Consider a linear programming problem with the following constraints and objective function, C.
y≥–x
y≤x+8
y≤–5x–4
C=–7x+2y

What is the maximum value of the objective function for the feasible region?

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 \( (-4, 4) \).

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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.

Consider a linear programming problem with the following constraints and objective function, C. y≥–x, y≤x+8, y≤–5x–4, C=–7x+2y. What is the maximum value of the objective function for the feasible region?

Solve a linear programming problem with constraints y≥–x, y≤x+8, and y≤–5x–4, and objective function C=–7x+2y. Find the maximum value of the objective function for the feasible region. This problem involves graphing inequalities, finding intersection points, and evaluating objective functions at the vertices. Ideal for high school students in Grades 11-12 studying optimization.

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