To solve this linear programming (LP) problem, we need to analyze the constraints and identify whether there exists an optimal solution for maximizing $p = x + y$.

Problem Setup

We are given:

1. Objective function: Maximize $p = x + y$
2. Constraints:
   • $x + 2y \leq 9$
   • $2x + y \leq 9$
   • $x \geq 0$
   • $y \geq 0$

Step-by-Step Solution

1. Identify the feasible region:

   • The feasible region is determined by the intersection of the inequalities.
   • Let's analyze each constraint to understand where this region lies.

2. Graphing the constraints:

   • $x + 2y = 9$: Rearranging, $y = \frac{9 - x}{2}$.
   • $2x + y = 9$: Rearranging, $y = 9 - 2x$.
   • $x \geq 0$ and $y \geq 0$ ensure we are in the first quadrant.

   By plotting these lines, we can identify where the feasible region is bounded by the constraints.

3. Finding the vertices of the feasible region:

   • Intersection of $x + 2y = 9$ and $2x + y = 9$:

     • Solving these equations simultaneously: $x + 2y = 9$ $2x + y = 9$
     • Solving for $x$ and $y$:
       • Multiply the first equation by 2 to get $2x + 4y = 18$.
       • Subtract the second equation from this result: $2x + 4y - (2x + y) = 18 - 9$ $3y = 9 \Rightarrow y = 3$
       • Substitute $y = 3$ into $x + 2y = 9$: $x + 2(3) = 9$ $x = 3$
     • So, one vertex is $(x, y) = (3, 3)$.

   • Intersection of $x + 2y = 9$ and $y = 0$:

     • Substitute $y = 0$ into $x + 2y = 9$: $x = 9$
     • So, another vertex is $(x, y) = (9, 0)$.

   • Intersection of $2x + y = 9$ and $x = 0$:

     • Substitute $x = 0$ into $2x + y = 9$: $y = 9$
     • So, another vertex is $(x, y) = (0, 9)$.

4. Evaluating the objective function at each vertex:

   • $(x, y) = (3, 3)$: $p = 3 + 3 = 6$
   • $(x, y) = (9, 0)$: $p = 9 + 0 = 9$
   • $(x, y) = (0, 9)$: $p = 0 + 9 = 9$

5. Conclusion:

   • The maximum value of $p = x + y$ is 9, which occurs at the points $(9, 0)$ and $(0, 9)$.
   • Therefore, the optimal solution is: $p = 9$ $(x, y) = (9, 0) \text{ or } (0, 9)$

Final Answer

• Optimal value of $p = 9$
• Points $(x, y) = (9, 0)$ or $(0, 9)$

Would you like more details on any part of this solution?

Here are some related questions to deepen your understanding:

1. How do we determine the feasibility of a region in linear programming?
2. What is the significance of corner points in linear programming problems?
3. Why do we check the objective function at vertices of the feasible region?
4. What steps would we take if there were more constraints?
5. Can a linear programming problem have multiple optimal solutions, and if so, when?

Tip: When graphing constraints, look for boundary intersections as potential optimal points for maximization or minimization of the objective function.