# 5.2 Solve systems of equations by substitution  (Page 5/5)

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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.

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.

$\left\{\begin{array}{c}2x+y=-4\hfill \\ 3x-2y=-6\hfill \end{array}$

$\left(-2,0\right)$

$\left\{\begin{array}{c}2x+y=-2\hfill \\ 3x-y=7\hfill \end{array}$

$\left\{\begin{array}{c}x-2y=-5\hfill \\ 2x-3y=-4\hfill \end{array}$

$\left(7,6\right)$

$\left\{\begin{array}{c}x-3y=-9\hfill \\ 2x+5y=4\hfill \end{array}$

$\left\{\begin{array}{c}5x-2y=-6\hfill \\ y=3x+3\hfill \end{array}$

$\left(0,3\right)$

$\left\{\begin{array}{c}-2x+2y=6\hfill \\ y=-3x+1\hfill \end{array}$

$\left\{\begin{array}{c}2x+3y=3\hfill \\ y=\text{−}x+3\hfill \end{array}$

$\left(6,-3\right)$

$\left\{\begin{array}{c}2x+5y=-14\hfill \\ y=-2x+2\hfill \end{array}$

$\left\{\begin{array}{c}2x+5y=1\hfill \\ y=\frac{1}{3}x-2\hfill \end{array}$

$\left(3,-1\right)$

$\left\{\begin{array}{c}3x+4y=1\hfill \\ y=-\frac{2}{5}x+2\hfill \end{array}$

$\left\{\begin{array}{c}3x-2y=6\hfill \\ y=\frac{2}{3}x+2\hfill \end{array}$

$\left(6,6\right)$

$\left\{\begin{array}{c}-3x-5y=3\hfill \\ y=\frac{1}{2}x-5\hfill \end{array}$

$\left\{\begin{array}{c}2x+y=10\hfill \\ -x+y=-5\hfill \end{array}$

$\left(5,0\right)$

$\left\{\begin{array}{c}-2x+y=10\hfill \\ -x+2y=16\hfill \end{array}$

$\left\{\begin{array}{c}3x+y=1\hfill \\ -4x+y=15\hfill \end{array}$

$\left(-2,7\right)$

$\left\{\begin{array}{c}x+y=0\hfill \\ 2x+3y=-4\hfill \end{array}$

$\left\{\begin{array}{c}x+3y=1\hfill \\ 3x+5y=-5\hfill \end{array}$

$\left(-5,2\right)$

$\left\{\begin{array}{c}x+2y=-1\hfill \\ 2x+3y=1\hfill \end{array}$

$\left\{\begin{array}{c}2x+y=5\hfill \\ x-2y=-15\hfill \end{array}$

$\left(-1,7\right)$

$\left\{\begin{array}{c}4x+y=10\hfill \\ x-2y=-20\hfill \end{array}$

$\left\{\begin{array}{c}y=-2x-1\hfill \\ y=-\frac{1}{3}x+4\hfill \end{array}$

$\left(-3,5\right)$

$\left\{\begin{array}{c}y=x-6\hfill \\ y=-\frac{3}{2}x+4\hfill \end{array}$

$\left\{\begin{array}{c}y=2x-8\hfill \\ y=\frac{3}{5}x+6\hfill \end{array}$

(10, 12)

$\left\{\begin{array}{c}y=\text{−}x-1\hfill \\ y=x+7\hfill \end{array}$

$\left\{\begin{array}{c}4x+2y=8\hfill \\ 8x-y=1\hfill \end{array}$

$\left(\frac{1}{2},3\right)$

$\left\{\begin{array}{c}-x-12y=-1\hfill \\ 2x-8y=-6\hfill \end{array}$

$\left\{\begin{array}{c}15x+2y=6\hfill \\ -5x+2y=-4\hfill \end{array}$

$\left(\frac{1}{2},-\frac{3}{4}\right)$

$\left\{\begin{array}{c}2x-15y=7\hfill \\ 12x+2y=-4\hfill \end{array}$

$\left\{\begin{array}{c}y=3x\hfill \\ 6x-2y=0\hfill \end{array}$

Infinitely many solutions

$\left\{\begin{array}{c}x=2y\hfill \\ 4x-8y=0\hfill \end{array}$

$\left\{\begin{array}{c}2x+16y=8\hfill \\ -x-8y=-4\hfill \end{array}$

Infinitely many solutions

$\left\{\begin{array}{c}15x+4y=6\hfill \\ -30x-8y=-12\hfill \end{array}$

$\left\{\begin{array}{c}y=-4x\hfill \\ 4x+y=1\hfill \end{array}$

No solution

$\left\{\begin{array}{c}y=-\frac{1}{4}x\hfill \\ x+4y=8\hfill \end{array}$

$\left\{\begin{array}{c}y=\frac{7}{8}x+4\hfill \\ -7x+8y=6\hfill \end{array}$

No solution

$\left\{\begin{array}{c}y=-\frac{2}{3}x+5\hfill \\ 2x+3y=11\hfill \end{array}$

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.

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

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

The numbers are −7 and −19.

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

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.

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

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.

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.

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°.

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.

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°.

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?

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.

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?

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.

## 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 $\left\{\begin{array}{c}15e+30c=435\hfill \\ 30e+40c=690\hfill \end{array}$ 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.

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 $\left\{\begin{array}{c}56s=70t\hfill \\ s=t+\frac{1}{2}\hfill \end{array}$ .

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 $s=2\frac{1}{2}$ hours

## Writing exercises

Solve the system of equations
$\left\{\begin{array}{c}x+y=10\hfill \\ x-y=6\hfill \end{array}$

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

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

## Self check

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

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

Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
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?
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.
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?
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?
90 minutes