Minimize $Z = - 3x + 4y$ subject to $x + 2y \le 8,3x + 2y \le 12,x \ge 0,y \ge 0.$

.: The system of constraints is :
$x + 2y \le 8$ ...(1)

$3x + 2y12$ ...(2)

and $x \ge 0,y \ge 0$ ...(3)

Let ${l_1}:x + 2y = 8;{l_2}:3x + 2y = 12$

The shaded region in the adjoining figure is

the feasible region determined by the system of constraints (1) to(3).

It is observed that the feasible region OCEB is bounded.

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

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

The co-ordinates of O,C, E and Bare (0, 0),

(4, 0), (2, 3) (on solving $x + 2y = 8$

and$3x + 2y = 12$) and (0, 4) respectively.

Hence, ${Z_{\min }} = - 12$ at the point (4, 0).