The minimum value of $Z = 3x + 4y$ subject to the constraints $x + y \leq 4, x \geq 0, y \geq 0$ is . . . . . . .

The feasible solution for a $LPP$ is shown in the figure. Let $z=3x-4y$ be the objective function. The maximum value of $Z$ occurs at $......$

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

The corner points of the feasible region determined by the system of linear inequalities are $(0,3), (1,1)$ and $(3,0)$. Let $Z = px + qy$ where $p, q > 0$. Find the condition on $p$ and $q$ such that the minimum of $Z$ occurs at both $(3,0)$ and $(1,1)$.

Minimize $Z = 13x - 15y$ subject to the constraints: $x + y \leq 7$,$2x - 3y + 6 \geq 0$,$x \geq 0$,$y \geq 0$.

$A$ feasible solution to an $LP$ problem . . . . . . .