Solve the following Linear Programming Problems graphically :
Minimize $Z = 3 x + 5 y$
subject to $x + y \leq 4, x \geq 0$ and $y \geq 0$

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Solution

solve these two equations
$x + y >= 2$ -->1
$x + 3 y >= 10$ -->1
We will get $(0, 2), (3, 0), (\frac{3}{2}, \frac{1}{2})$
From the graph the corner points are $(0, 2)$ , $(\frac{3}{2}, \frac{1}{2})$ and $(3, 0)$

and the corresponding values of $Z$ are $10$ , $7$ , $9$
manimum value is $7$

Hence the optimal solution is $7$

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