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 23 rd : 2, 4, 6  Thursday Jan 24 th : 1, 3, 5 Minimum Days Final Review sheet  due DAY OF FINAL Extra Credit: “Additional Practice Problems”

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 One Solution No Solution Infinite Solutions Same slope Different y-intercept “Run parallel  Never intersect” Same slope Same y-intercept “Same line  Intersect infinitely” Different slope Different y-intercept “Intersect at one point”

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 1) y = 3x x + 2y = -21 2) 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 x = -4(1) + 1 x = -3 Answer: (-3,1) Solve for x (because there is no number in front of it)

11. TOO 1) 2y = -3x2) 2x – y = -4 4x + y = 5 -3x + y = -9 Answers: 1) (2,-3) 2) (13,30)

12. Special Cases x + y = 16 x = 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

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

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

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

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