Q.Solve graphically: Minimize $Z = 7x + 5y$ subject to $2x + y \le 10,\ x + 3y \le 15,\ x, y \ge 0$, and explain why the minimum value occurs where it does.

Constraints: $2x+y\le10,\ x+3y\le15,\ x,y\ge0$.

At $y=0$: first gives $x\le5$, second gives $x\le15$; binding $x\le5$, vertex $(5,0)$.

At $x=0$: first gives $y\le10$, second gives $y\le5$; binding $y\le5$, vertex $(0,5)$.

Intersection: $2x+y=10,\ x+3y=15$. From the first, $y=10-2x$; substituting, $x+3(10-2x)=15\Rightarrow x+30-6x=15\Rightarrow-5x=-15\Rightarrow x=3,\ y=4$, giving $(3,4)$.

Corners: $(0,0),\ (5,0),\ (3,4),\ (0,5)$.

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