https://file.aiquickdraw.com/mathbot/file/1728400700047xdh396pt.jpg

The problem you uploaded involves solving a linear programming (LP) problem using the graphical method. Here's the breakdown:

### Objective:
Minimize the function \( Z = -4x + 5y \).

### Subject to constraints:
1. \( 2x + 3y \leq 30 \)
2. \( 2x + y \geq 4 \)
3. \( x - y \leq 10 \)
4. \( y \leq 8 \)
5. \( x, y \geq 0 \) (non-negativity constraints)

### Steps to solve graphically:
1. **Plot the constraint lines**:
   - Convert each inequality to an equation and find where they intersect with the axes.
   - Draw the lines on a graph and shade the feasible region where all constraints overlap.
   
2. **Find corner points**:
   - The feasible region will be a polygon. Identify the corner points by solving for the intersections of the constraint lines.

3. **Evaluate the objective function**:
   - Calculate the value of \( Z = -4x + 5y \) at each corner point.
   - The minimum value of \( Z \) at these points will give you the solution.

Would you like a more detailed step-by-step solution with graph plotting or any clarifications? 

Here are five related questions for further practice:
1. How do you identify feasible regions in graphical solutions?
2. What happens if the feasible region is unbounded?
3. Can linear programming problems have multiple optimal solutions?
4. How can you determine if a solution is infeasible?
5. What role does the objective function play in linear programming?

**Tip**: Always check that the solution lies within the feasible region, especially for boundary cases where equalities are involved.

Solve the linear programming problem: Minimize -4x + 5y subject to 2x + 3y <= 30, 2x + y >= 4, x - y <= 10, y <= 8, x, y >= 0.

Learn how to solve a linear programming problem using the graphical method. This problem focuses on minimizing the function -4x + 5y subject to several constraints, and it's suitable for students in higher education or advanced high school.

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