1. Solving Systems Using Elimination
When both linear equations of a system are in the form Ax + By = C, you can solve the system using elimination. You can add or subtract equations to eliminate a variable.

2. Check Solve the system using elimination. 5x – 6y = -32 3x + 6y = 48
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging x into one of the equations.
5x – 6y = -32
3x + 6y = 48
5x – 6y = -32
5(2) – 6y = -32
10 – 6y = -32
8x + 0y = 16
- 6y = -42
x = 2
y = 7
Since x = 2 and y = 7, the solution is (2,7).
See if (2,7) makes the other equation true.
Check
3x + 6y = 48
= 48
3(2) + 6(7) = 48
48 = 48

3. Check Solve the system using elimination. x + y = 12 x – y = 2
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging x into one of the equations.
x + y = 12
x - y = 2
x + y = 12
(7) + y = 12
y = 5
2x + 0y = 14
x = 7
Since x = 7 and y = 5, the solution is (7,5).
See if (7,5) makes the other equation true.
Check
x - y = 2
2 = 2
(7) - (5) = 2

4. Check Solve the system using elimination. -x + 2y = -1 x – 3y = -1
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging y into one of the equations.
-x + 2y = -1
x – 3y = -1
-x + 2(2) = -1
-x + 4 = -1
-x = -5
0x – y = -2
x = 5
y = 2
Since x = 5 and y = 2, the solution is (5,2).
See if (5,2) makes the other equation true.
Check
x – 3y = -1
5 – 6 = -1
(5) – 3(2) = -1
-1 = -1

5. Check Solve the system using elimination. 3x + y = 8 x – y = -12
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging x into one of the equations.
3x + y = 8
x - y = -12
3x + y = 8
3(-1) + y = 8
-3 + y = 8
4x + 0y = -4
y = 11
x = -1
Since x = -1 and y = 11, the solution is (-1,11).
See if (-1,11) makes the other equation true.
Check
x - y = -12
-12 = -12
(-1) – (11) = -12

6. Since 0 = 0 is a true statement, there are infinitely many solutions.
Solve the system using elimination.
x + 4y = 1
3x + 12y = 3
First, eliminate one variable.
-3( ) ( ) -3 Multiply by -3
x + 4y = 1
3x + 12y = 3
-3x – 12y = -3
3x + 12y = 3
Since 0 = 0 is a true statement, there are infinitely many solutions.
0x + 0y = 0
0 = 0
TRUE!

7. Check Solve the system using elimination. 3x + 4y = -10 5x – 2y = 18
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging x into one of the equations.
3x + 4y = -10
5x – 2y = 18
3x + 4y = -10
2( ) ( ) 2 Multiply by 2
3(2) + 4y = -10
3x + 4y = -10
10x – 4y = 36
6 + 4y = -10
4y = -16
13x + 0y = 26
y = -4
x = 2
Since x = 2 and y = -4, the solution is (2,-4).
See if (2,-4) makes the other equation true.
Check
5x – 2y = 18
= 18
5(2) – 2(-4) = 18
18 = 18

8. Check Solve the system using elimination. 2x – y = 6 -3x + 4y = 1
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging x into one of the equations.
4( ) ( ) 4 Multiply by 4
2x – y = 6
-3x + 4y = 1
2x – y = 6
2(5) - y = 6
8x – 4y = 24
-3x + 4y = 1
10 - y = 6
-y = -4
5x + 0y = 25
y = 4
x = 5
Since x = 5 and y = 4, the solution is (5,4).
See if (5,4) makes the other equation true.
Check
-3x + 4y = 1
= 1
-3(5) + 4(4) = 1
1 = 1

9. Since 0 = 0 is a true statement, there are infinitely many solutions.
Solve the system using elimination.
4x – 2y = 6
-2x + y = -3
First, eliminate one variable.
4x – 2y = 6
-2x + y = -3
2( ) ( ) 2 Multiply by 2
4x – 2y = 6
-4x + 2y = -6
Since 0 = 0 is a true statement, there are infinitely many solutions.
0x + 0y = 0
0 = 0
TRUE!

10. Check Solve the system using elimination. 2x – 3y = 4 3x + 2y = 6
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging x into one of the equations.
2( ) ( ) 2 Multiply by 2
2x – 3y = 4
3x + 2y = 6
2x – 3y = 4
3( ) ( ) 3 Multiply by 3
2(2) + 3y = 4
4x – 6y = 8
9x + 6y = 18
4 + 3y = 4
3y = 0
13x + 0y = 26
y = 0
x = 2
Since x = 2 and y = 0, the solution is (2,0).
See if (2,0) makes the other equation true.
Check
3x + 2y = 6
6 + 0 = 6
3(2) + 2(0) = 6
6 = 6

11. no solution. Since 0 = 8 is a false statement, there is
Solve the system using elimination.
x – 3y = 2
-2x + 6y = 4
First, eliminate one variable.
2( ) ( ) 2 Multiply by 2
x – 3y = 2
-2x + 6y = 4
2x – 6y = 4
-2x + 6y = 4
Since 0 = 8 is a false statement, there is
no solution.
0x + 0y = 8
0 = 8
FALSE!

12. Check Solve the system using elimination. x + y = 6 x + 3y = 10
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging y into one of the equations.
-1( ) ( ) Multiply by -1
x + y = 6
x + 3y = 10
x + y = 6
x + (2) = 6
-x - y = -6
x + 3y = 10
x = 4
0x + 2y = 4
y = 2
Since x = 4 and y = 2, the solution is (4,2).
See if (4,2) makes the other equation true.
Check
x + 3y = 10
4 + 6 = 10
(4) + 3(2) = 10
10 = 10

13. Check Solve the system using elimination. 5x + 7y = -1 4x – 2y = 22
First, eliminate one variable.
Then, find the value of the eliminated variable by plugging x into one of the equations.
2( ) ( ) 2 Multiply by 2
5x + 7y = -1
4x – 2y = 22
5x + 7y = -1
7( ) ( ) 7 Multiply by 7
5(4) + 7y = -1
10x + 14y = -2
28x – 14y = 154
20 + 7y = -1
7y = -21
38x + 0y = 152
y = -3
x = 4
Since x = 4 and y = -3, the solution is (4,-3).
See if (4,-3) makes the other equation true.
Check
4x - 2y = 22
= 22
4(4) – 2(-3) = 22
22 = 22

14. no solution. Since 0 = -9 is a false statement, there is
Solve the system using elimination.
2x – 3y = 6
6x – 9y = 9
First, eliminate one variable.
-3( ) ( ) Multiply by -3
2x – 3y = 6
6x – 9y = 9
-6x + 9y = -18
6x – 9y = 9
Since 0 = -9 is a false statement, there is
no solution.
0x + 0y = -9
0 = -9
FALSE!

15. Solving Systems Using Elimination
Time to Practice!

16. Solving Systems Using Elimination
2x – 3y = 5
x + 2y = -1
x + y = 10
x – y = 2
-x + 4y = 12
2x – 3y = 6
4x + 2y = 1
2x + y = 4
(1,-1)
2x + 5y = 20
3x – 10y = 37
3x + 2y = -19
x – 12y = 19
-2x + y = -1
6x – 3y = 3
4x + y = 8
-3x – y = 0
(11,-2/5)
(6,4)
(-5,-2)
(12,6)
infinitely
many
solutions no
solution
(8,-24)