3.4: Linear ProgrammingObjectives:Students will be able to…• Use linear inequalities to optimize the value of some

quantity• To solve linear programming problems

Linear Programming VocabOptimization: finding the max or min value of some quantity

Linear Programming: optimize a linear function that is subject to a system of linear inequalities

Objective Function: What you are trying to maximize or minimize

Constraints: All the linear inequalities that describe the restrictions in the problem

Feasible Region: The shaded region that satisfies all the constraints

The Key to Solving Linear Programming problems….•The max or min value of the objective

function will occur at one of the vertices of the feasible region

•If the feasible region is bounded, you are guaranteed both a max and a min

Steps to Linear Programming1. Identify all constraints (restrictions) on the variables.2. Write inequalities to model constraints (if not already done)3. Write the Objective Function (if not already done). What do you

want to maximize of minimize?4. Graph all constraints (inequalities) on the same coordinate

plane. Darken the area that satisfies all constraints. This is the feasible region.

5. Identify ALL vertices of the feasible region. (May have to solve a system of linear equations to find a vertex)

6. Substitute each ordered pair for each vertex into the Objective Function to find the max or min value.

Let’s start easy….

Objective Function:C= - x + 3y

Constraints: x> 2x< 5y> 0y< -2x +12Find the max and min value.

Good times….

Objective Function:C= 2x -y

Constraints:

Find the max value

6

2

0

yx

xy

x

Prize…

•What does having the constraints of y>0 and x>0 mean when you graph? Why would you use these constraints?

Handout problems

1. You have 30 acres to grow strawberries and tomatoes. You want at least 10 acres to grow strawberries and at least 5 acres to grow tomatoes. The profit for strawberries is $80 per acre; the profit for tomatoes is $60 per acre. How many acres of each crop should you plant to maximize your profit? What is your max profit?