The constraints $-x+y \leq 1, -x+3y \leq 9, x \geq 0, y \geq 0$ define a:

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

The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:

The coordinates of the corner points of the bounded feasible region are $(0, 10)$, $(5, 5)$, $(15, 15)$, and $(0, 20)$. The minimum value of the objective function $z = 3x + 9y$ is . . . . . . .

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

Show that the minimum of $Z$ occurs at more than two points.
Maximize $Z = -x + 2y$,subject to the constraints:
$x \geq 3, x + y \geq 5, x + 2y \geq 6, y \geq 0$