Question

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

Answer 1

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

Corner point	$Z=3x+4y$
$O(0,0)$	$0$
$A(4,0)$	$12$
$B(0,4)$	$16$	$\to$ Maximum

Therefore, the maximum value of $Z$ is $16$ at the point $B(0,4)$