1. Warm-up Solve each system of equations: 1. 2. 3.

2. 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.

3. 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.

4. Step 1: Develop constraints. Constraints: Inequalities that limit the variables in the situation.

5. Step 2: Develop the objective function. Also called profit or cost function.

6. Step 3: Graph the constraints to find the feasible region Feasible region: The solution region for the system of inequalities created by the constraints.

7. 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.

8. Step 5: Plug the vertex points into the objective function to find the maximum (profit) or minimum (cost).

9. 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?

10. Develop Constraints

11. Objective Function

12. Graph feasible region

13. Vertex Points

14. Max profit?

15. Assignment Classwork: #4, 6, 8 Homework: #1-3, 10a