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

For a linear programming problem,the objective function is $Z = 3x + 2y$. If the corner points of the bounded feasible region are $(12, 0)$,$(4, 2)$,$(1, 5)$,and $(1, 10)$,then the maximum value of $Z$ is . . . . . . .

For the objective function $Z = 4x + y$ subject to the constraints $x + y \leq 50$,$3x + y \leq 90$,$x \geq 0$,$y \geq 0$,whose corner points of the feasible region are $(0,0)$,$(30,0)$,$(20,30)$,and $(0,50)$,the maximum value of $Z$ is . . . . . . .

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

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 $\dots$

Show that the minimum of $Z$ occurs at more than two points.
Minimise and Maximise $Z = 5x + 10y$
subject to $x + 2y \leq 120, x + y \geq 60, x - 2y \geq 0, x, y \geq 0$.