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. Solving a system of equations by substitution
Pick the easier equation. The goal
is to get y= ; x= ; a= ; etc.
Step 1: Solve an equation for one variable.
Step 2: Substitute
Put the equation solved in Step 1
into the other equation.
Substitute the value of the variable
into the equation.
Step 3: Plug back in to find the other variable.
Step 4: Check your solution.
Substitute your ordered pair into
BOTH equations.

4. 1) Solve the system using substitution
x + y = 5
y = 3 + x
Step 1: Solve an equation for one variable.
The second equation is
already solved for y!
Step 2: Substitute
x + y = 5 x + (3 + x) = 5
2x + 3 = 5
2x = 2
x = 1

5. 1) Solve the system using substitution
x + y = 5
y = 3 + x
x + y = 5
(1) + y = 5
y = 4
Step 3: Plug back in to find the other variable.
(1, 4)
(1) + (4) = 5
(4) = 3 + (1)
Step 4: Check your solution.
The solution is (1, 4). What do you think the answer would be if you graphed the two equations?

6. 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)

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

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

13. 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.

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

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

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

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

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

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

20. 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

21. 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

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

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

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

25. 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)

26. 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

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

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

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

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

31. 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)

32. 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

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