \frac{1}{3} x - \frac{1}{2} x + \frac{2}{3} x

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2.1 Exercise 93

Solve each equation with fraction coefficients.

13w+54=w−14\frac{1}{3} w + \frac{5}{4} = w - \frac{1}{4}

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2.1 Exercise 95

Solve each equation with fraction coefficients.

13b+15=25b−35\frac{1}{3} b + \frac{1}{5} = \frac{2}{5} b - \frac{3}{5}

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2.1 Exercise 97

Solve each equation with fraction coefficients.

14(p−7)=13(p+5)\frac{1}{4} \left(p - 7\right) = \frac{1}{3} \left(p + 5\right)

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2.1 Exercise 99

Solve each equation with fraction coefficients.

12(x+4)=34\frac{1}{2} \left(x + 4\right) = \frac{3}{4}

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2.1 Exercise 101

Solve each equation with fraction coefficients.

4n+84=n3\frac{4 n + 8}{4} = \frac{n}{3}

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2.1 Exercise 103

Solve each equation with fraction coefficients.

3x+42+1=5x+108\frac{3 x + 4}{2} + 1 = \frac{5 x + 10}{8}

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2.1 Exercise 105

Solve each equation with fraction coefficients.

7u−14−1=4u+85\frac{7 u - 1}{4} - 1 = \frac{4 u + 8}{5}

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2.1 Exercise 107

Solve each equation with decimal coefficients.

0.4x+0.6=0.5x−1.20.4 x + 0.6 = 0.5 x - 1.2

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2.1 Exercise 109

Solve each equation with decimal coefficients.

0.9x−1.25=0.75x+1.750.9 x - 1.25 = 0.75 x + 1.75

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2.1 Exercise 111

Solve each equation with decimal coefficients.

0.05n+0.10(n+8)=2.150.05 n + 0.10 \left(n + 8\right) = 2.15

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2.1 Exercise 113

Solve each equation with decimal coefficients.

0.10d+0.25(d+5)=4.050.10 d + 0.25 \left(d + 5\right) = 4.05

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2.1 Exercise 115

Fencing 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)=74.2 L + 2 \left(L - 2.5\right) = 74 .

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2.1 Exercise 117

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

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2.1 Exercise 119

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

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2.1 Exercise 121

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

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Adapted from Intermediate Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Intermediate Algebra 2e. Imported at revision 77a933c.