Q. The maximum value of $2x + y$ subject to $3x + 5y \leq 26$ and $5x + 3y \leq 30, x \geq 0, y \geq 0$ is
MHT CETMHT CET 2018
Solution:
We have, Maximize
$Z=2 x+y$
Subject to constraints
$3 x+5 y \leq 26$
$5 x+3 y \leq 30 $
$x \geq 0, y \geq 0$
The graph of inequality are
Corner points
$ Z = 2x + y$
$(0 , 0)$
$Z = 0$
$(0, \frac{26}{5})$
$Z = \frac{26}{5} = 5.2$
$(6, 0)$
$ Z = 12$ (Maximum)
$( \frac{9}{2}, \frac{5}{2})$
$ Z = \frac{18}{2} + \frac{5}{2} = \frac{23}{2} = 11.5$
Maximum value $Z = 12$
| Corner points | $ Z = 2x + y$ |
|---|---|
| $(0 , 0)$ | $Z = 0$ |
| $(0, \frac{26}{5})$ | $Z = \frac{26}{5} = 5.2$ |
| $(6, 0)$ | $ Z = 12$ (Maximum) |
| $( \frac{9}{2}, \frac{5}{2})$ | $ Z = \frac{18}{2} + \frac{5}{2} = \frac{23}{2} = 11.5$ |