SOLVING
SYSTEMS USING
ELIMINATION
The goal is to ELIMINATE
one of the variables by
ADDING/SUBTRACTING
your equations.
Elimination Method - Steps
Step 1: Line up your variables, equal signs, and
constants.
Step 2: MULTIPLY one or both of the equations by
a constant to obtain coefficients that are the
same or differ only in sign for one of the
variables.
Step 3: ADD/Subtract the revised equations from
Step 1. Combining like terms will eliminate one
of the variables. Solve for the remaining
variables.
Step 4: SUBSTITUTE the value obtained in Step 2
into either of the original equations and solve
for the other variable.
Elimination Method –Example 1
4x - 2y = 80
x + 2y = 45
First, look to see if any terms can cancel out. In this
case, they can! Which one, x or y?
4x - 2y = 80
+ x + 2y = 45
5x + 0 = 125
5x = 125
x = 25
Now that you know x = 25, plug it back in to either
equation to solve for y:
4x - 2y = 80
-2y = -20
4(25) – 2y = 80
100 – 2y = 80
y!
y = 10
The solution is (25, 10)
Elimination Method –Example 2
6x – 2y = 20
3x + 3y = 18
In this case, neither variable can be cancelled out. Which
term is going to be easier to work with?
x!
6x – 2y = 20
-2(3x + 3y = 18)
To cancel out the “3x”, we will need to multiply it by a “-2”.
You should get….
6x – 2y = 20
-6x – 6y = -36
Now, you can add them like on the previous example!
6x – 2y = 20
+ -6x – 6y = -36
0 – 8y = -16
- 8y = -16
y=2
Now that you know y = 2 plug it back in to either
equation to solve for x:
3x + 3y = 18
3x + 3 (2) = 18
3x + 6 = 18
3x = 12
x =4
The solution to the systems of equations is (4, 2)
Elimination Method –Example 3
2x + 6y = 28
3x + 4y = 27
By looking at the variables, neither the x or the y is going to be
easy to cancel. In this case, we will have to multiply both
equations by something! Let’s cancel out the x…
2 and 3 are both factors of …
6
So let’s multiply the first equation by 3 and the
bottom by -2.
3 ( 2x + 6y = 28 )
-2 (3x + 4y = 27)
6x + 18y = 84
+
-6x – 8y = -54
0 + 10y = 30
10y = 30
y=3
Now that you know y = 3 plug it back in to either equation
to solve for x:
2x + 6y = 28
2x + 6 (3) = 28
2x + 18 = 28
x=5
The solution is (5, 3)
In one week, a music store sold 9 guitars for a
total of $3611. Electric guitars sold for $479
each and acoustic guitars sold for $339 each.
How many of each type of guitar were sold?
Step 1: Identify your variables
x = number of electric guitars
y = number of acoustic guitars
Step 2: Write a systems of equations
x+y=9
479x + 339y = 3611
In one week, a music store sold 9 guitars for a total of
$3611. Electric guitars sold for $479 each and acoustic
guitars sold for $339 each. How many of each type of
guitar were sold?
Step 3: Solve the systems using elimination.
-479( x +
y=
9)
479x + 339y = 3611
+
-479x – 479y = -4311
x + (5) = 9
479x + 339y = 3611
x = 4
0x - 140y = -700
y = 5 and x = 4
- 140y = -700
y = 5
The solution is 4 electric and
5 acoustical guitars!