1. Thinking Mathematically Algebra: Graphs, Functions and Linear Systems 7.5 Linear Programming

2. Objectives and Constraints Objective function – algebraic expression that describes the quantity to maximized or minimized. Constraints – restrictions on the variables written as a system of linear inequalities. (linear inequalities)

3. Solving a Linear Programming Problem 1.Graph the system of inequalities representing the constraints. 2.Find the value of the objective function at each corner, or vertex, of the graphed region. The maximum and minimum of the objective function occur at one or more of the corner points.

4. Examples Exercise Set 7.5 #1 Constraint vertices (2,10), (7,5), (8,3), (1, 2) Objective Function z = 5x + 6y

5. Examples Exercise Set 7.5 #7 Objective Function z = 10x + 6y Constraints x + y <= 12 x + 2y <= 20 x >= 0 y >= 0

6. Examples Exercise Set 7.5 #15 Each carton of food feeds 12 people. Each carton of clothing helps 5 people. Each 20-cubic foot carton of food weighs 50lbs. Each 10-cubic-foot carton of clothing weighs 20lbs. Total weight per carrier cannot exceed 19,000lbs Total volume per carrier must be no more than 8,000 cubic-feet. Maximize the number of people that can be helped.

7. Thinking Mathematically Chapter 7 Algebra: Graphs, Functions and Linear Systems