Solve the following Linear Programming Problem graphically:
Maximise $Z = 3x + 2y$
subject to the constraints:
$x + 2y \leq 10$
$3x + y \leq 15$
$x, y \geq 0$

The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:

The minimum value of $Z = 2x + 3y$ for the system of linear constraints: $2x + 4y \leq 12$,$x + y \leq 3$,$x \geq 0$,and $y \geq 0$ is . . . . . . .

The feasible region for an $LPP$ is shown in the figure. Let $z = 3x - 4y$ be the objective function. The minimum value of $Z$ is:

For the objective function $Z = 4x + y$ subject to the constraints $x + y \leq 50$,$3x + y \leq 90$,$x \geq 0$,$y \geq 0$,whose corner points of the feasible region are $(0,0)$,$(30,0)$,$(20,30)$,and $(0,50)$,the maximum value of $Z$ is . . . . . . .

The objective function of a Linear Programming Problem $(LPP)$ defined over a convex set attains its optimum value at: