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The problem provides a graph of a feasible region defined by three constraints:

1. \( 2x + y = 40 \)
2. \( x + 5y = 100 \)
3. \( 8x + 5y = 170 \)

Additionally, we have the objective function \( f = 7x + 4y \), which needs to be optimized (likely maximized or minimized) within the feasible region.

### Steps to solve:

1. **Identify the vertices**: The feasible region is the blue-shaded area where all three lines intersect. We need to find the points of intersection of the three lines because these points form the vertices of the feasible region.

2. **Solve the system of equations**: For each pair of lines, solve them simultaneously to find the intersection points. These points will be the vertices of the region.

3. **Evaluate the objective function**: Once we have the vertices, substitute their coordinates into the objective function \( f = 7x + 4y \) to determine the maximum or minimum value of \( f \).

Let’s first find the points of intersection:

- **Intersection of \( 2x + y = 40 \) and \( x + 5y = 100 \)**:
  
  Solve this system using substitution or elimination.
  
- **Intersection of \( 2x + y = 40 \) and \( 8x + 5y = 170 \)**:
  
  Solve this system.
  
- **Intersection of \( x + 5y = 100 \) and \( 8x + 5y = 170 \)**:
  
  Solve this system.

Would you like me to proceed with solving these intersections for the vertices, or do you have a specific question or direction in mind?

The graph of the feasible region is shown with the objective function f = 7x + 4y. Three constraints are provided: 2x + y = 40, x + 5y = 100, and 8x + 5y = 170. Solve for the optimal value of f within the feasible region.

This example shows how to solve a linear programming problem with three constraints: 2x + y = 40, x + 5y = 100, and 8x + 5y = 170. Learn how to find the vertices of the feasible region and optimize the objective function f = 7x + 4y. Suitable for Grades 10-12.

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