The corner points of the feasible region determined by the system of linear constraints are $(0, 10)$, $(5, 5)$, $(15, 15)$, and $(0, 20)$. Let $z = px + qy$ where $p, q > 0$. The condition on $p$ and $q$ such that the maximum value of $z$ occurs at both points $(15, 15)$ and $(0, 20)$ is

Solve the following linear programming problem graphically:
Maximise $Z = 4x + y$......$(1)$
subject to the constraints:
${x + y \leqslant 50}$.......$(2)$
${3x + y \leqslant 90}$......$(3)$
${x \geqslant 0, y \geqslant 0}$......$(4)$

$A$ toy company manufactures two types of dolls,$A$ and $B$. Market research and available resources have indicated that the combined production level should not exceed $1200$ dolls per week and the demand for dolls of type $B$ is at most half of that for dolls of type $A$. Further,the production level of dolls of type $A$ can exceed three times the production of dolls of type $B$ by at most $600$ units. If the company makes a profit of $Rs. 12$ and $Rs. 16$ per doll respectively on dolls $A$ and $B$,how many of each should be produced weekly in order to maximize the profit?

In the following figure, the shaded region represents the system of constraints:

For the following shaded region,the linear constraints are:

The objective function $z = 4x + 5y$ subject to the constraints $2x + y \geq 7$,$2x + 3y \leq 15$,$y \leq 3$,$x \geq 0$,and $y \geq 0$ has a minimum value at which point?