# Feasible region (shaded) for a LPP is shown. Maximise $z=5x+7y$

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:
The corner points of the feasible region are $O(0,0),A(7,0),B(3,4)$ and $C(0,2)$.
The lines are subjected to a constraint of $z=5x+7y$.
The values of the objective function at those points are as follows :
Step 2:
For the point $O(0,0)$ the value of the objective function is $z=5x+7y\Rightarrow 5\times 0+7\times 0=0$
For the point $A(7,0)$ the value of the objective function is $z=5x+7y\Rightarrow 5\times 7+7\times 0=35$
For the point $B(3,4)$ the value of the objective function is $z=5x+7y\Rightarrow 5\times 3+7\times 4=43$
For the point $C(0,2)$ the value of the objective function is $z=5x+7y\Rightarrow 5\times 0+7\times 2=14$
Step 3:
Clearly the maximum value is $43$ at $B(3,4)$