Solving Systems - Elimination NOTES
- 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)