Question

Solve the following using graphical method:
Minimize: $Z=3x+5y$ subject to $2x+3y\ge 12,$ $-x+y\le 3$, $x\le 4,y\ge 3,x\ge 0,y\ge 0$

Answer 1

$z=3x+5y$

corners points are $(0,0),(0,3),(4,0),(\frac{3}{2},3),(4,\frac{4}{3})$

$z$ at $(0,0)=0$

$z$ at $(0,3)=0+5\left(3\right)=15$

$z$ at $(4,0)=4\left(3\right)70=12$

$z$ at $(\frac{3}{2},3)=3\left(\frac{3}{2}\right)+5\left(3\right)=\frac{39}{2}$

$z$ at $(4,\frac{4}{3})=3\left(4\right)+5\left(\frac{4}{3}\right)=\frac{56}{3}$

$\therefore$ $z$ is minimum at $(0,0)$ and $z=0$