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Solving Systems Using Elimination

Learn how to solve systems of equations algebraically using the elimination method. Follow step-by-step examples to eliminate variables and find solutions in ordered pairs.

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Solving Systems Using Elimination

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  1. Solving Systems Using Elimination Objective: To solve systems of equations algebraically

  2. System of Equations • Remember: • 2 or more equations that use the same 2 or more variables. • Solved by graphing and substitution • Now … We have elimination - Just another method of solving systems

  3. Solving By Elimination • Also Known as the Addition/Subtraction Method • You’re going to ELIMINATE one of the two variables • Solve for the 2nd variable • Use the solution of the 2nd variable to solve for the 1st. • Answer comes in an ordered pair: (x, y)

  4. Example #1 2x + 5y = 17 6x – 5y = -9 • Step #1: Align Equals Sign • Step #2: Align Variables • Step #3: Ask yourself “If I add/subtract vertically, does something get eliminated?”

  5. Example #1 2x + 5y = 17 6x – 5y = -9 8x = 8 YES!!! Now, solve for x 8x = 8 8 8 x = 1

  6. x = 1 2x + 5y = 17 2(1) + 5y = 17 2 + 5y = 17 -2 -2 5y = 15 5 5 y = 3 Now plug 1 in for x and solve for y in either equation Our Solution is (1, 3) Example #1

  7. Example #2 2x +5y = -22 10x + 3y = 22 • Step #1: Align Equals • Step#2: Align Variables • Step #3: Ask Yourself “If I add/subtract vertically, does something get eliminated?”

  8. Example #2 2x +5y = -22 10x + 3y = 22 12x + 8y = 0 2x +5y = -22 10x + 3y = 22 -8x + 2y = -44 Nothing gets eliminated by adding … Try Subtracting Nothing there, either. What to do?

  9. Example #2 2x +5y = -22 10x + 3y = 22 5 (2x + 5y) = 5(-22) 10x + 25y = -110 10x + 25y = -110 10x + 3y = 22 • If you multiply the top equation by 5 … would that simplify things? • Don’t forget to multiply both sides. • How about now?Can something get eliminated?

  10. Example #2 10x + 25y = -110 10x + 3y = 22 22y = -132 22 22 y = -6 10x + 3(-6) = 22 10x – 18 = 22 + 18 +18 10x = 40 10 10 x = 4 • Yes! By Subtraction. • Solve for y • Now, plug in the answer to y in either equation • Solve for x • Our solution is (4, -6)

  11. 2x + 5y = 18 3x - 4y = 4 Step #1: Align equals sign Step #2: Align variables Step #3: Ask yourself If you add/subtract vertically, does something get eliminated? Example #3

  12. 2x + 5y = 18 3x - 4y = 4 5x + y = 22 2x + 5y = 18 3x - 4y = 4 -x + 9y = 14 Nothing gets eliminated by adding. Nothing gets eliminated by subtracting either Example #3

  13. 2x + 5y = 18 3x - 4y = 4 3(2x + 5y) = 3(18) 2(3x – 4y) = 2(4) 6x + 15y = 54 6x – 8y = 8 In this system, both equations must be multiplied. We’re going to multiply the 1st equation by 3 and the 2nd equation by 2 Now we can subtract!!! Example #3

  14. 6x + 15y = 54 6x – 8y = 8 23y = 46 23 23 y = 2 2x + 5y = 18 Don’t forget 15 – (-8) is 23. Now solve for x. This is one of our original equations Example #3

  15. 2x + 5y = 18 2x + 5(2) = 18 2x + 10 = 18 -10 -10 2x = 8 2 2 x = 4 Replace y with 2 Subtract 10 from both sides Divide by 2 Our Solution is (4,2) Example #3

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