# 9.    Solve the following Linear Programming Problems graphically:           Maximise $Z = -x+2y$           Subject to the constraints:$x\geq 3,x+y\geq 5,x+2y\geq 6,y\geq 0.$           Show that the minimum of Z occurs at more than two points.

S seema garhwal

The region determined by constraints $x\geq 3,x+y\geq 5,x+2y\geq 6,y\geq 0.$is as follows,

The corner points of the feasible region are $A(6,0),B(4,1),C(3,2)$

The value of these points at these corner points are :

 Corner points $Z = -x+2y$ $A(6,0)$ - 6 minimum $B(4,1)$ -2 $C(3,2)$ 1 maximum

The feasible region is unbounded, therefore 1 may or may not be the maximum value of Z.

For this, we draw $-x+2y> 1$ and check whether resulting half plane has a point in common with a feasible region or not.

We can see the resulting feasible region has a common point with a feasible region.

Hence , Z =1 is  not maximum value , Z has no maximum value.

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