https://file.aiquickdraw.com/mathbot/file/172862495465772dlklwo.jpg

The problem asks to minimize \( x - 3y \) for the bounded feasible region shown in the graph.

### Steps to solve the problem:

1. **Identify the vertices of the feasible region**:
   - From the graph, the vertices of the region are where the boundary lines intersect.
   - The vertices are approximately at these points:
     - (1, 0)
     - (4, 7)
     - (7, 7)
     - (10, 0)

2. **Evaluate the objective function \( x - 3y \) at each vertex**:
   - At \( (1, 0) \):  
     \( x - 3y = 1 - 3(0) = 1 \)
   - At \( (4, 7) \):  
     \( x - 3y = 4 - 3(7) = 4 - 21 = -17 \)
   - At \( (7, 7) \):  
     \( x - 3y = 7 - 3(7) = 7 - 21 = -14 \)
   - At \( (10, 0) \):  
     \( x - 3y = 10 - 3(0) = 10 \)

3. **Find the minimum value**:
   - The values of \( x - 3y \) are:
     - 1 at \( (1, 0) \)
     - -17 at \( (4, 7) \)
     - -14 at \( (7, 7) \)
     - 10 at \( (10, 0) \)

   The minimum value is \( -17 \), which occurs at the vertex \( (4, 7) \).

### Final Answer:
The minimum value of \( x - 3y \) is \( -17 \).

Let me know if you'd like further details or explanations!

### Related Questions:
1. How do you determine the vertices of a bounded feasible region in general?
2. What is the geometric interpretation of minimizing or maximizing a linear objective function?
3. Can you explain how linear programming works for more complex feasible regions?
4. How would the problem change if the objective function was \( x + 3y \) instead?
5. How do constraints in optimization problems affect the shape of the feasible region?

### Tip:
Always evaluate the objective function at the vertices of the feasible region in a linear programming problem, as the optimal solution lies at one of the vertices.

Minimize x - 3y for the bounded feasible region displayed above.

Solve the linear programming problem by minimizing x - 3y within the bounded feasible region. Using vertices and optimization techniques, this problem is suitable for high school students in Grades 10-12.

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