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7-2 and 7-3 Solving Systems of Equations Algebraically

7-2 and 7-3 Solving Systems of Equations Algebraically. The Substitution and Elimination Method. Solving Systems of Equations.

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7-2 and 7-3 Solving Systems of Equations Algebraically

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  1. 7-2 and 7-3Solving Systems of Equations Algebraically The Substitution and Elimination Method

  2. Solving Systems of Equations • You have already seen how to solve a system of equations by graphing but you won’t necessarily always have a graphing calculator. So there are 2 other ways to solve systems by using algebra… • The 2 other ways to solve systems of equations: • Substitution • Elimination

  3. Solving Systems of Equations • Substitution is used when the equation is already solved for one variable then plug-n-chug that value into the other equation. • Elimination is when you solve for one variable by adding, subtracting, or multiplying to the original equations.

  4. Substitution Ex. 1 Consider the system Substitution is desired if one variable is already by itself or can be solved for very easily. What is already by itself on one side of the equation? x - 2y = 5 y = 2x + 3 x – 2y = 5 x – 2(2x+3) = 5 x – 4x – 6 = 5 -3x – 6 = 5 x = -3 or -11/3 So take y = 2x + 3 and plug it in for y in the other equation. REMEMBER: We are trying to find the Point of Intersection. (x, y)

  5. Substitution Consider the system Now that you’ve found the answer for x = -3 or -11/3 , how do you think you’ll find the answer for y? x - 2y = 5 y = 2x + 3 y = 2x + 3 y = 2(-11/3) + 3 y = -4 or -13/3 So your answer is (-11/3, -13/3) So what you got for x and plug it in and find y. You’ll now have your point of intersection.  REMEMBER: We are trying to find the Point of Intersection. (x, y)

  6. Substitution Ex. 2 Consider the system Substitution is desired if one variable is already by itself or can be solved for very easily. What is already by itself on one side of the equation? y = x + 1 y = 2x - 1 x + 1 = 2x – 1 -x -x 1 = x – 1 +1 +1 x = 2 So take both equations and set them equal to each other since both equal y. REMEMBER: We are trying to find the Point of Intersection. (x, y)

  7. Substitution Consider the system Now that you’ve found the answer for x = 2, how do you think you’ll find the answer for y? y = x + 1 y = 2x - 1 y = x + 1 y = 2 + 1 y = 3 Your answer is (2,3) So what you got for x and plug it in and find y. You’ll now have your point of intersection.  REMEMBER: We are trying to find the Point of Intersection. (x, y)

  8. Substitution Ex. 3 Consider the system Now you try this one all on your own. x - y = 1 x = ½ y + 2 What is your final answer? (3,2) REMEMBER: We are trying to find the Point of Intersection. (x, y)

  9. Now we’re going to look at the Elimination Method

  10. Lets add both equations to each other Elimination using Addition Ex. 4 Consider the system x - 2y = 5 2x + 2y = 7 REMEMBER: We are trying to find the Point of Intersection. (x, y)

  11. + Elimination using Addition Consider the system…what makes it different than the other systems we just solved for? x - 2y = 5 Lets add both equations to each other 2x + 2y = 7 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  12. Elimination using Addition Consider the system x - 2y = 5 Lets add both equations to each other + 2x + 2y = 7 = 12 3x x = 4  ANS: (4, y) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  13. 1 y = 2 Elimination using Addition Consider the system x - 2y = 5 Lets substitute x = 4 into this equation. 2x + 2y = 7 4 - 2y = 5 Solve for y - 2y = 1  ANS: (4, y) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  14. 1 2 1 2 Elimination using Addition Consider the system x - 2y = 5 Lets substitute x = 4 into this equation. 2x + 2y = 7 4 - 2y = 5 Solve for y - 2y = 1 y =  ANS: (4, ) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  15. Elimination using Addition Ex. 5 Consider the system 3x + y = 14 4x - y = 7 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  16. + Elimination using Addition Consider the system 3x + y = 14 4x - y = 7 7x = 21 x = 3  ANS: (3, y)

  17. Elimination using Addition Consider the system 3x + y = 14 Substitute x = 3 into this equation 4x - y = 7 3(3) + y = 14 9 + y = 14 y = 5  ANS: (3, ) 5 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  18. Elimination using Multiplication Ex. 6 Consider the system 6x + 11y = -5 + 6x + 9y = -3 12x + 20y = -8 When we add equations together, nothing cancels out

  19. Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = -3

  20. Elimination using Multiplication Consider the system -1 ( ) 6x + 11y = -5 6x + 9y = -3

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

  22. +9 +9 Elimination using Multiplication Consider the system 6x + 11y = -5 Lets substitute y = -1 into this equation 6x + 9y = -3 y = -1 6x + 9(-1) = -3 6x + -9 = -3 6x = 6  x = 1 -1 ANS: (x, )

  23. +9 +9 Elimination using Multiplication Consider the system 6x + 11y = -5 Lets substitute y = -1 into this equation 6x + 9y = -3 y = -1 6x + 9(-1) = -3 6x + -9 = -3 6x = 6  x = 1 -1 ANS: ( , ) 1

  24. Elimination using Multiplication Ex. 7 Consider the system x + 2y = 6 Multiply by -3 to eliminate the x term 3x + 3y = -6

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

  26. Elimination using Multiplication Consider the system -3x + -6y = -18 + 3x + 3y = -6 -3y = -24 y = 8  ANS: (x, 8)

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

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

  29. Elimination using Multiplication Ex. 8 Consider the system x + 2y = 5 2x + 6y = 12 ANS: (3,1 )

  30. Homework You have a worksheet for classwork/homework TOMORROW. The worksheet does not tell you which method to use. You must figure out if substitution or elimination is the way you want to solve the systems. Whatever you choose is not wrong. It may take longer but whatever you choose is not wrong!  My advice is to go ahead and label S or E on the problems tonight so you have a plan on how to solve them. Make you life easier for tomorrow!!!

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