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

 Page 5 / 5

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?

integer greater than 2 and less than 12
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400? Alicia Reply at least 20 Ayla what are the steps? Alicia 6.4 jobs Grahame 32 Grahame 1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32 Orlando He must work 32 jobs to make a profit POP what is algebra Azhar Reply repeated addition and subtraction of the order of operations. i love algebra I'm obsessed. Shemiah hi Krekar One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag? Leanna Reply they are 92 candies in the bag POP rectangular field solutions Navin Reply What is this? Donna t muqtaar the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is anas Reply ? Choli a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190 Usman Stella bought a dinette set on sale for$725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount? Manhwa Reply 44.19% Scott 40.22% Terence 44.2% Orlando I don't know Donna if you want the discounted price subtract$725 from $1299. then divide the answer by$1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif
Tausif
how
muqtaar
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce