# Determine the maximum value of $z=3x+4y$ if the feasible region (shaded) for a LPP is shown in fig:

Toolbox:
• Let $R$ be the feasible region for a linear programming problem and let $z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region.
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R
Step 1:
Given lines are $x+2y=76$ and $2x+y=104$.
Now the corner points that can be obtained from the given graph are :
$O(0,0),A(52,0),E(44,16)$ and $D(0,38)$
Step 2:
It is clear that when $2x+y=104$ meet the $x$-axis $2x+0=104\Rightarrow x=52$
Hence point $A$ is $(52,0)$
The point $E$ can be obtained solving the two equations
$2x+4y=152$
$2x+y=104$
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$3y=48$
$y=16$
$2x+y=104$
Hence $2x=104-16$
$2x=88$
$x=44$
Step 3:
The values of objective function at there points are as follows :
For the point $O(0,0)$ the value of the objective function is $z=3x+4y\Rightarrow 3\times 0+4\times 0=0$
For the point $A(52,0)$ the value of the objective function is $z=3x+4y\Rightarrow 3\times 52+4\times 0=156$
For the point $E(44,16)$ the value of the objective function is $z=3x+4y\Rightarrow 3\times 44+4\times 16=196$
For the point $D(0,38)$ the value of the objective function is $z=3x+4y\Rightarrow 3\times 0+4\times 38=152$
Hence the maximum value is $196$ at $E(44,16)$