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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

Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Sam Reply
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Mckenzie Reply
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?
Jake Reply
enrique borrowed $23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total $23,500
Nakiya Reply
13.5
Pervaiz
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 cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Bridget Reply
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
help me understand graphs
Marlene Reply
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
what's the formula given? f(x)=?
Melissa
It shows a graph that I wish I could send photo of to you on here
Marlene
Which problem specifically?
Melissa
which problem?
Melissa
I don't know any to be honest. But whatever you can help me with for I can practice will help
Marlene
I got it. sorry, was out and about. I'll look at it now.
Melissa
Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things.
Marlene
I just responded.
Melissa
Thank you
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
WENDY Reply
State the question clearly please
Rich
write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
0.81818
Rich
5/100 = .05 but Rich is right that 9/11 = .81818
Melissa
Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
write in simplest form 3 4/2
August
definition of quadratic formula
Ahmed Reply
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
What Reply
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Mckenzie Reply
x > $110,000
bruce
greater than $110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use?
nawal Reply
$1.38 worth of strawberries + $1.14 worth of blueberries which= $2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times $2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa

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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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