The feasible solution for a Linear Programming Problem $(LPP)$ is shown in the figure. Let $z = 3x - 4y$ be the objective function. The value of (Maximum value of $z$ + Minimum value of $z$) is equal to $....$

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 feasible region for a $LPP$ is shown in the following figure. Evaluate $Z = 4x + y$ at each of the corner points of this region. Find the minimum value of $Z$,if it exists.

The corner points of the feasible region determined by $A (20, 10)$,$B (18, 12)$,and $C (12, 12)$. The maximum value of the objective function $Z = 2x + 3y$ is . . . . . . .

The corner points of the feasible region determined by the following system of linear inequalities: $2x + y \leq 10$,$x + 3y \leq 15$,$x, y \geq 0$ are $(0,0)$,$(5,0)$,$(3,4)$,and $(0,5)$. Let $Z = px + qy$,where $p, q > 0$. The condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$ is:

If $x+y \leq 2, x \geq 0, y \geq 0$,the point at which the maximum value of $3x+2y$ is attained will be: