Question

Solve each linear programming problem.

Minimize$z=2x+3y$subject to$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 3\\ x+y\le 9\\ x+3y\ge 6\end{array}\right.$

Short answer

The minimum value is $7.5$at$(1.5,1.5)$

Step-by-step solution

Step 1. Given information

The given system of linear inequalities is :

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 3\\ x+y\le 9\\ x+3y\ge 6\end{array}\right.$

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)	$z=2x+3y$
$(0,9)$	$z=27$
$(0,3)$	$z=9$
$(1.5,1.5)$	$z=7.5$
$(6,0)$	$z=12$
$(9,0)$	$z=18$