Chapter 11, Problem 27CT

To determine

The maximum value of $z=5x+8y$ subject to constraints $x\ge 0,2x+y\le 8,$ and $x-3y\le -3$.

Answer to Problem 27CT

Solution:

The maximum value of $z$ is $64$ and occurs at point $\left(0,8\right)$.

Explanation of Solution

Given information:

Maximize $z=5x+8y$

Subject to constraints

$x\ge 0$

$2x+y\le 8$

$x-3y\le -3$

Explanation:

Step 1: The objective function is $z=5x+8y$, the largest value of $z$ can occur if $x$ and $y$ are the solution of the system of linear inequalities.

Step 2: The constraints from the system of linear inequalities

$\{\begin{array}{l}x\ge 0\\ 2x+y\le 8\\ x-3y\le -3\end{array}$

Step 3: The graph of the system of linear inequalities with shaded region with corner points islabeled is shown below:

From the above graph the corner points are $\left(0,1\right),\left(3,2\right),\left(0,8\right)$.

Step 4: The below table shows the value of $z$ using the corner points:

$\begin{array}{cc}\text{Corner}\text{points}& \text{Value}\text{of}\text{objective}\text{function, z}\\ \left(0,1\right)& z=5\left(0\right)+8\left(1\right)=8\\ \left(3,2\right)& z=5\left(3\right)+8\left(2\right)=31\\ \left(0,8\right)& z=5\left(0\right)+8\left(8\right)=64\end{array}$

From the table, the maximum value of $z$ is $64$, and it occurs at a point $\left(0,8\right)$.

Therefore, the maximum value of $z$ occurs at point $\left(0,8\right)$ and it is $64$.