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5.2 Solving Systems of Equations by Substitution

Isolating one variable in one equation and putting that expression into the other, reducing a two-variable system to a single equation.

Elementary Algebra 2e, 5.2 Solving Systems of Equations by Substitution

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56 exercises

5.2 Exercise 1

Before you get started, take this readiness quiz.

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

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5.2 Exercise 2

Simplify 4−2(n+5)4 - 2 \left(n + 5\right).
If you missed this problem, review the reference.

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5.2 Exercise 3

Solve for yy: 8y−8=32−2y8 y - 8 = 32 - 2 y
If you missed this problem, review the reference.

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5.2 Exercise 4

Solve for xx: 3x−9y=−33 x - 9 y = −3
If you missed this problem, review the reference.

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5.2 Exercise 6

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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5.2 Exercise 7

Solve the system by substitution. {x+3y=104x+y=18\begin{cases} x + 3 y = 10 \\ 4 x + y = 18 \end{cases}

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5.2 Exercise 9

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

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5.2 Exercise 10

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

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5.2 Exercise 12

Solve the system by substitution. {4x+y=23x+2y=−1\begin{cases} 4 x + y = 2 \\ 3 x + 2 y = −1 \end{cases}

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5.2 Exercise 13

Solve the system by substitution. {−x+y=44x−y=2\begin{cases} - x + y = 4 \\ 4 x - y = 2 \end{cases}

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5.2 Exercise 15

Solve the system by substitution. {x−5y=134x−3y=1\begin{cases} x - 5 y = 13 \\ 4 x - 3 y = 1 \end{cases}

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5.2 Exercise 16

Solve the system by substitution. {x−6y=−62x−4y=4\begin{cases} x - 6 y = −6 \\ 2 x - 4 y = 4 \end{cases}

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5.2 Exercise 18

Solve the system by substitution. {y=3x−16y=13x\begin{cases} y = 3 x - 16 \\ y = \frac{1}{3} x \end{cases}

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5.2 Exercise 19

Solve the system by substitution. {y=−x+10y=14x\begin{cases} y = − x + 10 \\ y = \frac{1}{4} x \end{cases}

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5.2 Exercise 21

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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5.2 Exercise 22

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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5.2 Exercise 24

Solve the system by substitution. {2x−3y=12−12y+8x=48\begin{cases} 2 x - 3 y = 12 \\ −12 y + 8 x = 48 \end{cases}

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5.2 Exercise 25

Solve the system by substitution. {5x+2y=12−4y−10x=−24\begin{cases} 5 x + 2 y = 12 \\ −4 y - 10 x = −24 \end{cases}

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5.2 Exercise 27

Solve the system by substitution. {3x+2y=9y=−32x+1\begin{cases} 3 x + 2 y = 9 \\ y = - \frac{3}{2} x + 1 \end{cases}

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5.2 Exercise 28

Solve the system by substitution. {5x−3y=2y=53x−4\begin{cases} 5 x - 3 y = 2 \\ y = \frac{5}{3} x - 4 \end{cases}

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5.2 Exercise 30

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

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5.2 Exercise 31

The sum of two number is −6. One number is 10 less than the other. Find the numbers.

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5.2 Exercise 33

The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

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5.2 Exercise 34

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

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5.2 Exercise 36

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

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5.2 Exercise 37

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

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5.2 Exercise 39

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

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5.2 Exercise 40

Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

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5.2 Exercise 41

Solve the systems of equations by substitution.

{2x+y=−43x−2y=−6\begin{cases} 2 x + y = −4 \\ 3 x - 2 y = −6 \end{cases}

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5.2 Exercise 43

Solve the systems of equations by substitution.

{x−2y=−52x−3y=−4\begin{cases} x - 2 y = −5 \\ 2 x - 3 y = −4 \end{cases}

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5.2 Exercise 45

Solve the systems of equations by substitution.

{5x−2y=−6y=3x+3\begin{cases} 5 x - 2 y = −6 \\ y = 3 x + 3 \end{cases}

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5.2 Exercise 47

Solve the systems of equations by substitution.

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

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5.2 Exercise 49

Solve the systems of equations by substitution.

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

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5.2 Exercise 51

Solve the systems of equations by substitution.

{3x−2y=6y=23x+2\begin{cases} 3 x - 2 y = 6 \\ y = \frac{2}{3} x + 2 \end{cases}

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5.2 Exercise 53

Solve the systems of equations by substitution.

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

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5.2 Exercise 55

Solve the systems of equations by substitution.

{3x+y=1−4x+y=15\begin{cases} 3 x + y = 1 \\ −4 x + y = 15 \end{cases}

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5.2 Exercise 57

Solve the systems of equations by substitution.

{x+3y=13x+5y=−5\begin{cases} x + 3 y = 1 \\ 3 x + 5 y = −5 \end{cases}

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5.2 Exercise 59

Solve the systems of equations by substitution.

{2x+y=5x−2y=−15\begin{cases} 2 x + y = 5 \\ x - 2 y = −15 \end{cases}

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5.2 Exercise 61

Solve the systems of equations by substitution.

{y=−2x−1y=−13x+4\begin{cases} y = −2 x - 1 \\ y = - \frac{1}{3} x + 4 \end{cases}

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5.2 Exercise 63

Solve the systems of equations by substitution.

{y=2x−8y=35x+6\begin{cases} y = 2 x - 8 \\ y = \frac{3}{5} x + 6 \end{cases}

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5.2 Exercise 65

Solve the systems of equations by substitution.

{4x+2y=88x−y=1\begin{cases} 4 x + 2 y = 8 \\ 8 x - y = 1 \end{cases}

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5.2 Exercise 67

Solve the systems of equations by substitution.

{15x+2y=6−5x+2y=−4\begin{cases} 15 x + 2 y = 6 \\ −5 x + 2 y = −4 \end{cases}

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5.2 Exercise 69

Solve the systems of equations by substitution.

{y=3x6x−2y=0\begin{cases} y = 3 x \\ 6 x - 2 y = 0 \end{cases}

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5.2 Exercise 71

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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5.2 Exercise 73

Solve the systems of equations by substitution.

{y=−4x4x+y=1\begin{cases} y = −4 x \\ 4 x + y = 1 \end{cases}

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5.2 Exercise 75

Solve the systems of equations by substitution.

{y=78x+4−7x+8y=6\begin{cases} y = \frac{7}{8} x + 4 \\ −7 x + 8 y = 6 \end{cases}

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5.2 Exercise 77

Translate to a system of equations and solve.

The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

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5.2 Exercise 79

Translate to a system of equations and solve.

The sum of two numbers is −26. One number is 12 less than the other. Find the numbers.

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5.2 Exercise 81

Translate to a system of equations and solve.

The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

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5.2 Exercise 83

Translate to a system of equations and solve.

The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

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5.2 Exercise 85

Translate to a system of equations and solve.

The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

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5.2 Exercise 87

Translate to a system of equations and solve.

The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

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5.2 Exercise 89

Translate to a system of equations and solve.

Jackie has been offered positions by two cable companies. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

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5.2 Exercise 91

Translate to a system of equations and solve.

Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

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5.2 Exercise 93

Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system {56s=70ts=t+12\begin{cases} 56 s = 70 t \\ s = t + \frac{1}{2} \end{cases}.

  1. ⓐ for tt to find out how long it will take Tina to catch up to Stephanie.
  2. ⓑ what is the value of ss, the number of hours Stephanie w…
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5.2 Exercise 95

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

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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.