Question

If $z=5x+y$ subject to the constraints $3x+y\le 15,4x+3y\le 30$ where $x,y\ge 0,$ then the value of $z_{max}$ is equal to

Answer 1

Given, $z=5x+y$ and constraints $3x+y\le 15,4x+3y\le 30$ where $x,y\ge 0$ $\Rightarrow \frac{x}{5}+\frac{y}{15}\le 1,\frac{x}{\frac{15}{2}}+\frac{y}{10}\le 1,x,y\ge 0$ We get common area as marked in figure
Now, finally

Corner points	$z=5x+y$
$(5,0)$	$25$ (Maximum)
$(3,6)$	$21$
$(0,10)$	$10$
$(0,0)$	$0$

Thus maximum value of the function $z$ is $25$