1. Do Now The cost of renting a pool at an aquatic center id either $30 an hr. or $20 an hr. with a $40 non refundable deposit. Use algebra to find for how many hours the cost of renting the pool is the same for both plans. Let x = # of hrs 30x = 20x + 40 10x = 40 X = 4

2. Do Now Solve using any method. Determine if the system is consistent, inconsistent, dependent and independent. 2x – 3y = 15 y = -3x + 3 2x – 3y = 15 x = -3y + 3

3. 3-4: Linear Programming Target: I can solve problems using linear programming

4. Linear Programming Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy.

5. Find the minimum and maximum value of the function f(x, y) = 3x - 2y. We are given the constraints: y 2 1 x 5 y x + 3

6. Linear Programming Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed. Substitute the vertices into the function and find the largest and smallest values.

7. 6 4 2 2 3 4 3 1 1 5 5 7 8 y x + 3 y 2 1 x 5

8. Linear Programming The vertices of the quadrilateral formed are: (1, 2) (1, 4) (5, 2) (5, 8) Plug these points into the function f(x, y) = 3x - 2y

9. Linear Programming f(x, y) = 3x - 2y f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1 f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5 f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11 f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

10. Linear Programming f(1, 4) = -5 minimum f(5, 2) = 11 maximum

11. Example 2 Find the minimum and maximum value of the function f(x, y) = 4x + 3y We are given the constraints: y -x + 2 y x + 2 y 2x -5

12. 6 4 2 5 34 5 1 1 2 3 y -x + 2 y 2x -5

13. Vertices f(x, y) = 4x + 3y f(0, 2) = 4(0) + 3(2) = 6 f(4, 3) = 4(4) + 3(3) = 25 f(, - ) = 4( ) + 3(- ) = -1 =

14. Linear Programming f(0, 2) = 6 minimum f(4, 3) = 25 maximum

15. Homework Homework P. 160– #10 – 17 Challenge - 24