Q. 14 - Precalculus Enhanced with Graphing Utilities

Question

Solve each linear programming problem.

Maximize $z=5x+3y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x+y\le 8\\ 2x+y\le 10\end{array}\right.$

Step-by-Step Solution

Answer

The maximum value is $28$.We get this value at $(2,6)$

Step 1. Given information

The given system of linear inequalities is :

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x+y\le 8\\ 2x+y\le 10\end{array}\right.$

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)	$z=5x+3y$
$(0,8)$	$z=24$
$(2,6)$	$z=28$
$(5,0)$	$z=25$
$(2,0)$	$z=10$
$(0,2)$	$z=6$