https://file.aiquickdraw.com/mathbot/file/17286236679950xj4mxpp.jpg

The problem asks to minimize the expression \( -5x + 2y \) over a bounded feasible region, which is represented graphically.

### Steps to solve:
1. **Identify the vertices of the feasible region**: The solution to linear programming problems on a bounded region typically occurs at one of the vertices of the polygon formed by the constraints.

   From the graph, the vertices of the polygon appear to be:
   - \( (1, 0) \)
   - \( (4, 7) \)
   - \( (8, 7) \)
   - \( (9, 0) \)

2. **Evaluate the objective function at each vertex**:
   - At \( (1, 0) \): \( -5(1) + 2(0) = -5 \)
   - At \( (4, 7) \): \( -5(4) + 2(7) = -20 + 14 = -6 \)
   - At \( (8, 7) \): \( -5(8) + 2(7) = -40 + 14 = -26 \)
   - At \( (9, 0) \): \( -5(9) + 2(0) = -45 \)

3. **Determine the minimum value**:
   - The values of the objective function at each vertex are:
     - \( (1, 0): -5 \)
     - \( (4, 7): -6 \)
     - \( (8, 7): -26 \)
     - \( (9, 0): -45 \)
   
   The minimum value is \( -45 \), which occurs at the vertex \( (9, 0) \).

### Conclusion:
The minimum value of \( -5x + 2y \) is \( -45 \), and it occurs at \( (9, 0) \).

Would you like any further explanation or details on this? 

Here are five related questions:
1. What is linear programming, and how is it applied in this context?
2. Why are the vertices of the feasible region the key to solving such optimization problems?
3. What role do constraints play in forming the feasible region?
4. Can you explain the graphical method used in linear programming?
5. How does the objective function differ in minimization versus maximization problems?

**Tip:** In linear programming, always check the values of the objective function at the vertices of the feasible region since these are the potential candidates for optimal solutions.

Minimize -5x + 2y for the bounded feasible region displayed above.

Learn how to minimize the objective function -5x + 2y over a bounded feasible region. This problem involves linear programming concepts such as feasible regions, vertices, and the fundamental theorem of linear programming. Suitable for students in Grades 10-12.

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