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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?

Kenneth would need to sell 1,000 suits.

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Access these online resources for additional instruction and practice with solving systems of equations by substitution.

Key concepts

  • Solve a system of equations by substitution
    1. Solve one of the equations for either variable.
    2. Substitute the expression from Step 1 into the other equation.
    3. Solve the resulting equation.
    4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
    5. Write the solution as an ordered pair.
    6. Check that the ordered pair is a solution to both original equations.

Practice makes perfect

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

{ 2 x + y = −4 3 x 2 y = −6

( −2 , 0 )

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{ 2 x + y = −2 3 x y = 7

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{ x 2 y = −5 2 x 3 y = −4

( 7 , 6 )

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{ x 3 y = −9 2 x + 5 y = 4

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{ 5 x 2 y = −6 y = 3 x + 3

( 0 , 3 )

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{ −2 x + 2 y = 6 y = −3 x + 1

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{ 2 x + 3 y = 3 y = x + 3

( 6 , −3 )

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{ 2 x + 5 y = −14 y = −2 x + 2

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{ 2 x + 5 y = 1 y = 1 3 x 2

( 3 , −1 )

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{ 3 x + 4 y = 1 y = 2 5 x + 2

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{ 3 x 2 y = 6 y = 2 3 x + 2

( 6 , 6 )

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{ −3 x 5 y = 3 y = 1 2 x 5

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{ 2 x + y = 10 x + y = −5

( 5 , 0 )

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{ −2 x + y = 10 x + 2 y = 16

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{ 3 x + y = 1 −4 x + y = 15

( −2 , 7 )

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{ x + y = 0 2 x + 3 y = −4

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{ x + 3 y = 1 3 x + 5 y = −5

( −5 , 2 )

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{ x + 2 y = −1 2 x + 3 y = 1

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{ 2 x + y = 5 x 2 y = −15

( −1 , 7 )

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{ 4 x + y = 10 x 2 y = −20

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{ y = −2 x 1 y = 1 3 x + 4

( −3 , 5 )

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{ y = x 6 y = 3 2 x + 4

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{ y = 2 x 8 y = 3 5 x + 6

(10, 12)

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{ y = x 1 y = x + 7

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{ 4 x + 2 y = 8 8 x y = 1

( 1 2 , 3 )

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{ x 12 y = −1 2 x 8 y = −6

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{ 15 x + 2 y = 6 −5 x + 2 y = −4

( 1 2 , 3 4 )

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{ 2 x 15 y = 7 12 x + 2 y = −4

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{ y = 3 x 6 x 2 y = 0

Infinitely many solutions

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{ x = 2 y 4 x 8 y = 0

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{ 2 x + 16 y = 8 x 8 y = −4

Infinitely many solutions

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{ 15 x + 4 y = 6 −30 x 8 y = −12

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{ y = −4 x 4 x + y = 1

No solution

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{ y = 1 4 x x + 4 y = 8

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{ y = 7 8 x + 4 −7 x + 8 y = 6

No solution

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{ y = 2 3 x + 5 2 x + 3 y = 11

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Solve Applications of Systems of Equations by Substitution

In the following exercises, 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.

The numbers are 13 and 17.

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The sum of two numbers is 30. One number is 4 less than the other. Find the numbers.

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

The numbers are −7 and −19.

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The perimeter of a rectangle is 50. The length is 5 more than the width. Find the length and width.

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

The length is 20 and the width is 10.

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

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

The length is 34 and the width is 8.

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The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.

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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.

The measures are 16° and 74°.

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The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

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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.

The measures are 45° and 45°.

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Maxim has been offered positions by two car dealers. The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. The second pays a salary of $20,000 plus a commission of $500 for each car sold. How many cars would need to be sold to make the total pay the same?

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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?

80 cable packages would need to be sold.

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Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?

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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?

Mitchell would need to sell 120 stoves.

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Everyday math

When Gloria spent 15 minutes on the elliptical trainer and then did circuit training for 30 minutes, her fitness app says she burned 435 calories. When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. Solve the system { 15 e + 30 c = 435 30 e + 40 c = 690 for e , the number of calories she burns for each minute on the elliptical trainer, and c , the number of calories she burns for each minute of circuit training.

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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 { 56 s = 70 t s = t + 1 2 .

  1. for t to find out how long it will take Tina to catch up to Stephanie.
  2. what is the value of s , the number of hours Stephanie will have driven before Tina catches up to her?

t = 2 hours s = 2 1 2 hours

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Writing exercises

Solve the system of equations
{ x + y = 10 x y = 6

by graphing.
by substitution.
Which method do you prefer? Why?

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Solve the system of equations
{ 3 x + y = 12 x = y 8 by substitution and explain all your steps in words.

Answers will vary.

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Self check

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

This figure shows a table with three rows and four columns. The columns are labeled, “I can…,” “Confidently.” “With some help.” and “No - I don’t get it.” The only column with filled in cells below it is labeled “I can…” It reads, “solve a system of equations by substitution.” “solve applications of systems of equations by substitution.”

After reviewing this checklist, what will you do to become confident for all objectives?

Questions & Answers

what is the answer for this 3×9+28÷4-8
What Reply
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What
In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
Jenise Reply
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
jamie Reply
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary?
Iris Reply
let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is $500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound?
Valeria Reply
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be $10 per square foot?
Nia Reply
I need help with maths can someone help me plz.. is there a wats app group?
Cindy Reply
WY need
Fernando
How did you get $750?
Laura Reply
if y= 2x+sinx what is dy÷dx
formon25 Reply
does it teach you how to do algebra if you don't know how
Kate Reply
Liam borrowed a total of $35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total $35,000? (Round your answer to the nearest tenth of a percent.)
Lara Reply
exact definition of length by bilbao
francemichael Reply
the definition of length
francemichael Reply
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
Chris Reply
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
Chris Reply

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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