For the $LP$ problem,"Maximize $z = x + 4y$ subject to $3x + 6y \leq 6$,$4x + 8y \geq 16$ and $x \geq 0, y \geq 0$."

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 . . . . . . .

The point at which the maximum value of $Z = 3x + 2y$ subject to the constraints $x + 2y \leq 2$,$x \geq 0$,$y \geq 0$ occurs is $.....$

For a linear programming problem,the objective function is $Z = 8000x + 12000y$. If the corner points of the feasible region are $(0,0)$,$(20,0)$,$(12,6)$,and $(0,10)$,then the maximum value of $Z$ occurs at which corner point?

An oil company has two depots $A$ and $B$ with capacities of $7000 \, L$ and $4000 \, L$ respectively. The company is to supply oil to three petrol pumps,$D, E$ and $F$ whose requirements are $4500 \, L, 3000 \, L$ and $3500 \, L$ respectively. The distances (in $km$) between the depots and the petrol pumps are given in the following table:

From/To	$A$	$B$
$D$	$7$	$3$
$E$	$6$	$4$
$F$	$3$	$2$

Assuming that the transportation cost of $10 \, L$ of oil is $Rs. \, 1$ per $km$,how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?

The constraints $-x_{1} + x_{2} \leq 1$,$-x_{1} + 3x_{2} \leq 9$,and $x_{1}, x_{2} \geq 0$ define: