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This is a linear programming problem where the objective function \( P = 5x + 6y \) needs to be maximized under the given constraints:

\[
\begin{aligned}
1. & \quad x + y \leq 6 \\
2. & \quad 2x + 3y \leq 16 \\
3. & \quad x \geq 0 \\
4. & \quad y \geq 0
\end{aligned}
\]

### Steps to solve the problem:

1. **Graphical Representation**:
   - Plot the constraints as linear equations on the xy-plane.
   - Identify the feasible region, which is the intersection of the half-planes defined by the inequalities.
   - The vertices (corner points) of this feasible region are potential candidates for the maximum value of \( P \).

2. **Find the Corner Points**:
   - Solve the system of equations corresponding to the boundary lines of the constraints to find the intersection points (vertices).

3. **Evaluate \( P \) at Each Vertex**:
   - Substitute the coordinates of each vertex into the objective function \( P = 5x + 6y \) to calculate the value of \( P \) at those points.

4. **Determine the Maximum**:
   - The maximum value of \( P \) will occur at one of these vertices.

I will now calculate the vertices of the feasible region and evaluate \( P \) at each vertex.The vertices of the feasible region and the corresponding values of the objective function \( P = 5x + 6y \) at each vertex are as follows:

1. Vertex \( (2, 4) \), \( P = 34 \)
2. Vertex \( (0, 6) \), \( P = 36 \)
3. Vertex \( (0, \frac{16}{3}) \approx (0, 5.33) \), \( P \approx 32 \)
4. Vertex \( (6, 0) \), \( P = 30 \)

### Maximum Value
The maximum value of \( P \) is **36**, which occurs at the vertex \( (0, 6) \).

Would you like further details on any step or have additional questions?

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Here are 5 related questions:
1. How would the solution change if the objective function were \( P = 4x + 3y \)?
2. What are the implications of changing one of the constraints to a strict inequality, such as \( x + y < 6 \)?
3. How can the simplex method be applied to solve this problem instead of the graphical method?
4. What happens if we remove the non-negativity constraints \( x \geq 0 \) and \( y \geq 0 \)?
5. Can this problem be extended to three variables, and how would that affect the solution process?

### Tip:
When graphically solving linear programming problems, always double-check the vertices of the feasible region, as the maximum or minimum of the objective function must occur at one of these points.

Find the maximum value of P = 5x + 6y subject to the following constraints: x + y ≤ 6, 2x + 3y ≤ 16, x ≥ 0, y ≥ 0.

Learn how to solve a linear programming problem by maximizing P = 5x + 6y under the constraints x + y ≤ 6, 2x + 3y ≤ 16, x ≥ 0, and y ≥ 0. This step-by-step guide uses the graphical method to find the feasible region and the maximum value of P.

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