Question

Find graphically, the maximum value of $z=2x+5y$, subject to constraints given below:
$2x+4y\le 8$
$3x+y\le 6$
$x+y\le 4$
$x\ge 0,y\le 0.6$

Answer 1

Maximise $z=2x+5y$, subject to the constraints
$2x+4y\le 8\Rightarrow x+2y\le 4$
$3x+y\le 6,x+y\le 4,x\ge 0,y\ge 0$.

Draw the lines $x+2y=4$ (passes through $(4,0),(0,2)$);
$3x+y=6$ (passes through $(2,0),(0,6)$) and
$x+y=4$ (passes through $(4,0),(0,4)$).
Shade the region satisfied by the given inequations.

The shaded region in the figure gives the feasible region determined by the given inequations.

Solving $3x+y=6$ and $x+2y=4$ simultaneously, we get
$x=\frac{8}{5}$ and $y=\frac{6}{5}$

We observe that the feasible region OABC is a convex polygon and bounded and has corner points.
$O(0,0),A(2,0),B(\frac{8}{5},\frac{6}{5}),C(0,2)$

The optimal solution occurs at one of the corner points.
At $O(0,0),z=2.0+5.0=0;$
At $A(2,0),z=2.2+5.0=4;$

At $B(\frac{8}{5},\frac{6}{5}),z=2.\frac{8}{5}+5.\frac{6}{5}=\frac{46}{5};$
At $C(0,2),z=2.0+5.2=10;$

Therefore, $z$ maximum value at $C$ and maximum value $=10.$