Introduction

Linear programming is a method for finding minimum or maximum value of a function, given a set of constraints, or limits, for the variables in the function.

Here is a simple example. The function you want to maximize is P = -2x + 3y.

The constraints on x and y are:
1 ≤ x ≤ 3
0 ≤ y ≤ 2

You must find the values of x and y, within the given constraints, that give the maximum value for P. To do this, choose a value for each variable and plug it in to the function P. Keep trying until you find the maximum value for P.

For example, suppose you choose x = 2 and y = 1. When you plug it in to P, we get P = -2(2) +3(1) = -1.

So P = -1 (the value of P is -1) when x = 2 and y = 1. Now try other values for x and y until you get the maximum value for P. Then complete the following:

The maximum value for P occurs when x = ______ and y = _____.

For example, suppose you choose x = 2 and y = 1. When you plug it in to P, we get P = -2(2) +3(1) = -1.

So, P = -1 (the value of P is -1) when x -2 and y = 1. Now, try other values for x and y until you get the maximum value for P. Then complete the following:

The maximum value for P occurs when x = 1 and y = 2. The maximum value of P is ______.

 

Lesson Objectives

Following successful completion of this lesson, students will be able to...

  • Represent constraints by inequalities and by systems of inequalities.

Enduring Understandings

  • Optimization is finding the best solution within given constraints.

The above objectives correspond with the Alabama Course of Study Algebra II and Algebra II with Trig standards: 22

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