Elimination

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 BY ELIMINATION
***Line up like terms vertically between the two
equations before starting elimination**
•Step 1: Choose a variable to eliminate
•Step 2: Eliminate that variable by adding, subtracting
one equation from the other. (Sometimes you have to
multiply first)
•Step 3: Solve the new equation
•Step 4: Plug in your answer to find the other variable
•Step 5: Check your answer

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

5. Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7
Lets add both equations
to each other+
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
2x + 2y = 7
Lets add both equations
to each other+
3x = 12
x = 4
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
2x + 2y = 7
ANS: (4, y)
Lets substitute x = 4 into this
equation.
4 - 2y = 5 Solve for y
- 2y = 1
y =
1
2
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
2x + 2y = 7
ANS: (4, )
Lets substitute x = 4 into this
equation.
4 - 2y = 5 Solve for y
- 2y = 1
y =
1
2 1
2
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
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.

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

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

12. Your Turn…
2x y+ 5=
3x y− 15=
1. 2.
2y x− 5=
6y x+ 11=
ANS: (4, -3) ANS: (-1, 2)

13. One more example
4x = -8y + 20
-4x + 2y = -30

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

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

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

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

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

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

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

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

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

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

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

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

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

27. More complex Problems
Consider the system
3x + 4y = -25
2x - 3y = 6
Multiply by 2
Multiply by -3

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

29. More complex Problems
Consider the system
6x + 8y = -50
-6x + 9y = -18+
17y = -68
y = -4
ANS: (x, -4)

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

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

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