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The problem provided is a linear programming problem where you are asked to find the maximum value of \( P = x + 4y \), subject to several constraints. Let's break it down:

### Objective Function:
\[
P = x + 4y
\]

### Constraints:
1. \( 2x + 3y \geq 24 \)
2. \( x \geq 0 \) (Non-negativity constraint for \( x \))
3. \( x \leq 12 \) 
4. \( y \leq 10 \)

### Steps:
1. **Plot the constraints:** The constraints form a feasible region on the \( xy \)-plane. 
   - The first constraint, \( 2x + 3y \geq 24 \), represents a line that separates the plane into two regions.
   - The second constraint, \( x \geq 0 \), keeps the values in the right half-plane.
   - The third constraint, \( x \leq 12 \), limits \( x \) on the right side.
   - The fourth constraint, \( y \leq 10 \), limits \( y \) from going above 10.

2. **Find the corner points:** These are where the lines from the constraints intersect, and the maximum value of the objective function often lies at one of these vertices.

3. **Evaluate the objective function at each vertex:** Plug the corner points into \( P = x + 4y \) to find the value of \( P \) at each point.

### Let's calculate the corner points and solve for the maximum \( P \).

I'll compute the corner points based on the constraints and find the maximum value of \( P \).The corner points from the intersection of the constraints are:

1. \( (-3, 10) \) (obtained from the intersection of \( 2x + 3y = 24 \) and \( y = 10 \), but it is outside the feasible region as \( x \geq 0 \)).
2. \( (12, 0) \) (intersection of \( 2x + 3y = 24 \) and \( x = 12 \)).
3. \( (0, 8) \) (intersection of \( 2x + 3y = 24 \) and \( x = 0 \)).

Now, I will evaluate the objective function \( P = x + 4y \) at the feasible points \( (12, 0) \) and \( (0, 8) \), since \( (-3, 10) \) violates the constraint \( x \geq 0 \). Let's find the maximum value of \( P \).The maximum value of \( P = x + 4y \) is \( 32 \), which occurs at the point \( (0, 8) \).

### Summary:
- Maximum \( P = 32 \) at \( (x, y) = (0, 8) \).

Would you like more details on any step of the solution?

Here are 5 related questions to deepen understanding:
1. How can the graphical method of solving linear programming problems be applied in other situations?
2. What role do corner points play in finding the optimal solution in linear programming?
3. How would the solution change if the constraints were slightly modified?
4. Can we use other methods, like the Simplex algorithm, to solve this problem?
5. How does linear programming apply to real-world optimization problems?

**Tip:** When solving linear programming problems, always check all feasible corner points, as the optimal solution often lies at one of them.

Find the maximum value of P = x + 4y subject to the following constraints: 2x + 3y ≥ 24, x ≥ 0, x ≤ 12, y ≤ 10.

Solve the linear programming problem to find the maximum value of P = x + 4y, subject to constraints like 2x + 3y ≥ 24. This exercise involves key concepts of linear programming and optimization, suitable for high school students in Grades 10-12.

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