1. Elimination

2. Take a look at these two equations. 2x + 5y = 17 6x – 5y = -9 Notice that both equations has a 5y term in it. One that is positive and one that is negative. This makes these terms opposites. What happens to opposite terms when you combine them?

3. Opposite terms cancel out when you combine them. Or they are “eliminated”. Our second method for solving a system of equations is elimination. This method works by eliminating one of the variables from both equations.

4. When you are solving two equations, a system of equations, you are looking for the (x, y) that is a solution for both equations. One way to do that is a method called Substitution. How do you think substitution works? Discuss your answer with your elbow partner.

5. Elimination– How it Works Eliminate the terms with the same variable in both equations. The terms must have the same coefficient, sign does not matter. If the terms have the same sign you will subtract the equations together to eliminate the terms. If the terms are opposites you will add the equations together to eliminate the terms. 2x + 5y = 176x – 5y = -9 The terms are 2x, 5y, 17, 6x, -5y, and -9 The coefficients are 2, 5, 6, and -6

6. Elimination –2x + 5y = 17 Example 1 6x – 5y = -9 Since the y terms have the same coefficient, you will eliminate y. Since the coefficients are OPPOSITES you will ADD the equations together. Solve this equation for x. 2x + 5y = 17 6x – 5y = -9 2x + 5y = 17 + 6x – 5y = -9 8x = 8 x = 1

7. Elimination – 2x + 5y = 17 Example 1 Continued6x – 5y = -9 Use your answer for x to find y. Substitute x into either equation. Solve this equation for y. Write your answer as (x, y). 2(1) + 5y = 17 (I used the first equation because it was all positive) 2(1) + 5y = 17 2 + 5y = 17 5y = 15 y = 3 (1, 3)

8. Elimination – 3x + y = 20 Example 2 x + y = 12 Since the __ terms have the same coefficient, you will eliminate __. Since the coefficients have the SAME signs you will SUB the equations together. (Same Sign Subtract) Solve this equation for ___. 3x + y = 20 x + y = 12 3x + y = 20 - x + y = 12 2x = 8 x = 4

9. Elimination – 3x + y = 20 Example 2 Continued x + y = 12 Use your answer for x to find y. Substitute x into either equation. Solve this equation for y. Write your answer as (x, y). 4 + y = 12 y = 8 (4, 8 )

10. What if you have two equations like these.-2x + 15y = -32 7x – 5y = 17 None of the terms have the same coefficients. What could you do to change that?

11. In this case, you multiply one or both equations by a constant so that the coefficients of one variable are the same or opposites. Then eliminate the variable.

12. Elimination – -2x + 15y = -32 Example 3 7x – 5y = 17 To eliminate one variable, you can multiply the second equation by 3. Since the coefficients for y are now opposites, add to eliminate. Solve this equation for ___. 3(7x – 5y = 17) 21x – 15y = 51 (use this equation as second one now) -2x + 15y = -32 +21x – 15y = 51 19x= 19 x = 1

13. Elimination – -2x + 15y = -32 Example 3 Continued 7x – 5y = 17 Use your answer for x to find y. Substitute x into __________ equation. Solve this equation for ___. Write your answer as (x, y). 7(1) – 5y = 17 7 – 5y = 17 -5y = 10 y = -2 (___, ___)

14. Got it? Solve the system using elimination. -3x – 3y = 9 3x – 4y = 5 (Which variable will you eliminate because the coefficients are opposites? Opposite Signs Add!)

15. Got it? Solve the system using elimination. 5x + 7y = 77 5x + 3y = 53 (Which variable will you eliminate because the coefficients are the same? Same Sign Subtract!)

16. Got it? Solve the system using elimination. 3x + 2y = 1 4x + 3y = -2 Do you need to multiply one or both equations to eliminate a variable? What are you going to multiply by?

17. How do you feel about elimination? Does it make sense to you? Do you need to talk through it? Do you have any questions? Make sure you understand before starting your assignment.

18. Assignment: TenMarks.com Finish 8 th Grade Pre-Test. Do your homework if you have finished the Pre-Test.