To find the maximum value of the objective function $C = -7x + 2y$, we need to follow these steps:

Step 1: Analyze the constraints

1. $y \geq -x$ — Line with slope $-1$, shaded above the line.
2. $y \leq x + 8$ — Line with slope $1$, shaded below the line.
3. $y \leq -5x - 4$ — Line with slope $-5$, shaded below the line.

Step 2: Find the intersection points

To determine the vertices of the feasible region, we need to solve the systems formed by pairs of intersecting lines.

Intersection of $y = -x$ and $y = x + 8$

$-x = x + 8 \implies 2x = -8 \implies x = -4$ Substitute back into $y = -x$: $y = 4$ So the intersection point is $(-4, 4)$.

Intersection of $y = -x$ and $y = -5x - 4$

$-x = -5x - 4 \implies 4x = -4 \implies x = -1$ Substitute back into $y = -x$: $y = 1$ So the intersection point is $(-1, 1)$.

Intersection of $y = x + 8$ and $y = -5x - 4$

$x + 8 = -5x - 4 \implies 6x = -12 \implies x = -2$ Substitute back into $y = x + 8$: $y = -2 + 8 = 6$ So the intersection point is $(-2, 6)$.

Step 3: Evaluate the objective function at each vertex

$C = -7x + 2y$

1. At $(-4, 4)$: $C = -7(-4) + 2(4) = 28 + 8 = 36$

2. At $(-1, 1)$: $C = -7(-1) + 2(1) = 7 + 2 = 9$

3. At $(-2, 6)$: $C = -7(-2) + 2(6) = 14 + 12 = 26$

Conclusion:

The maximum value of the objective function is 36, which occurs at the vertex $(-4, 4)$.

---

Would you like a detailed plot of the feasible region?

Here are 5 follow-up questions to extend learning:

1. How would you determine the feasible region visually?
2. What happens if the objective function coefficients change?
3. How would the solution change if one of the constraints was removed?
4. How do you check whether the feasible region is bounded or unbounded?
5. Can linear programming be applied to non-linear systems?

Tip: Always evaluate the objective function at all vertices of the feasible region to ensure you've found the optimal solution.