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Course: 8th grade (Eureka Math/EngageNY) > Unit 4

Lesson 4: Topic D: Systems of linear equations and their solutions
Math
8th grade (Eureka Math/EngageNY)
Module 4: Linear equations
Topic D: Systems of linear equations and their solutions
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Systems of equations with elimination

Google Classroom
Walk through examples of solving systems of equations with elimination.
In this article, we're going to be solving systems of linear equations using a strategy called elimination. First, we need to understand that it's okay to add equations together.
Key idea: Anytime we have two true equations, we can add or subtract them to create another true equation.
For example, here are two very basic true equations:
2=2
5=5
We can add these equations together to create another true equation:
2=2+    5=57=7
Or we can subtract these equations to create another true equation:
2=2−    5=5−3=−3
Here's another example with more complicated equations:
2x+3=7+    4x+1=96x+4=16
Great, now that we see that it's okay to add or subtract equations, we can move onto solving a system of equations using elimination.

Solving a system of equations using elimination

We'll solve this system of equations as an example:
x+3y=8        Equation 1
4x−3y=17        Equation 2
The hard thing about solving is that there are two variables x and y. If only we could get rid of one of the variables...
Here's an idea! Let's add the two equations together to cancel out the y variable:
x+3y=8+    4x−3y=175x+0=25
Brilliant! Now we have an equation with just the x variable that we know how to solve:
5x+0=255x=25 x=5Divide each side by 5
Baller! Let's use the first equation to find y when x equals 5:
x+3y=8Equation 15+3y=8Substitute 5 for x3y=3Subtract 5 from each sidey=1Divide each side by 3
Sweet! So the solution to the system of equations is (5,1).
Use elimination to solve the following system of equations.
4y−2x=4
5y+2x=23
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Multiplying one of the equations by a constant, then using elimination

That last example worked out great because the y variable was eliminated when we added the equations. Sometimes it isn't quite that easy.
Take this system of equation as an example:
6x+5y=28        Equation 1
3x−4y=1        Equation 2
If we add these equations, neither the x or y variable will be eliminated, so that won't work. Here are the steps for problems like this:
Step 1: Multiply one of the equations by a constant so that when we add it to the other equation, one of the variables is eliminated.
−2(3x−4y)=−2(1)Multiply the second equation by−2−6x+8y=−2Simplify to get a new equation
Step 2: Add the new equation to the equation we didn't use in step 1 in order to eliminate one of the variables.
6x+5y=28Equation 1+ −6x+8y=−2The new equation13y=26
Step 3: Solve for y.
13y=26y=2Divide each side by 13
Step 4: Substitute y=2 into one of the original equations, and solve for x.
3x−4y=1Equation 23x−4(2)=1Substitute 2 for y3x−8=13x=9Add 8 to each sidex=3Divide each side by 3
So our solution is (3,2).
Use elimination to solve the following system of equations.
8x+14y=12
−6x−7y=−16
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Multiplying both of the equations by a constant, then using elimination

Sometimes we'll need to multiply both equations by a constant when using elimination.
For example, consider this system of equations:
5x+3y=14        Equation 1
3x+2y=8        Equation 2
Here are the steps to solve a system of equations like this one:
Step 1: Multiply each equation by a constant so we can eliminate one variable.
5x+3y=14⇒Multiply by 2⇒10x+6y=28
3x+2y=8⇒Multiply by 3⇒9x+6y=24
Step 2: Combine the new equations to eliminate one variable.
10x+6y=289x+6y=24x+0=4Subtract the equations
Step 3: Substitute x=4 into one of the original equations, and solve for y.
3x+2y=8Equation 23(4)+2y=8Substitute 4 for x12+2y=82y=−4Subtract 12 from each sidey=−2Divide each side by 2
So our solution is (4,−2).
Use elimination to solve the following system of equations.
5x+4y=−14
3x+6y=6
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Let's practice!

1) Use elimination to solve the following system of equations.
3y+x=7
2y−x=−2
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

2) Use elimination to solve the following system of equations.
−7y−4x=1
7y−2x=53
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

3) Use elimination to solve the following system of equations.
−9y+4x−20=0
−7y+16x−80=0
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

4) Use elimination to solve the following system of equations.
3x−11y=−1
2x−5y=−3
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Challenge problem

A school is selling tickets to a play. On the first day of ticket sales, the school sold 6 adult tickets and 10 student tickets for a total of $140. On the second day of ticket sales, the school sold 7 adult tickets and 3 student tickets for a total of $94.
Write and solve a system of equations to find the cost of an adult ticket and the cost of a student ticket.
The cost of an adult ticket is $
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
.
The cost of a student ticket is $
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
.

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