The feasible region determined by some constraints is represented by the shaded region in the graph…

The boundary lines of the feasible region are: 1. Line through $A(1,1)$ and $B(3,3)$: equation is $y = x \Rightarrow x - y \le 0$ (or $y - x \ge 0$). 2. Line through $B(3,3)$ and $C(1,7)$: slope is $\frac{7-3}{1-3} = -2$, equation is $y - 3 = -2(x - 3) \Rightarrow 2x + y \le 9$. 3. Vertical line through $A(1,1)$ and $C(1,7)$: $x \ge 1$. 4. Non-negativity constraints: $x \ge 0, y \ge 0$. Therefore, the constraints representing the feasible region are: $2x + y \le 9$, $x - y \le 0$, $x \ge 1$, $x \ge 0, y \ge 0$.

The corner points of the bounded feasible region are $A(1,1)$, $B(3,3)$, and $C(1,7)$. Evaluating the objective function $Z = x + y$ at each vertex: - At $A(1,1)$: $Z = 1 + 1 = 2$ - At $B(3,3)$: $Z = 3 + 3 = 6$ - At $C(1,7)$: $Z = 1 + 7 = 8$ Therefore, the maximum value of $Z$ is $8$, which occurs at $C(1,7)$.

Changing the constraint $2x + y \le 9$ to $2x + y \ge 9$ (or removing the upper bound constraint $2x + y \le 9$) will make the feasible region unbounded.