The maximum value of $z = 4x + y$ subject to the constraints $x + y \leq 5$, $2x + y \leq 7$, $3x + 2y \leq 11$, $x \geq 0$, $y \geq 0$ is:

Determine graphically the minimum value of the objective function
$Z = -50x + 20y$ .....$(1)$
subject to the constraints:
${2x - y \geqslant -5}$ .....$(2)$
${3x + y \geqslant 3}$ .....$(3)$
${2x - 3y \leqslant 12}$ .....$(4)$
${x \geqslant 0, y \geqslant 0}$ .....$(5)$

In a Linear Programming Problem ($L$.$P$.$P$.), the corner points of the feasible region defined by the constraints $3x - y \geq 6$, $x \leq 3$, $y \leq 2$, $y \geq 0$, and $x \geq 0$ are:

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 corner points of the bounded feasible region are $(60,0), (120,0), (60,40), (40,20)$ and $(20,30)$. For the objective function $z=5x+10y$:
$(i)$ Maximum value of $z$.
$(ii)$ Minimum value of $z$.
$(iii)$ Maximum value of $z$ occurs at.
$(iv)$ Minimum value of $z$ occurs at.

Maximize the function $Z = 11x + 7y$,subject to the constraints:
$x \leq 3, y \leq 2, x \geq 0, y \geq 0$