Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

Solve each linear programming problem.

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

Step-by-Step Solution

Answer

The maximum value is $20$. We get this value at point $(0,4)$

Step 1. Given information

The given system of linear inequalities is :

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

The goal of the task is to solve the problem of linear programming. We are looking for the largest value that can occur if z 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)	$z=3x+5y$
$(0,2)$	$z=10$
$(0,4)$	$z=20$
$(2,0)$	$z=6$
$(4,0)$	$z=12$
$(\frac{12}{5},\frac{12}{5})$	$z=19.2$