Question

Minimise $Z=3x+5y$
such that $x+3y\ge 3,x+y\ge 2,x,y\ge 0$

Answer 1

The feasible region determined by the system of constraints $x+3y\ge 3,x+y\ge 2,x,y\ge 0$, and $x,y\ge 0$ is as follows.
It can be seen that the feasible region is unbounded.
The corner points of the feasible region are $A(3,0),B(\frac{3}{2},\frac{1}{2})$ and $C(0,2)$
The values of $Z$ at these corner points are as follows.

Corner point	$Z=3x+5y$
$A(3,0)$	$9$
$B(\frac{3}{2},\frac{1}{2})$	$7$	$\to$ Smallest
$C(0,2)$	$10$

As the feasible region is unbounded, therefore, $7$ may or may not be the minimum value of $Z$
For this, we draw the graph of the inequality, $3x+5y<7$ and check whether the resulting half plane has points in common with the feasible region or not It can be seen that the feasible region has no common point with $3x+5y<7$
Therefore, the minimum value of $Z$ is $7$ at $(\frac{3}{2},\frac{1}{2})$