Q. 10 - Precalculus Enhanced with Graphing Utilities

Question

Solve each linear programming problem.

Maximize $z=x+3y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 3\\ x\le 5\\ y\le 7\end{array}\right.$

Step-by-Step Solution

Answer

Maximum value is $26$, we get this value at $(5,7)$

Step 1. Given information

The given system of linear inequalities is :

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 3\\ x\le 5\\ y\le 7\end{array}\right.$

The goal of the task is to solve the problem of linear programming. We are looking for the largest value that can occur if 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)	$z=x+3y$
$(0,3)$	$z=9$
$(0,7)$	$z=21$
$(3,0)$	$z=3$
$(5,0)$	$z=5$
$(5,7)$	$z=26$