For the $LP$ problem,maximize $z = 2x + 3y$,the coordinates of the corner points of the bounded feasible region are $A(3, 3), B(20, 3), C(20, 10), D(18, 12),$ and $E(12, 12)$. The maximum value of $z$ is $\dots$

The feasible region represented by the constraints $y - 2x \leq 4$, $x + y \geq 5$, $x \leq 4$, $y \geq 2$, and $x, y \geq 0$ is

Maximum value of $z = 3x + 4y$ subject to the constraints $x - y \leqslant -1$,$-x + y \leqslant 0$,and $x, y \geqslant 0$ is:

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:

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

For a linear programming problem, the objective function is $z = px + qy$, where $p, q > 0$. If at the corner points $(0, 10)$ and $(5, 5)$ the values of $z$ are $90$ and $60$ respectively, then the relation between $p$ and $q$ is . . . . . . .