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Let ’ s Warm Up!

Let ’ s Warm Up!. 1) Solve the system of equations by graphing: 2x + 3y = 12 2x – y = 4 Answer: 2) Find the slope-intercept form for the equation of a line that passes through (0, 5) and is parallel to a line whose equation is 4x – y = 3? Answer: 3)Solve 3│x – 5│= 12 Answer:.

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Let ’ s Warm Up!

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  1. Let’s Warm Up! 1) Solve the system of equations by graphing: 2x + 3y = 12 2x – y = 4 Answer: 2) Find the slope-intercept form for the equation of a line that passes through (0, 5) and is parallel to a line whose equation is 4x – y = 3? Answer: 3)Solve 3│x – 5│= 12 Answer: (3, 2) y=4x+5 x= 1, 9

  2. Let’s chat about finals • Wednesday Jan 22nd : 2, 4, 6 • Thursday Jan 23rd : 1, 3, 5 • Minimum Days • Final Review sheet • due DAY OF FINAL

  3. Mini Quiz Time! • 3 graphing Questions • Get out a pencil please.

  4. 8-2 Substitution Objective: To use the substitution method to solve systems of equations.

  5. Two Algebraic Methods: • Substitution Method • Elimination Method  will learn about next 

  6. RECALL…Three Types of Solutions: Intersection is Solution Infinite Solutions One Solution No Solution Same slope Same y-intercept “Same line  Intersect infinitely” Different slope Different y-intercept “Intersect at one point” Same slope Different y-intercept “Run parallel  Never intersect”

  7. Substitution Method • Use the substitution method when: • one equation is set equal to a variable • y = 2x + 1 or x = 3y - 2

  8. Example 1 • Instead of x = 2 we have: x = y + 2 x + 2y = 11 (y + 2) + 2y = 11 3y + 2 = 11 3y = 9 y = 3 These are all the same! x = 3 + 2 x = 5 Answer: (5,3)

  9. Try with a Mathlete • y = 3x x + 2y = -21 • y = 2x – 6 3x + 2y = 9 Answers: 1) (-3,-9) 2) (3,0)

  10. Example 2 x + 4y = 1 2x – 3y = -9 • First, solve for a variable x = -4y + 1 2(-4y + 1) – 3y = -9 -8y + 2 – 3y = -9 -11y + 2 = -9 -11y = -11 y = 1 Solve for x (because there is no number in front of it) x = -4(1) + 1 x = -3 Answer: (-3,1)

  11. TOO • 2y = -3x 2) 2x – y = -4 4x + y = 5 -3x + y = -9 Answers: 1) (2,-3) 2) (13,30)

  12. x + y = 16x = 16 – y 2y = -2x + 2 2y = -2(16 – y) + 2 2y = -32 + 2y + 2 2y = -30 + 2y 0 = -30 False NO SOLUTION 6x – 2y = -4 y = 3x + 2 6x – 2(3x + 2) = -4 6x – 6x – 4 = -4 -4 = -4 True INFINITELY MANY Special Cases

  13. TOO for Homework 1) y = -x + 3, 2y + 2x = 4 2) x + y = 0, 3x + y = -8 3) y = 3x – 7, 3x – y = 7

  14. Homework Pg. 467 #17-32 left column

  15. Pg. 467 #17-32 Left ColumnSolve using substitution. 17. 26. 20. 29. 23. 32.

  16. MORE Explanations The following slides have more examples and explanations of the substitution method.

  17. Examples: Use the substitution method to solve the system of equations. (use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed) 1) 2x + 3y = 2 x – 3y = –17 1st: Transform one equation to isolate a variable 2nd: Substitute into the other equation and solve for variable #1 3rd: Substitute into transformed equation from 1st step and solve for variable #2 2x + 3y = 2 “x = 3y – 17” 2(3y – 17) + 3y = 2 6y – 34 + 3y = 2 –34 + 9y = 2 +34 +34 9y = 36 9 9 y = 4 x = 3y – 17 “y = 4” x = 3(4) – 17 x = 12 – 17 x = –5 x – 3y = –17 +3y +3y x = 3y – 17 (we picked x – 3y = – 17 because x has a coefficient of 1 and can easily be transformed) One Solution (–5 , 4) Write answer as an ordered pair (x, y):

  18. Examples: Use the substitution method to solve the system of equations. (use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed) 2) –9x + 3y = –21 3x – y = 7 1st: Transform one equation to isolate a variable 2nd: Substitute it into the other equation and solve for variable #1 3rd: Substitute into the transformed equation from 1st step and solve for variable #2 –9x + 3y = –21 “y = 3x – 7” –9x + 3(3x – 7) = –21 –9x + 9x – 21= –21 –21 = –21 3x – y = 7 -3x -3x –y = –3x + 7 -1 -1 -1 y = 3x – 7 True!! (we picked 3x – y = 7 because y has a coefficient of -1 and can easily be transformed) Infinite Solutions Write answer as an ordered pair (x, y):

  19. Examples: Use the substitution method to solve the system of equations. (use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed) 3) 4x – 2y = 5 y = 2x + 1 1st: Transform one equation to isolate a variable 2nd: Substitute into the other equation and solve for variable #1 3rd: Substitute into transformed equation from 1st step and solve for variable #2 4x – 2y = 5 “y = 2x + 1” 4x – 2(2x + 1) = 5 4x – 4x – 2 = 5 – 2 = 5 y = 2x + 1 (already isolated) False!! No Solution

  20. Chapter 8 Systems of Equations 8-2 Substitution We will become experts at solving systems of equations with substitution.

  21. Math Lab • Solve the system with substitution: x = y + 2 x + 2y = 11 (y + 2) + 2y = 11 3y + 2 = 11 3y = 9 y = 3 These are all the same! x = 3 + 2 x = 5 Answer: (5,3)

  22. Math Lab ReviewSubstitution • y = -x + 3 2y + 2x = 4 y = 3x – 7 3x – y = 7 x + y = 0 3x + y = -8

  23. x + y = 16x = 16 – y 2y = -2x + 2 2y = -2(16 – y) + 2 2y = -32 + 2y + 2 2y = -30 + 2y 0 = -30 False NO SOLUTION 6x – 2y = -4 y = 3x + 2 6x – 2(3x + 2) = -4 6x – 6x – 4 = -4 -4 = -4 True INFINITELY MANY Special Cases

  24. Partner Big Yellow Dice Game

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