Solve the following Linear Programming Problem graphically:
Minimise $Z = -3x + 4y$
Subject to the constraints:
$x + 2y \leq 8$
$3x + 2y \leq 12$
$x \geq 0, y \geq 0$

The feasible region (shaded) for a $LPP$ is shown in the adjacent figure. Maximize $Z = 5x + 7y$.

Corner points of the feasible region determined by the system of linear constraints are $(0,3)$,$(1,1)$ and $(3,0)$. Let $Z = px + qy$,where $p, q > 0$. The condition on $p$ and $q$ so that the minimum value of $Z$ occurs at $(3,0)$ and $(1,1)$ is . . . . . . .

The solution for minimizing the function $z = x + y$ under an $L$.$P$.$P$. with constraints $x + y \geqslant 2$,$x + 2y \leqslant 8$,$y \leqslant 3$,$x, y \geqslant 0$ is

Maximize $Z=3x+4y$,subject to the constraints: $x+y \leq 1, x \geq 0, y \geq 0$.

The corner points of the feasible region are $(0, 6)$,$(3, 3)$,$(9, 9)$,and $(0, 12)$. What is the maximum value of the objective function $z = 6x + 12y$?