# 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?

DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike
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?
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra