Question

Solve the following Linear Programming Problems graphically:
Minimise $Z=x+2y$
subject to $2x+y\ge 3,x+2y\ge 6,x,y\ge 0$.

Answer 1

Given objective function is $Z=x+2y$

We have to minimize Z on constraints

$2x+y\ge 3$

$x+2y\ge 6$

$x\ge 0,y\ge 0$

After plotting the inequalities we got the feasible region as shown in the image

Now there are two corner points $(0,3)and(6,0)$ lying on same line $x+2y=6$

Value at corner points are :

Corner Points	Value of $Z=x+2y$
(0,3)	6 (minimum)
(6,0)	6 (minimum)

Since, feasible region is unbounded. So, 6 may or may not be minimum value.

Now to check if 6 is minimum or not, we have to draw $Z<6\Rightarrow x+2y<6$

Since this region doesn't have any common region with feasible region.

So, 6 is the minimum value of Z.