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5.3 Solve Systems of Equations by Elimination

Adding scaled copies of the two equations so one variable cancels, and judging when this beats graphing or substitution.

Elementary Algebra 2e, 5.3 Solve Systems of Equations by Elimination

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53 exercises

5.3 Exercise 1

Before you get started, take this readiness quiz.

Simplify −5(6−3a)−5 \left(6 - 3 a\right).
If you missed this problem, review the reference.

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5.3 Exercise 2

Solve the equation 13x+58=3124\frac{1}{3} x + \frac{5}{8} = \frac{31}{24}.
If you missed this problem, review the reference.

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5.3 Exercise 4

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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5.3 Exercise 5

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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5.3 Exercise 7

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

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5.3 Exercise 8

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

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5.3 Exercise 10

Solve the system by elimination. {4x−3y=15x−9y=−4\begin{cases} 4 x - 3 y = 1 \\ 5 x - 9 y = −4 \end{cases}

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5.3 Exercise 11

Solve the system by elimination. {3x+2y=26x+5y=8\begin{cases} 3 x + 2 y = 2 \\ 6 x + 5 y = 8 \end{cases}

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5.3 Exercise 13

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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5.3 Exercise 14

Solve the 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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5.3 Exercise 16

Solve the 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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5.3 Exercise 17

Solve the 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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5.3 Exercise 18

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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5.3 Exercise 19

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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5.3 Exercise 20

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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5.3 Exercise 21

Solve the system by elimination. {−6x+15y=102x−5y=−5\begin{cases} −6 x + 15 y = 10 \\ 2 x - 5 y = −5 \end{cases}

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5.3 Exercise 22

Solve the system by elimination. {−3x+2y=89x−6y=13\begin{cases} −3 x + 2 y = 8 \\ 9 x - 6 y = 13 \end{cases}

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5.3 Exercise 23

Solve the system by elimination. {7x−3y=−2−14x+6y=8\begin{cases} 7 x - 3 y = - 2 \\ −14 x + 6 y = 8 \end{cases}

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5.3 Exercise 24

The sum of two numbers is 39. Their difference is 9. Find the numbers.

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5.3 Exercise 25

The sum of two numbers is 42. Their difference is 8. Find the numbers.

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5.3 Exercise 26

The sum of two numbers is −15. Their difference is −35. Find the numbers.

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5.3 Exercise 28

Malik stops at the grocery store to buy a bag of diapers and 2 cans of formula. He spends a total of $37. The next week he stops and buys 2 bags of diapers and 5 cans of formula for a total of $87. How much does a bag of diapers cost? How much is one can of formula?

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5.3 Exercise 29

To get her daily intake of fruit for the day, Sasha eats a banana and 8 strawberries on Wednesday for a calorie count of 145. On the following Wednesday, she eats two bananas and 5 strawberries for a total of 235 calories for the fruit. How many calories are there in a banana? How many calories are in a strawberry?

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5.3 Exercise 30

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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5.3 Exercise 31

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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5.3 Exercise 32

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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5.3 Exercise 34

Solve the systems of equations by elimination.

{−3x+y=−9x−2y=−12\begin{cases} −3 x + y = −9 \\ x - 2 y = −12 \end{cases}

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5.3 Exercise 36

Solve the systems of equations by elimination.

{3x−y=−74x+2y=−6\begin{cases} 3 x - y = −7 \\ 4 x + 2 y = −6 \end{cases}

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5.3 Exercise 38

Solve the systems of equations by elimination.

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

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5.3 Exercise 40

Solve the systems of equations by elimination.

{−7x+6y=−10x−6y=22\begin{cases} −7 x + 6 y = −10 \\ x - 6 y = 22 \end{cases}

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5.3 Exercise 42

Solve the systems of equations by elimination.

{5x+2y=1−5x−4y=−7\begin{cases} 5 x + 2 y = 1 \\ −5 x - 4 y = −7 \end{cases}

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5.3 Exercise 44

Solve the systems of equations by elimination.

{3x−4y=−11x−2y=−5\begin{cases} 3 x - 4 y = −11 \\ x - 2 y = −5 \end{cases}

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5.3 Exercise 46

Solve the systems of equations by elimination.

{6x−5y=−75−x−2y=−13\begin{cases} 6 x - 5 y = −75 \\ - x - 2 y = −13 \end{cases}

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5.3 Exercise 48

Solve the systems of equations by elimination.

{2x−5y=73x−y=17\begin{cases} 2 x - 5 y = 7 \\ 3 x - y = 17 \end{cases}

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5.3 Exercise 50

Solve the systems of equations by elimination.

{7x+y=−413x+3y=4\begin{cases} 7 x + y = −4 \\ 13 x + 3 y = 4 \end{cases}

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5.3 Exercise 52

Solve the systems of equations by elimination.

{3x−5y=−95x+2y=16\begin{cases} 3 x - 5 y = −9 \\ 5 x + 2 y = 16 \end{cases}

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5.3 Exercise 54

Solve the systems of equations by elimination.

{4x+7y=14−2x+3y=32\begin{cases} 4 x + 7 y = 14 \\ −2 x + 3 y = 32 \end{cases}

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5.3 Exercise 56

Solve the systems of equations by elimination.

{3x+8y=−32x+5y=−3\begin{cases} 3 x + 8 y = −3 \\ 2 x + 5 y = −3 \end{cases}

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5.3 Exercise 58

Solve the systems of equations by elimination.

{3x+8y=675x+3y=60\begin{cases} 3 x + 8 y = 67 \\ 5 x + 3 y = 60 \end{cases}

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5.3 Exercise 60

Solve the systems of equations by elimination.

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

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5.3 Exercise 62

Solve the systems of equations by elimination.

{x+13y=−112x−13y=−2\begin{cases} x + \frac{1}{3} y = −1 \\ \frac{1}{2} x - \frac{1}{3} y = −2 \end{cases}

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5.3 Exercise 64

Solve the systems of equations by elimination.

{2x+y=36x+3y=9\begin{cases} 2 x + y = 3 \\ 6 x + 3 y = 9 \end{cases}

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5.3 Exercise 66

Solve the systems of equations by elimination.

{−3x−y=86x+2y=−16\begin{cases} −3 x - y = 8 \\ 6 x + 2 y = −16 \end{cases}

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5.3 Exercise 68

Solve the systems of equations by elimination.

{3x+2y=6−6x−4y=−12\begin{cases} 3 x + 2 y = 6 \\ −6 x - 4 y = −12 \end{cases}

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5.3 Exercise 70

Solve the systems of equations by elimination.

{−11x+12y=60−22x+24y=90\begin{cases} −11 x + 12 y = 60 \\ −22 x + 24 y = 90 \end{cases}

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5.3 Exercise 72

Solve the systems of equations by elimination.

{5x−3y=15y=53x−2\begin{cases} 5 x - 3 y = 15 \\ y = \frac{5}{3} x - 2 \end{cases}

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5.3 Exercise 74

Translate to a system of equations and solve.

The sum of two numbers is 65. Their difference is 25. Find the numbers.

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5.3 Exercise 76

Translate to a system of equations and solve.

The sum of two numbers is −27. Their difference is −59. Find the numbers.

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5.3 Exercise 78

Translate to a system of equations and solve.

Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for $114 or she is able to buy 2 shirts and 4 sweaters for $164. How much does a shirt cost? How much does a sweater cost?

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5.3 Exercise 80

Translate to a system of equations and solve.

The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?

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5.3 Exercise 82

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


ⓐ {8x−15y=−326x+3y=−5\begin{cases} 8 x - 15 y = −32 \\ 6 x + 3 y = −5 \end{cases} ⓑ {x=4y−34x−2y=−6\begin{cases} x = 4 y - 3 \\ 4 x - 2 y = −6 \end{cases}

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5.3 Exercise 84

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


ⓐ {y=4x+95x−2y=−21\begin{cases} y = 4 x + 9 \\ 5 x - 2 y = −21 \end{cases} ⓑ {9x−4y=243x+5y=−14\begin{cases} 9 x - 4 y = 24 \\ 3 x + 5 y = −14 \end{cases}

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5.3 Exercise 86

Norris can row 3 miles upstream against the current in 1 hour, the same amount of time it takes him to row 5 miles downstream, with the current. Solve the system. {r−c=3r+c=5\begin{cases} r - c = 3 \\ r + c = 5 \end{cases}

  1. ⓐ for rr, his rowing speed in still water.
  2. ⓑ Then solve for cc, the speed of the river current.
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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.