1. 1. What does a need to be if there were infinitely many solutions to the system of equations. y + 2x = 12 2y + 4x = 2a

2. Lesson 3.5 Linear Programming

3. A chauffer must decide between driving his Rolls Royce and his Mercedez Benz. He must make his decision based on the following constraints:  The Rolls costs $1.75 per mile to operate  The Mercedez costs $2 per mile to operate  His expenses can be no more than $200 per day  He can charge $4 per mile with the Rolls  He can charge $6.50 per mile with the Mercedez  He would like his total charges to be at least $600 per day  The Rolls can travel at most 90 miles per day  The mercedez must travel at least 30 miles per day What is the maximum profit in a day?

4. 1. A method called linear programming is used to find optimal solutions. 2. The inequalities contained in the problem are called constraints. 3. The solution to the set of constraints is called the feasible region. This is a graph of the inequalities. 4. The function to be maximized or minimized is called the objective function.

5. Let x = miles for Rolls Let y = miles for Mercedez 1.75x + 2y ≤ 200 4x + 6.5y ≥ 600 x ≤ 90 y ≥ 30 Constraints Feasible Region

6. What is our objective in the problem? To maximize the profit What is the equation for the profit? Objective Function

7. At the corners of the feasible region (30, 74) (80, 30) (90, 30) (90, 37)

8. (30, 74) P = 2.25x30 + 4.50x74 = 400.50 (80, 30) (90, 30) (90, 37) (30, 74) (80, 30) P=2.25x80 + 4.50x30 = 315 (90, 30) P=2.25x90 + 4.50x30 = 337.5 (90, 37) P=2.25x90 + 4.50x37 = 369

9.  Write the constraints (inequalities)  Write the objective function  Graph the constraints  Find the corner points  Plug the corner points into the objective function to find the max or min DON’T PANIC!!!!

10.  Graph the feasible region for each set of constraints. Then, identify the vertices of the feasible region. y ≤ x - 2 y ≥ -x + 4 x ≤ 6 y≥ 0

11.  Graph the feasible region for each set of constraints. Then, identify the vertices of the feasible region. 3x + 2y ≤ 12 3x + 7y ≤ 21 x ≥ 0 y≥ 0

12. The feasible region for a set of constraints has vertices at (-2,0), (3,3), (6,2) and (5,1). Given this feasible region, find the maximum and minimum values of each objective function. #2 P = 2x – 4y #3 M = 4y - x

13. Lesson 3.5.1 Worksheet