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This is a linear programming problem where we aim to maximize the objective function \( P = 4x + 5y \), subject to the constraints:

\[
\begin{aligned}
1. &\quad x + 3y \leq 13 \\
2. &\quad 3x + 2y \leq 25 \\
3. &\quad x \geq 0 \\
4. &\quad y \geq 0
\end{aligned}
\]

### Step 1: Plot the constraints
We'll plot the inequalities to find the feasible region:

1. \( x + 3y = 13 \) → This is a straight line passing through points \( (0, \frac{13}{3}) \) and \( (13, 0) \).
2. \( 3x + 2y = 25 \) → This line passes through \( (0, 12.5) \) and \( ( \frac{25}{3}, 0) \).
3. \( x \geq 0 \) and \( y \geq 0 \) are non-negativity constraints, meaning the feasible region lies in the first quadrant.

### Step 2: Find corner points
The feasible region is bounded by the intersections of these lines. We solve for the intersection points:

1. **Intersection of \( x + 3y = 13 \) and \( 3x + 2y = 25 \)**:
   - Multiply the first equation by 3: \( 3x + 9y = 39 \).
   - Subtract the second equation from this: \( (3x + 9y) - (3x + 2y) = 39 - 25 \).
   - This simplifies to: \( 7y = 14 \Rightarrow y = 2 \).
   - Substitute \( y = 2 \) into \( x + 3y = 13 \): \( x + 6 = 13 \Rightarrow x = 7 \).
   - So, one intersection point is \( (7, 2) \).

2. **Intersection of \( x + 3y = 13 \) and the x-axis ( \( y = 0 \) )**:
   - \( x + 3(0) = 13 \Rightarrow x = 13 \).
   - So, another corner point is \( (13, 0) \).

3. **Intersection of \( 3x + 2y = 25 \) and the y-axis ( \( x = 0 \) )**:
   - \( 3(0) + 2y = 25 \Rightarrow y = 12.5 \).
   - So, this corner point is \( (0, 12.5) \).

4. **Intersection of \( 3x + 2y = 25 \) and the x-axis ( \( y = 0 \) )**:
   - \( 3x + 2(0) = 25 \Rightarrow x = \frac{25}{3} \approx 8.33 \).
   - So, this corner point is \( (8.33, 0) \).

### Step 3: Evaluate the objective function at corner points
We now evaluate \( P = 4x + 5y \) at the feasible corner points:

1. \( (7, 2) \): \( P = 4(7) + 5(2) = 28 + 10 = 38 \).
2. \( (13, 0) \): \( P = 4(13) + 5(0) = 52 \).
3. \( (0, 12.5) \): \( P = 4(0) + 5(12.5) = 62.5 \).
4. \( (8.33, 0) \): \( P = 4(8.33) + 5(0) = 33.32 \).

### Step 4: Conclusion
The maximum value of \( P \) occurs at \( (0, 12.5) \), and the maximum value is \( P = 62.5 \).

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the graphical interpretation of linear programming?
2. How do constraints affect the feasible region in linear programming?
3. What is the Simplex method for solving linear programming problems?
4. How can you handle linear programming problems with more than two variables?
5. What is the significance of corner points in linear programming?

**Tip:** In linear programming, the optimal solution always lies at a corner point of the feasible region.

Find the maximum value of P = 4x + 5y subject to the constraints: x + 3y ≤ 13, 3x + 2y ≤ 25, x ≥ 0, y ≥ 0

Solve a linear programming problem by maximizing the objective function P = 4x + 5y, subject to given constraints. Learn the process of solving systems of inequalities and applying optimization techniques. Suitable for students in Grades 10-12.

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