https://file.aiquickdraw.com/mathbot/file/1725460397332kvrw0nmg.jpg

The image contains a mathematical problem that appears to be related to a maximization or minimization problem in the context of linear programming, involving constraints.

### Extracted Problem:
The problem is to **maximize \( Z = 3x + 5y \)** subject to the constraints:
\[
\begin{aligned}
2x + 3y &\leq 24, \\
3x + 2y &\leq 21, \\
x, y &\geq 0.
\end{aligned}
\]

### Solution Outline:
The steps to solve this linear programming problem using the graphical method or simplex method would be:

1. **Graph the Constraints**: Plot the lines corresponding to the equality in each inequality.
   - \( 2x + 3y = 24 \)
   - \( 3x + 2y = 21 \)
   - \( x = 0 \) and \( y = 0 \) are the axes.

2. **Identify the Feasible Region**: The feasible region is the intersection of the half-planes defined by the inequalities.

3. **Find the Corner Points**: Identify the vertices (corner points) of the feasible region.

4. **Evaluate the Objective Function**: Calculate the value of \( Z = 3x + 5y \) at each vertex.

5. **Determine the Maximum Value**: The maximum value of \( Z \) will occur at one of the vertices of the feasible region.

Would you like me to solve this problem step by step, or provide more details on any specific part?

### Related Questions:
1. How do you plot the inequalities on a graph?
2. What is the significance of the feasible region in linear programming?
3. How do you find the intersection points of the lines representing the constraints?
4. What is the Simplex method, and when is it preferred over the graphical method?
5. Can there be cases where the feasible region is unbounded, and how does it affect the solution?

### Tip:
In linear programming, always verify that all constraints are linear (straight lines) and ensure that the feasible region is properly identified before finding the optimal solution.

Maximize Z = 3x + 5y subject to the constraints 2x + 3y ≤ 24, 3x + 2y ≤ 21, x ≥ 0, y ≥ 0.

Learn how to maximize the linear programming function Z = 3x + 5y subject to constraints such as 2x + 3y ≤ 24 and 3x + 2y ≤ 21. This problem explores key concepts in optimization and linear programming. Suitable for students in Grades 10-12.

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