Question

In Problems 51 and 52, solve each linear programming problem.

Maximize $z=3x+4y$ when $x\ge 0,y\ge 0,3x+2y\ge 6,x+y\le 8$

Short answer

By maximizing $z=3x+4y$when $x\ge 0,y\ge 0,3x+2y\ge 6,x+y\le 8$, the maximum value is $z=32$which occurs at $x=0;y=8$

Step-by-step solution

Step 1. Given

The problem:

$z=3x+4y$

The constraints:

$$\begin{gathered} x\ge 0, \\ y\ge 0, \\ 3x+2y\ge 6, \\ x+y\le 8 \end{gathered}$$

Step 2. Graph the constraints.

The graph of the constraints (the feasible points) is illustrated:

Step 3. List the corner points

We list the corner points and evaluate the objective function at each.

S.No.	Corner points (x,y)	Value of the objective function $z=3x+4y$
1.	$(0,3)$	$$\begin{gathered} z=3\left(0\right)+4\left(3\right) \\ =12 \end{gathered}$$
2.	$(0,8)$	$$\begin{gathered} z=3\left(0\right)+4\left(8\right) \\ =32 \end{gathered}$$
3.	$(2,0)$	$$\begin{gathered} z=3\left(2\right)+4\left(0\right) \\ =6 \end{gathered}$$
4.	$(8,0)$	$$\begin{gathered} z=3\left(8\right)+4\left(0\right) \\ =24 \end{gathered}$$

Step 4. Maximize the objective function.

From the table, the maximum value of the given objective function is $32$ and it occurs at $(0,8)$