Warm-up• Solve each system of equations: • 1.

• 2.

• 3.

34

75

2

xy

xy

1034

632

yx

yx

1162

5

yx

x

Section 3-4: Linear Programming

Goal 2.10: Use systems of 2 or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix

operations, and algebraic properties.

Vertex Principle of Linear Programming

If there is a maximum or a minimum value of the linear objective function, it occurs at one or more vertices of the feasible region.

Step 1:

• Develop constraints.• Constraints: Inequalities that limit the

variables in the situation.

Step 2:

• Develop the objective function.

• Also called profit or cost function.

Step 3:

• Graph the constraints to find the feasible region

• Feasible region: The solution region for the system of inequalities created by the constraints.

Step 4:

• Find the vertices of the feasible region

• Vertices can be found by graphing the boundary lines of the feasible region to find where they intersect.

Step 5:

• Plug the vertex points into the objective function to find the maximum (profit) or minimum (cost).

Example: • Suppose you are selling cases of mixed nuts

(x) and roasted peanuts (y). You can order no more than a total of 500 cans and packages and spend no more than $600. Mixed nuts cost you $24 per case up front, but sell for $18 profit per case. Roasted peanuts cost $15 per case, but sell for $15 profit per case. How can you maximize your profit? How much is the maximum profit?

Develop Constraints

Objective Function

Graph feasible region

Vertex Points

Max profit?

Assignment

• Classwork: #4, 6, 8

• Homework: #1-3, 10a