Question

Maximize $Z=12x+24y$ subject to the constraints $x+y\ge 5,5x+7y\le 35,x-y\ge 0,x,y\ge 0$ graphically.

Answer 1

$Z=12x+24y\left(\text{Given}\right)$

Subject to the constraints:

$x+y\ge 5$

$5x+7y\le 35$

$x-y\ge 0$

$x,y\ge 0$

Case I:- $x+y\ge 5$

$x$	$0$	$5$
$y$	$5$	$0$

Case II:- $5x+7y\le 35$

$x$	$0$	$7$
$y$	$5$	$0$

Case II:- $x-y\ge 0$

$x$	$0$	$1$
$y$	$0$	$1$

$MaxZ=12x+24y$

Corner points	Value of $Z$
$(\frac{5}{2},\frac{5}{2})$	$90$
$(2.9,2.9)$	$104.4$
$(5,0)$	$60$
$(7,0)$	$84$

$MaxZ=104.4$ when $x=y=2.9$.

Hence the maximum value of $Z$ is $104.4$.