Q. 15 - Precalculus Enhanced with Graphing Utilities

Question

Solve each linear programming problem.

Minimize $z=5x+4y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ 2x+3y\le 12\\ 3x+y\le 12\end{array}\right.$

Step-by-Step Solution

Answer

The minimum value is at $8$ at $(0,2)$

Step 1. Given information

The given system of linear inequalities is :

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ 2x+3y\le 12\\ 3x+y\le 12\end{array}\right.$

The goal of the task is to solve the problem of linear programming. We are looking for the minimum value that can occur if x and y are solutions of the system. We draw a graph of the system and read the corner points from the graph.

Step 2. Draw the graph of the system

Step 3. Make a table for the corner points

Corner point (x,y)	$z=5x+4y$
$(0,2)$	$z=8$
$(0,4)$	$z=16$
$(2,0)$	$z=10$
$(4,0)$	$z=20$
$(\frac{24}{7},\frac{12}{7})$	$z=24$