1. 7.3 Solving Systems of Equations The Elimination Method

2. Three Ways to solve a system Graphing Method Substitution Method Elimination Method

3. Review Quickly – Solve System using Substitution 1.) x + 4y = 8 2.) -3x + 2y = -11 2x + 3y = 1 5x – y = 23 Answer: (-4, 3)Answer: (5, 2) Step 1 Step 2 x + 4y = 8 -4y x = -4y +8 2(-4y + 8) + 3y = 1 -8y + 16 + 3y = 1 -5y + 16 = 1 -16 -5y = -15 y = 3

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. (either equation works) 4 - 2y = 5Solve 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 = 5Solve 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. 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 = -8When 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 6x + 11y = -5 6x + 9y = -3 -1 ( )

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

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

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

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

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

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 3x + 3y = -6 ANS: (x, 8) Substitute y =14 into equation y =8 x + 2(8) = 6 x + 16 = 6 x = -10 

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

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

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

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

27. 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 

28. 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

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

30. Examples… 1.2. ANS: (4, -3)ANS: (-1, 2)

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