Question

Maximize $f=4x-y$, subject to the constraints
$7x+4y\le 28$, $2y\le 7$, $x\ge 0$, $y\ge 0$.

Answer 1

Converting inequations into equations we get,

$7x+4y=28$

$2y=7⟹y=\frac{7}{2}=3.5$

$x=0,y=0$

Consider, $7x+4y=28$

$x$	$0$	$4$
$y$	$7$	$0$

The feasible region is $OABC.$

The values of the objective functions are given below:

Points	Value of objective function, $4x-y$
$O(0,0)$	$4\times 0-0=0$
$A(4,0)$	$4\times 4-0=16$
$B(2,3.5)$	$4\times 2-3.5=4.5$
$(0,3.5)$	$4\times 0-3.5=-3.5$

Thus, $f$ is maximum at $A(4,0)$ i.e. $f_{max}=16$