1. Solving Systems of Equations
The Elimination Method

2. Objectives
Learn the procedure of the Elimination Method using addition
Learn the procedure of the Elimination Method using multiplication
Solving systems of equations using the Elimination Method

3. Elimination using Addition
Consider the system
x - 2y = 5
Lets add both equations
to each other
2x + 2y = 7
REMEMBER: We are trying to find the
Point of Intersection. (x, y)

4. Elimination using Addition
Consider the system
x - 2y = 5
Lets add both equations
to each other +
2x + 2y = 7
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.

5. Elimination using Addition
Consider the system
x - 2y = 5
Lets add both equations
to each other +
2x + 2y = 7 3x
= 12
x = 4 
ANS: (4, y)
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.

6. Elimination using Addition
Consider the system
x - 2y = 5
Lets substitute x = 4 into this
equation.
2x + 2y = 7
4 - 2y = 5
Solve for y
- 2y = 1
y = 1 2 
ANS: (4, y)
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.

7. Elimination using Addition
Consider the system
x - 2y = 5
Lets substitute x = 4 into this
equation.
2x + 2y = 7
4 - 2y = 5
Solve for y
- 2y = 1 1 2
y =  1 2
ANS: (4, )
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.

8. Elimination using Addition
Consider the system
3x + y = 14
4x - y = 7
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.

9. Elimination using Addition
Consider the system
3x + y = 14 +
4x - y = 7 7x
= 21
x = 3 
ANS: (3, y)

10. Elimination using Addition
Consider the system
3x + y = 14
Substitute x = 3 into this equation
4x - y = 7
3(3) + y = 14
9 + y = 14
y = 5 
ANS: (3, ) 5
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.

11. Examples… 1. 2.
ANS: (4, -3)
ANS: (-1, 2)

12. Elimination using Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3

13. Elimination using Multiplication
Consider the system
6x + 11y = -5 +
6x + 9y = -3
12x + 20y = -8
When we add equations together,
nothing cancels out

14. Elimination using Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3

15. Elimination using Multiplication
Consider the system
-1 ( )
6x + 11y = -5
6x + 9y = -3

16. Elimination using Multiplication
Consider the system
- 6x - 11y = 5 +
6x + 9y = -3
-2y = 2 
y = -1
ANS: (x, ) -1

17. Elimination using Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
Lets substitute y = -1 into this equation
y = -1
6x + 9(-1) = -3
6x + -9 = -3 +9
6x = 6 
x = 1
ANS: (x, ) -1

18. Elimination using Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
Lets substitute y = -1 into this equation
y = -1
6x + 9(-1) = -3
6x + -9 = -3 +9
6x = 6 
x = 1
ANS: ( , ) 1 -1

19. Elimination using Multiplication
Consider the system
x + 2y = 6
Multiply by -3 to eliminate the x term
3x + 3y = -6

20. Elimination using Multiplication
Consider the system
-3 ( )
x + 2y = 6
3x + 3y = -6

21. Elimination using Multiplication
Consider the system
-3x + -6y = -18 +
3x + 3y = -6
-3y = -24
y = 8 
ANS: (x, 8)

22. Elimination using Multiplication
Consider the system
x + 2y = 6
Substitute y =14 into equation
3x + 3y = -6
y =8
x + 2(8) = 6
x + 16 = 6
x = -10 
ANS: (x, 8)

23. Elimination using Multiplication
Consider the system
x + 2y = 6
Substitute y =14 into equation
3x + 3y = -6
y =8
x + 2(8) = 6
x + 16 = 6 
x = -10
ANS: ( , 8)
-10

24. Examples 1. 2. x + 2y = 5 x + 2y = 4 2x + 6y = 12 x - 4y = 16
ANS: (3, 1)
ANS: (8, -2)

25. More complex Problems Consider the system 3x + 4y = -25 Multiply by 2

26. More complex Problems Consider the system 2( ) 3x + 4y = -25 -3( )
2( )
3x + 4y = -25
-3( )
2x - 3y = 6

27. More complex Problems + Consider the system 6x + 8y = -50
ANS: (x, -4)

28. More complex Problems Consider the system 3x + 4y = -25 2x - 3y = 6
Substitute y = -4
2x - 3(-4) = 6
2x = 6
2x + 12 = 6
2x = -6
x = -3 
ANS: (x, -4)

29. More complex Problems Consider the system 3x + 4y = -25 2x - 3y = 6
Substitute y = -4
2x - 3(-4) = 6
2x = 6
2x + 12 = 6
2x = -6
x = -3 
ANS: ( , -4) -3

30. Examples… 2. 1. 2x + 3y = 1 4x + y = 9 5x + 7y = 3 3x + 2y = 8
ANS: (2, 1)
ANS: (2, -1)