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Method #1: Substitution Method

Method #1: Substitution Method. Example 1. Solve using substitution. x= 4y x+3y=-21. x + 3y = -21. x = 4y. 4y + 3y = -21. x = 4(-3). 7y = -21. x = -12. y = -3. (-12,-3). EXAMPLE 1:. x= 4y x+3y=-21. Check solution ( -12,-3). x=4y -12=4(-3) -12=-12. x+3y=-21 -12+3(-3)=-21

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Method #1: Substitution Method

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  1. Method #1: Substitution Method

  2. Example 1 Solve using substitution. x= 4y x+3y=-21 x + 3y = -21 x = 4y 4y + 3y = -21 x = 4(-3) 7y = -21 x = -12 y = -3 (-12,-3)

  3. EXAMPLE 1: x= 4y x+3y=-21 Check solution ( -12,-3) x=4y -12=4(-3) -12=-12 x+3y=-21 -12+3(-3)=-21 -12+-9=-21 -21=-21

  4. Example 2 Solve using substitution. y= 4x+1 x - 2y = 12 x - 2y = 12 y = 4x + 1 x– 2(4x + 1) = 12 y = 4(-2) + 1 x – 8x - 2 = 12 y = -8 + 1 -7x - 2= 12 y = -7 -7x = 14 (-2,-7) x = -2

  5. EXAMPLE 3: 2x + y = 7 3x + 2y = 11 3x + 2y = 11 3x + 2(-2x + 7) = 11 3x – 4x + 14 = 11 -x + 14 = 11 -x = -3 x = 3 2x + y = 7 2(3) + y = 7 6 + y = 7 y = 1 2x + y = 7 -2x -2x y = -2x + 7 Why do you think it’s wise to solve for the y in the top equation? The solution is (3, 1)!

  6. EXAMPLE 4: x + y=10 5x - y=2 5x – y = 2 5(-y + 10) – y= 2 -5y + 50 – y = 2 -6y + 50 = 2 -6y = -48 y = 8 x + y = 10 -y -y x = -y + 10 x + y = 10 x + 8 = 10 x = 2 The solution is (2, 8)!

  7. EXAMPLE 4: x + y=10 5x - y=2 Check solution (2,8) 5x-y=2 5(2)-(8)=2 10-8=2 2=2 x + y=10 2+8=10 10=10

  8. EXAMPLE 5: 2x + 3y = 6 5x - 2y = -4 STEP1:In order for us to work together on this example, let’s all solve for the bottom x. ( )5 (0, 2)

  9. 2x + 3y = 6 5x - 2y = -4 EXAMPLE 5: What would this look like if I solved for the y on bottom first? ( )2 (0, 2)

  10. EXAMPLE 6: 3x + 4y = 7 2x - 3y = -18 Let’s start by solving for the top y. ( )4 (-3, 4)

  11. THE END OF NOTES! Here’s some quick practice: 1. 2. 3x – 2y = 12 4x + 3y = 16 Answers: 1) (2, -1) 2) (4, 0)

  12. SYSTEMS OF EQUATIONS Method #2- Elimination

  13. Solving by Elimination Example 1: 2x + 3y = 12 x – 3y = 6 2x + 3y = 12 2(6) + 3y = 12 12 + 3y = 12 3y = 0 y = 0 3x = 18 x = 6 (6,0)

  14. Solving by Elimination Example 2: 3x + 2y = 16 3x + 2(2) = 16 3x + 4 = 16 3x = 12 x = 4 3x + 2y = 16 3x – y = 10 3x + 2y = 16 -3x + y = -10 -1( ) 3y = 6 y = 2 Notice, these aren’t going to eliminate when I add them together. So what next? Just multiply one of the rows by whatever it takes to make them eliminate. (4,2)

  15. Solving by Elimination Example 3: x + 4y = 17 x + 4(3) = 17 x + 12 = 17 x = 5 -3( ) x + 4y = 17 3x – 2y = 9 -3x – 12y = -51 3x – 2y = 9 -14y = -42 y = 3 Notice, these aren’t even close to eliminating. So what next? Just multiply one of the rows by whatever it takes to make them eliminate. (5,3)

  16. Solving by Elimination Example 3 continued: x + 4y = 17 5 + 4y = 17 4y = 12 y = 3 x + 4y = 17 3x – 2y = 9 x + 4y = 17 6x – 4y = 18 2( ) 7x = 35 x = 5 Could you have eliminated the y’s? Yes- If you had multiplied the bottom row by 2. (5,3)

  17. Solving by Elimination Example 4: You try this one. x + 2y = 14 x + 2(4) = 14 x + 8 = 14 x = 6 4x – 3y = 12 x + 2y = 14 4x – 3y = 12 -4x – 8y = -56 -4( ) -11y = -44 y = 4 (6,4)

  18. Solving by Elimination Example 5: 2x + 3y = 6 2x + 3(4) = 6 2x + 12 = 6 2x = -6 x = -3 3( ) 2x + 3y = 6 3x – 4y = -25 6x + 9y = 18 -6x + 8y = 50 -2( ) 17y = 68 y = 4 What are we supposed to do when I can’t multiply one row by something to make it eliminate? Just multiply both of the rows by whatever it takes to make them eliminate. (-3,4)

  19. Solving by Elimination Example 6: This one is for you to try. 5x + 3y = -18 5(-3) + 3y = -18 -15 + 3y = -18 3y = -3 y = -1 4( ) 5x + 3y = -18 7x – 4y = -17 20x + 12y = -72 21x – 12y = -51 3( ) 41x = -123 x = -3 (-3,-1)

  20. Practice: Solve the following systems of equations. For the odd problems, solve by using substitution. For the even problems, solve by using elimination. 1) 2x + y = 6 x – 4y = -15 2) 3x – y = -3 2x + 5y = -19 3) x + 2y = -1 2x + 3y = 2 4) 3x + 5y = 7 6x + y = 5 5) 2x + 3y = 1 3x + 4y = 2 6) 3x – 2y = -10 5x + 2y = 10 7) 3x + 4y = 0 5x – 4y = -8 8) 5x – 3y = -27 2x + 7y = 22 Answers: 1) (1, 4) 2) (-2, -3) 3) (7, -4) 4) (2/3, 1) 5) (2, -1) 6) (0, 5) 7) (-1, ¾) 8) (-3, 4)

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