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Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.

Now we’ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.

How to solve a system of equations by elimination

Solve the system by elimination. { 2 x + y = 7 x 2 y = 6

Solution

This figure has seven rows and three columns. The first row reads, “Step 1. Write both equations in standard form. If any coefficients are fractions, clear them.” It also says, “Both equations are in standard form, A x + B y = C. There are no fractions.” It also gives the two equations as 2x + y = 7 and x – 2y = 6. The second row reads, “Step 2: Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites.” It also says, “We can eliminate the y’s by multiplying the first equation by 2. Multiply both sides of 2x + y = 7 by 2.” It also shows the steps with equations. Initially the equations are ex + y = 7 and x – 2y = 6. Then they become 2(2x + y) = 2 times 7 and x – 2y = 6. They then become 4x + 2y = 14 and x – 2y = 6. The third row says, “Step 3: Add the equations resulting from step 2 to eliminate one variable.” It also says, “We add the x’s, y’s, and constants.” It then gives the equation as 5x = 20. The fourth row says, “Step 4: Solve for the remaining variable.” It also says, “Solve for x.” It gives the equation as x = 4. The fifth row says, “Step 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.” It also says, “Substitute x = 4 into the second equation, x – 2y = 6. Then solve for y.” It then gives the equations as x – 2y = 6 which becomes 4 – 2y = 6. This is then −2y = 2, and thus, y = −1. The sixth row says, “Step 6: Write the solution as an order pair.” It also says, “Write it as (x, y).” It gives the ordered pair as (4, −1). The seventh row says, “Step 7: Check that the ordered pair is a solution to both original equations.” It also says, “Substitute (4, −1) into 2x + y = 7 and x – 2y = 6. Do they make both equations true? Yes!” It then gives the equations. 2x + y = 7 becomes 2 times 4 + −1 = 7 which is 7 = 7. x – 2y = 6 becomes 4 – 2 times −1 = 6 which is 6 = 6. The row then says, “The solution is (4, −1).”
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Solve the system by elimination. { 3 x + y = 5 2 x 3 y = 7

( 2 , −1 )

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Solve the system by elimination. { 4 x + y = −5 −2 x 2 y = −2

( −2 , 3 )

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The steps are listed below for easy reference.

How to solve a system of equations by elimination.

  1. Write both equations in standard form. If any coefficients are fractions, clear them.
  2. Make the coefficients of one variable opposites.
    • Decide which variable you will eliminate.
    • Multiply one or both equations so that the coefficients of that variable are opposites.
  3. Add the equations resulting from Step 2 to eliminate one variable.
  4. Solve for the remaining variable.
  5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.
  6. Write the solution as an ordered pair.
  7. Check that the ordered pair is a solution to both original equations.

First we’ll do an example where we can eliminate one variable right away.

Solve the system by elimination. { x + y = 10 x y = 12

Solution

.
Both equations are in standard form.
The coefficients of y are already opposites.
Add the two equations to eliminate y .
The resulting equation has only 1 variable, x .
.
Solve for x , the remaining variable.

Substitute x = 11 into one of the original equations.
.
.
Solve for the other variable, y . .
Write the solution as an ordered pair. The ordered pair is (11, −1).
Check that the ordered pair is a solution
to both original equations.

x + y = 10 11 + ( 1 ) = ? 10 10 = 10 x y = 12 11 ( 1 ) = ? 12 12 = 12
The solution is (11, −1).
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Solve the system by elimination. { 2 x + y = 5 x y = 4

( 3 , −1 )

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Solve the system by elimination. { x + y = 3 −2 x y = −1

( −2 , 5 )

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In [link] , we will be able to make the coefficients of one variable opposites by multiplying one equation by a constant.

Solve the system by elimination. { 3 x 2 y = −2 5 x 6 y = 10

Solution

.
Both equations are in standard form.
None of the coefficients are opposites.
We can make the coefficients of y opposites by multiplying
the first equation by −3.
.
Simplify. .
Add the two equations to eliminate y . .
Solve for the remaining variable, x .
Substitute x = −4 into one of the original equations.
.
.
Solve for y . .
.
.
Write the solution as an ordered pair. The ordered pair is (−4, −5).
Check that the ordered pair is a solution to
both original equations.

3 x 2 y = 2 3 ( 4 ) 2 ( 5 ) = ? 2 12 + 10 = ? 2 2 y = 2 5 x 6 y = 10 3 ( 4 ) 6 ( 5 ) = ? 10 20 + 30 = ? 10 10 = 10
The solution is (−4, −5).
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Solve the system by elimination. { 4 x 3 y = 1 5 x 9 y = −4

( 1 , 1 )

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Solve the system by elimination. { 3 x + 2 y = 2 6 x + 5 y = 8

( −2 , 4 )

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Now we’ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.

Solve the system by elimination. { 4 x 3 y = 9 7 x + 2 y = −6

Solution

In this example, we cannot multiply just one equation by any constant to get opposite coefficients. So we will strategically multiply both equations by a constant to get the opposites.

.
Both equations are in standard form. To get opposite
coefficients of y , we will multiply the first equation by 2
and the second equation by 3.
.
Simplify. .
Add the two equations to eliminate y . .
Solve for x .

Substitute x = 0 into one of the original equations.
.
.
Solve for y . .
.
Write the solution as an ordered pair. The ordered pair is (0, −3).
Check that the ordered pair is a solution to
both original equations.

4 x 3 y = 9 4 ( 0 ) 3 ( 3 ) = ? 9 9 = 9 7 x + 2 y = 6 7 ( 0 ) + 2 ( 3 ) = ? 6 6 = 6
The solution is (0, −3).

What other constants could we have chosen to eliminate one of the variables? Would the solution be the same?

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Questions & Answers

integer greater than 2 and less than 12
Emily Reply
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 Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
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?
Ariana Reply
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?
Kirisma Reply
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.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
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
Niazmohammad
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
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce

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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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