Warm-up Solve each system of equations:

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