Question

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

Answer 1

The feasible region determined by the system of constraints $x+2y\le 8$, $3x+2y\le 12$, $x\ge 0$, $y\ge 0$ is as follows.
The corner points of the feasible region are $O(0,0),A(4,0),B(2,3)$ and $C(0,4)$
The values of $Z$ at these corner points are as follows.

Corner point	$Z=-3x+4y$
$O(0,0)$	$0$
$A(4,0)$	$-12$	$\to$ Minimum
$B(2,3)$	$6$
$C(0,4)$	$16$

Therefore, the minimum value of $Z$ is $-12$ at the point $(4,0)$