Question

Solve the following problem graphically:
Minimise and Maximise
$z=3x+9y$
Subject to the constraints:
$x+3y\le 60$
$x+y\ge 10$
$x\le y$
$x\ge 0,y\ge 0$.

Answer 1

Let $Z=3x+9y....\left(1\right)$
Converting inequalities to equalities
$x+3y=60$

$x$	$0$	$60$
$y$	$20$	$0$

Points are $(0,20),(60,0)$
$x+y=10$

$x$	$0$	$10$
$y$	$10$	$0$

Points are $(0,10),(10,0)$
$x-y=0$

$x$	$0$	$10$	$20$
$y$	$0$	$10$	$20$

Points are $(0,0),(10,10),(20,20)$
Plot the graph for the set of points
The graph shows the bounded feasible region. $ABCD$, with corner points $A=(10,0),B=(5,5),C=(15,15)$ and $D=(0,20)$
To find maximum and minimum

Corner point	$Z=3x+9y$
$A=(0,10)$	$90$
$B=(5,5)$	$60$
$C=(15,15)$	$180$
$D=(0,20)$	$180$

From the graph maximum value of $X$ occurs at two corner points $C(15,5)$ and $D(0,20)$ with value $180$
and minimum occurs at point $B(5,5)$ with value $60$.