1. 3.2 Solving Systems Algebraically 2. Solving Systems by Elimination

2. 2) Solving Systems by Elimination By adding and subtracting linear systems, you can “eliminate” a variable and solve for an unknown

3. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 2x – y = 7 {

4. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 2x – y = 7 { Elimination – add or subtract the equations of a linear system until you “eliminate” a variable

5. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 1 2x – y = 7 2 Step 1: Number the equations. {

6. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 1 2x – y = 7 2 Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations. {

7. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 1 2x – y = 7 2 multiply by 2 Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations. {

8. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 1 2x – y = 7 2 multiply by 2 4x – 2y = 14 Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations. { 2

9. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 1 2x – y = 7 2 multiply by 2 4x – 2y = 14 Step 3: Equation subtract { 221

10. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 4x – 2y = 14 Step 3: Equation subtract 2112 - Use subtraction to eliminate x

11. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 4x – 2y = 14 5y = -10 y = -2 Step 3: Equation subtract 2112 -

12. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 4x – 2y = 14 5y = -10 y = -2 Step 4: Use substitution to solve for the remaining unknown. 12 -

13. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. 4x + 3y = 4 Sub y = -2 into 4x + 3(-2) = 4 4x – 6 = 4 4x = 10 x = 2.5 Step 4: Use substitution to solve for the remaining unknown. 11

14. 2) Solving Systems by Elimination Example 1: Solve the system by elimination. Therefore, the solution is (2.5, -2).

15. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. x + 6y = 2 5x + 4y = 36 {

16. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. x + 6y = 2multiply by 5 5x + 4y = 36 { 1 12

17. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. x + 6y = 2multiply by 5 5x + 4y = 36 becomes… 5x + 30y = 10 { 1 121

18. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. x + 6y = 2multiply by 5 5x + 4y = 36 becomes… 5x + 30y = 10 subtract { 1 12112

19. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. 5x + 30y = 10 5x + 4y = 36 1 12 -

20. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. 5x + 30y = 10 5x + 4y = 36 26y = -26 y = -1 1 12 -

21. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. 5x + 30y = 10 5x + 4y = 36 26y = -26 y = -1Sub y = -1 in either equation. 1 12 -

22. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. 5x + 4y = 36 5x + 4(-1) = 36 5x – 4 = 36 5x = 40 x = 8 1 2

23. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. x + 6y = 2 5x + 4y = 36 Therefore, the solution to the system is (8, -1). {

24. 2) Solving Systems by Elimination Example 2: Solve the system by elimination. x + 6y = 2 5x + 4y = 36 Check: 8 + 6(-1) = 25(8) + 4(-1) = 36 2 = 2 36 = 36 {

25. 2) Solving Systems by Elimination Example 3: Solve each system by elimination. a) -3x + 5y = 7b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6 {{

26. 2) Solving Systems by Elimination Example 3: Solve each system by elimination. a) -3x + 5y = 7b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6 {{ Multiply by 2

27. 2) Solving Systems by Elimination Example 3: Solve each system by elimination. a) -6x + 10y = 14b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6 0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12 {{ + +

28. 2) Solving Systems by Elimination Example 3: Solve each system by elimination. a) -6x + 10y = 14b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6 0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12 {{ Always true. The equations represent the same line. The system is dependent. There is an infinite number of solutions. + +

29. 2) Solving Systems by Elimination Example 3: Solve each system by elimination. a) -6x + 10y = 14b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6 0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12 {{ Never true. The equations represent parallel lines. The system is inconsistent. There is no solution. + + Always true. The equations represent the same line. The system is dependent. There is an infinite number of solutions.

30. 2) Solving Systems by Elimination Example 3: Solve each system by elimination. a) -6x + 10y = 14b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6 0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12 {{ Never true. The equations represent parallel lines. The system is inconsistent. There is no solution. + + Always true. The equations represent the same line. The system is dependent. There is an infinite number of solutions. Homework p.128 #18-21, 33, 36, 46, 55, 56, 57, 62

31. Homework p.128 #18-21, 33, 36, 46, 55, 56, 57, 62 Tomorrow: In-class assignment…come prepared!