Maximize $Z = 3x + 4y$
subject to the constraints : $x + y \le 4,x \ge 0,y \ge 0.$

The system of constraints is
$x + y \le 4$ ...(1)

and $x \ge 0,y \ge 0$ ...(2)

Let $l:x + y = 4$

The shaded region in the adjoining figure is

the feasible region determined by the system of constraints (1) and (2).

It is observed that the feasible region OAB is bounded.

Thus, we use Comer Point Method to determine the maximum value of Z.

We have : $Z = 3x + 4y$ ...(3)

The co-ordinates of O, A and B are (0, 0), (4, 0) and (0, 4) respectively.

Hence, ${Z_{\max }} = 16$ at the point (0, 4).