Minimize $Z = 13x - 15y$ subject to the constraints: $x + y \leq 7$,$2x - 3y + 6 \geq 0$,$x \geq 0$,$y \geq 0$.

(C) We need to minimize $Z = 13x - 15y$ subject to the constraints $x + y \leq 7$,$2x - 3y + 6 \geq 0$,$x \geq 0$,and $y \geq 0$. The feasible region is determined by these inequalities as shown in the figure.
The shaded region is a polygon with vertices $O(0, 0)$,$A(7, 0)$,$B(3, 4)$,and $C(0, 2)$.

Corner Point	Value of $Z = 13x - 15y$
$O(0, 0)$	$13(0) - 15(0) = 0$
$A(7, 0)$	$13(7) - 15(0) = 91$
$B(3, 4)$	$13(3) - 15(4) = 39 - 60 = -21$
$C(0, 2)$	$13(0) - 15(2) = -30$

Comparing the values of $Z$ at these corner points,the minimum value of $Z$ is $-30$ at the point $(0, 2)$.