Solve the Linear Programming Problem graphically:
Minimise $Z = 3x + 5y$
subject to the constraints:
$x + 3y \geq 3$
$x + y \geq 2$
$x, y \geq 0$

Maximize $Z=3x+4y$,subject to the constraints: $x+y \leq 1, x \geq 0, y \geq 0$.

In the following figure,the feasible region (shaded) for a $LPP$ is shown. Determine the maximum and minimum value of $Z=x+2y$.

Show that the minimum of $Z$ occurs at more than two points.
Minimise and Maximise $Z = x + 2y$
subject to $x + 2y \geq 100, 2x - y \leq 0, 2x + y \leq 200; x, y \geq 0$.

The corner points of the feasible region determined by the system of linear constraints are $(0,10), (5,5), (15,15), (5,25)$. Let $z = px + qy$ where $p, q > 0$. The condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points $(15,15)$ and $(5,25)$ is . . . . . . .

The linear programming problem $(LPP)$ to maximize $z = 2x + 5y$ subject to the constraints $x + 3y \leq 6$, $2x + 6y \leq 18$, $x \geq 0$, and $y \geq 0$ has: