Sullivan Algebra and Trigonometry: Section 12.9 Objectives of this Section Set Up a Linear Programming Problem Solve a Linear Programming Problem.

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Sullivan Algebra and Trigonometry: Section 12.9 Objectives of this Section Set Up a Linear Programming Problem Solve a Linear Programming Problem

A linear programming problem in two variables x and y consists of maximizing (or minimizing) a linear objective function z = Ax + By, A and B are real numbers, not both 0 subject to certain constraints expressible as linear inequalities in x and y.

Begin by graphing each inequality. Find the feasible region, that is, all the points in the plane that satisfy all constraints.

y = -3x + 16 (0,0) (0, 20/3) (16/3, 0) (4, 4) Feasible Region

The maximum value of z is 24 and occurs at the point (4, 4).

Theorem: Location of the Solution of a Linear Programming Problem If a linear programming problem has a solution, it is located at a corner point of the graph of the feasible points. If a linear programming problem has multiple solutions, at least one of them is located at a corner point of the graph of the feasible points. In either case, the corresponding value of the objective function is unique.