Chapter 11, Problem 96RE

To determine

Solve each linear programming problem.

Answer to Problem 96RE

  $\boxed{z=4at\left(0,4\right)}$

Explanation of Solution

Given information:

Solve each linear programming problem.

Minimize $z=3x+y$ subject to $x\ge 0,y\ge 0,x\le 8,y\le 6,2x+y\ge 4$ .

Calculation:

Consider the function.

  $z=3x+y$

To minimize the given objective function first graph the given inequalities and label the corner points.

Now evaluate the given function $z=3x+y$ , and tabulate as follows,

  $\begin{array}{cc}\begin{array}{l}CornerPo\mathrm{int}\\ \left(x,y\right)\end{array}& \begin{array}{l}ValueOfTheObjectiveFunction\\ z=3x+y\end{array}\\ \left(0,4\right)& z=3\left(0\right)+4=4\\ \left(0,6\right)& z=3\left(0\right)+6=6\\ \left(8,6\right)& z=3\left(8\right)+6=30\\ \left(8,0\right)& z=3\left(8\right)+0=24\\ \left(2,0\right)& z=3\left(2\right)+0=6\end{array}$

The minimum value of the objective function is $4$ at the coordinates $\left(0,4\right)$ .

Hence $\boxed{z=4at\left(0,4\right)}$ .