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4.1 Solve Systems of Linear Equations with Two Variables

Graphing, substitution, and elimination compared side by side, with inconsistent and dependent systems and guidance on which method a given system rewards.

Intermediate Algebra 2e, 4.1 Solve Systems of Linear Equations with Two Variables

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68 exercises

4.1 Exercise 1

Before you get started, take this readiness quiz.

For the equation y=23x−4,y = \frac{2}{3} x - 4 ,
ⓐ Is (6,0)\left(6 , 0\right) a solution? ⓑ Is (−3,−2)\left(−3 , −2\right) a solution?
If you missed this problem, review the reference.

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4.1 Exercise 2

Find the slope and y-intercept of the line 3x−y=12.3 x - y = 12 .
If you missed this problem, review the reference.

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4.1 Exercise 3

Find the x- and y-intercepts of the line 2x−3y=12.2 x - 3 y = 12 .
If you missed this problem, review the reference.

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4.1 Exercise 5

Determine whether the ordered pair is a solution to the system {3x+y=0x+2y=−5.\begin{matrix}\begin{cases} 3 x + y = 0 \\ x + 2 y = −5 \end{cases} . \\ \end{matrix}

ⓐ (1,−3)\left(1 , −3\right) ⓑ (0,0)\left(0 , 0\right)

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4.1 Exercise 6

Determine whether the ordered pair is a solution to the system {x−3y=−8−3x−y=4.\begin{matrix}\begin{cases} x - 3 y = −8 \\ - 3 x - y = 4 \end{cases} . \\ \end{matrix}

ⓐ (2,−2)\left(2 , −2\right) ⓑ (−2,2)\left(−2 , 2\right)

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4.1 Exercise 8

Solve the system by graphing: {x−3y=−3x+y=5.\begin{cases} x - 3 y = −3 \\ x + y = 5 \end{cases} .

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4.1 Exercise 9

Solve the system by graphing: {−x+y=13x+2y=12.\begin{cases} - x + y = 1 \\ 3 x + 2 y = 12 \end{cases} .

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4.1 Exercise 11

Solve the system by graphing: {−x+y=12x+y=10.\begin{cases} - x + y = 1 \\ 2 x + y = 10 \end{cases} .

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4.1 Exercise 12

Solve the system by graphing: {2x+y=6x+y=1.\begin{cases} 2 x + y = 6 \\ x + y = 1 \end{cases} .

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4.1 Exercise 14

Solve the system by graphing: {y=−14x+2x+4y=−8.\begin{cases} y = - \frac{1}{4} x + 2 \\ x + 4 y = - 8 \end{cases} .

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4.1 Exercise 15

Solve the system by graphing: {y=3x−16x−2y=6.\begin{cases} y = 3 x - 1 \\ 6 x - 2 y = 6 \end{cases} .

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4.1 Exercise 17

Solve the system by graphing: { y=−3x−66x+2y=−12.\begin{cases} y = - 3 x - 6 \\ 6 x + 2 y = - 12 \end{cases} .

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4.1 Exercise 18

Solve the system by graphing: {y=12x−42x−4y=16.\begin{cases} y = \frac{1}{2} x - 4 \\ 2 x - 4 y = 16 \end{cases} .

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4.1 Exercise 19

Without graphing, determine the number of solutions and then classify the system of equations.

ⓐ {y=3x−16x−2y=12\begin{cases} y = 3 x - 1 \\ 6 x - 2 y = 12 \end{cases} ⓑ {2x+y=−3x−5y=5\begin{cases} 2 x + y = - 3 \\ x - 5 y = 5 \end{cases}

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4.1 Exercise 20

Without graphing, determine the number of solutions and then classify the system of equations.

ⓐ {y=−2x−44x+2y=9\begin{cases} y = −2 x - 4 \\ 4 x + 2 y = 9 \end{cases} ⓑ {3x+2y=22x+y=1\begin{cases} 3 x + 2 y = 2 \\ 2 x + y = 1 \end{cases}

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4.1 Exercise 21

Without graphing, determine the number of solutions and then classify the system of equations.

ⓐ {y=13x−5x−3y=6\begin{cases} y = \frac{1}{3} x - 5 \\ x - 3 y = 6 \end{cases} ⓑ {x+4y=12−x+y=3\begin{cases} x + 4 y = 12 \\ - x + y = 3 \end{cases}

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4.1 Exercise 23

Solve the system by substitution: {−2x+y=−11x+3y=9.\begin{cases} - 2 x + y = −11 \\ x + 3 y = 9 \end{cases} .

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4.1 Exercise 24

Solve the system by substitution: {2x+y=−14x+3y=3.\begin{cases} 2 x + y = −1 \\ 4 x + 3 y = 3 \end{cases} .

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4.1 Exercise 26

Solve the system by substitution: {x−4y=−4−3x+4y=0.\begin{cases} x - 4 y = −4 \\ - 3 x + 4 y = 0 \end{cases} .

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4.1 Exercise 27

Solve the system by substitution: {4x−y=02x−3y=5.\begin{cases} 4 x - y = 0 \\ 2 x - 3 y = 5 \end{cases} .

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4.1 Exercise 29

Solve the system by elimination: {3x+y=52x−3y=7.\begin{cases} 3 x + y = 5 \\ 2 x - 3 y = 7 \end{cases} .

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4.1 Exercise 30

Solve the system by elimination: { 4x+y=−5−2x−2y=−2.\begin{cases} 4 x + y = - 5 \\ - 2 x - 2 y = - 2 \end{cases} .

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4.1 Exercise 32

Solve the system by elimination: { 3x−4y=−95x+3y=14.\begin{cases} 3 x - 4 y = - 9 \\ 5 x + 3 y = 14 \end{cases} .

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4.1 Exercise 33

Solve each system by elimination: { 7x+8y=43x−5y=27.\begin{cases} 7 x + 8 y = 4 \\ 3 x - 5 y = 27 \end{cases} .

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4.1 Exercise 35

Solve each system by elimination: { 13x−12y=134x−y=52 .\begin{cases} \frac{1}{3} x - \frac{1}{2} y = 1 \\ \frac{3}{4} x - y = \frac{5}{2} \end{cases} .

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4.1 Exercise 36

Solve each system by elimination: { x+35y=−15 −12x−23y=56 .\begin{cases} x + \frac{3}{5} y = - \frac{1}{5} \\ - \frac{1}{2} x - \frac{2}{3} y = \frac{5}{6} \end{cases} .

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4.1 Exercise 37

Solve the system by elimination: {3x+4y=12y=3−34x.\begin{cases} 3 x + 4 y = 12 \\ y = 3 - \frac{3}{4} x \end{cases} .

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4.1 Exercise 38

Solve the system by elimination: { 5x−3y=15y=−5+53x.\begin{cases} 5 x - 3 y = 15 \\ y = - 5 + \frac{5}{3} x \end{cases} .

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4.1 Exercise 39

Solve the system by elimination: { x+2y=6y=−12x+3.\begin{cases} x + 2 y = 6 \\ y = - \frac{1}{2} x + 3 \end{cases} .

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4.1 Exercise 40

For each system of linear equations, decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.

ⓐ {3x+8y=407x−4y=−32\begin{cases} 3 x + 8 y = 40 \\ 7 x - 4 y = −32 \end{cases} ⓑ {5x+6y=12y=23x−1\begin{cases} 5 x + 6 y = 12 \\ y = \frac{2}{3} x - 1 \end{cases}

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4.1 Exercise 41

For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.

ⓐ {4x−5y=−323x+2y=−1\begin{cases} 4 x - 5 y = −32 \\ 3 x + 2 y = −1 \end{cases} ⓑ {x=2y−13x−5y=−7\begin{cases} x = 2 y - 1 \\ 3 x - 5 y = −7 \end{cases}

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4.1 Exercise 42

For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.

ⓐ {y=2x−13x−4y=−6\begin{cases} y = 2 x - 1 \\ 3 x - 4 y = - 6 \end{cases} ⓑ {6x−2y=123x+7y=−13\begin{cases} 6 x - 2 y = 12 \\ 3 x + 7 y = −13 \end{cases}

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4.1 Exercise 43

Determine if the following points are solutions to the given system of equations.

{2x−6y=03x−4y=5\begin{cases} 2 x - 6 y = 0 \\ 3 x - 4 y = 5 \end{cases}

ⓐ (3,1)\left(3 , 1\right) ⓑ (−3,4)\left(−3 , 4\right)

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4.1 Exercise 45

Determine if the following points are solutions to the given system of equations.

{x+y=2y=34x\begin{cases} x + y = 2 \\ y = \frac{3}{4} x \end{cases}

ⓐ (87,67)\left(\frac{8}{7} , \frac{6}{7}\right) ⓑ (1,34)\left(1 , \frac{3}{4}\right)

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4.1 Exercise 47

Solve the following systems of equations by graphing.

{3x+y=−32x+3y=5\begin{cases} 3 x + y = −3 \\ 2 x + 3 y = 5 \end{cases}

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4.1 Exercise 49

Solve the following systems of equations by graphing.

{y=x+2y=−2x+2\begin{cases} y = x + 2 \\ y = −2 x + 2 \end{cases}

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4.1 Exercise 51

Solve the following systems of equations by graphing.

{y=32x+1y=−12x+5\begin{cases} y = \frac{3}{2} x + 1 \\ y = - \frac{1}{2} x + 5 \end{cases}

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4.1 Exercise 53

Solve the following systems of equations by graphing.

{x+y=−4−x+2y=−2\begin{cases} x + y = −4 \\ - x + 2 y = −2 \end{cases}

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4.1 Exercise 55

Solve the following systems of equations by graphing.

{−2x+3y=3x+3y=12\begin{cases} - 2 x + 3 y = 3 \\ x + 3 y = 12 \end{cases}

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4.1 Exercise 57

Solve the following systems of equations by graphing.

{x+3y=−6y=−43x+4\begin{cases} x + 3 y = −6 \\ y = - \frac{4}{3} x + 4 \end{cases}

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4.1 Exercise 59

Solve the following systems of equations by graphing.

{−2x+4y=4y=12x\begin{cases} - 2 x + 4 y = 4 \\ y = \frac{1}{2} x \end{cases}

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4.1 Exercise 61

Solve the following systems of equations by graphing.

{4x−3y=88x−6y=14\begin{cases} 4 x - 3 y = 8 \\ 8 x - 6 y = 14 \end{cases}

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4.1 Exercise 63

Solve the following systems of equations by graphing.

{x=−3y+42x+6y=8\begin{cases} x = −3 y + 4 \\ 2 x + 6 y = 8 \end{cases}

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4.1 Exercise 65

Solve the following systems of equations by graphing.

{2x+y=6−8x−4y=−24\begin{cases} 2 x + y = 6 \\ - 8 x - 4 y = −24 \end{cases}

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4.1 Exercise 67

Without graphing, determine the number of solutions and then classify the system of equations.

{y=23x+1−2x+3y=5\begin{cases} y = \frac{2}{3} x + 1 \\ - 2 x + 3 y = 5 \end{cases}

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4.1 Exercise 69

Without graphing, determine the number of solutions and then classify the system of equations.

{5x+3y=42x−3y=5\begin{cases} 5 x + 3 y = 4 \\ 2 x - 3 y = 5 \end{cases}

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4.1 Exercise 71

Without graphing, determine the number of solutions and then classify the system of equations.

{5x−2y=10y=52x−5\begin{cases} 5 x - 2 y = 10 \\ y = \frac{5}{2} x - 5 \end{cases}

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4.1 Exercise 73

Solve the systems of equations by substitution.

{2x+y=−23x−y=7\begin{cases} 2 x + y = −2 \\ 3 x - y = 7 \end{cases}

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4.1 Exercise 75

Solve the systems of equations by substitution.

{x−3y=−92x+5y=4\begin{cases} x - 3 y = −9 \\ 2 x + 5 y = 4 \end{cases}

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4.1 Exercise 77

Solve the systems of equations by substitution.

{−2x+2y=6y=−3x+1\begin{cases} - 2 x + 2 y = 6 \\ y = −3 x + 1 \end{cases}

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4.1 Exercise 79

Solve the systems of equations by substitution.

{3x+4y=1y=−25x+2\begin{cases} 3 x + 4 y = 1 \\ y = - \frac{2}{5} x + 2 \end{cases}

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4.1 Exercise 81

Solve the systems of equations by substitution.

{4x+y=10x−2y=−20\begin{cases} 4 x + y = 10 \\ x - 2 y = −20 \end{cases}

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4.1 Exercise 83

Solve the systems of equations by substitution.

{y=x−6y=−32x+4\begin{cases} y = x - 6 \\ y = - \frac{3}{2} x + 4 \end{cases}

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4.1 Exercise 85

Solve the systems of equations by substitution.

{2x−16y=8−x−8y=−4\begin{cases} 2 x - 16 y = 8 \\ - x - 8 y = −4 \end{cases}

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4.1 Exercise 87

Solve the systems of equations by substitution.

{y=−23x+52x+3y=11\begin{cases} y = - \frac{2}{3} x + 5 \\ 2 x + 3 y = 11 \end{cases}

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4.1 Exercise 89

Solve the systems of equations by elimination.

{6x−5y=−12x+y=13\begin{cases} 6 x - 5 y = −1 \\ 2 x + y = 13 \end{cases}

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4.1 Exercise 91

Solve the systems of equations by elimination.

{5x−3y=−12x−y=2\begin{cases} 5 x - 3 y = −1 \\ 2 x - y = 2 \end{cases}

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4.1 Exercise 93

Solve the systems of equations by elimination.

{4x−3y=32x+5y=−31\begin{cases} 4 x - 3 y = 3 \\ 2 x + 5 y = −31 \end{cases}

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4.1 Exercise 95

Solve the systems of equations by elimination.

{11x+9y=−57x+5y=−1\begin{cases} 11 x + 9 y = −5 \\ 7 x + 5 y = −1 \end{cases}

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4.1 Exercise 97

Solve the systems of equations by elimination.

{2x+9y=−43x+13y=−7\begin{cases} 2 x + 9 y = −4 \\ 3 x + 13 y = −7 \end{cases}

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4.1 Exercise 99

Solve the systems of equations by elimination.

{x+12y=3215x−15y=3\begin{cases} x + \frac{1}{2} y = \frac{3}{2} \\ \frac{1}{5} x - \frac{1}{5} y = 3 \end{cases}

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4.1 Exercise 101

Solve the systems of equations by elimination.

{13x−y=−323x+52y=3\begin{cases} \frac{1}{3} x - y = −3 \\ \frac{2}{3} x + \frac{5}{2} y = 3 \end{cases}

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4.1 Exercise 103

Solve the systems of equations by elimination.

{x−4y=−1−3x+12y=3\begin{cases} x - 4 y = −1 \\ - 3 x + 12 y = 3 \end{cases}

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4.1 Exercise 105

Solve the systems of equations by elimination.

{4x+3y=220x+15y=10\begin{cases} 4 x + 3 y = 2 \\ 20 x + 15 y = 10 \end{cases}

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4.1 Exercise 107

Decide whether it would be more convenient to solve the system of equations by substitution or elimination.


ⓐ {y=7x−53x−2y=16\begin{cases} y = 7 x - 5 \\ 3 x - 2 y = 16 \end{cases}
ⓑ {12x−5y=−423x+7y=−15\begin{cases} 12 x - 5 y = −42 \\ 3 x + 7 y = −15 \end{cases}

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4.1 Exercise 109

Decide whether it would be more convenient to solve the system of equations by substitution or elimination.


ⓐ {14x−15y=−307x+2y=10\begin{cases} 14 x - 15 y = −30 \\ 7 x + 2 y = 10 \end{cases}
ⓑ {x=9y−112x−7y=−27\begin{cases} x = 9 y - 11 \\ 2 x - 7 y = −27 \end{cases}

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4.1 Exercise 111

Solve the system of equations by substitution and explain all your steps in words: {3x+y=12x=y−8.\begin{cases} 3 x + y = 12 \\ x = y - 8 \end{cases} .

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4.1 Exercise 113

Solve the system of equations {x+y=10x−y=6\begin{cases} x + y = 10 \\ x - y = 6 \end{cases}

ⓐ by graphing ⓑ by substitution
ⓒ Which method do you prefer? Why?

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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.