Let's solve the linear programming problem shown in the image.

The objective function is $P = 9x + 8y$, where you need to maximize $P$, subject to the following constraints:

1. $8x + 6y \leq 48$
2. $7x + 7y \leq 49$
3. $x \geq 0$
4. $y \geq 0$

The corner points provided are:

• $(0, 0)$
• $(6, 0)$
• $(0, 7)$
• $(3, 4)$

Step 1: Calculate $P$ at each corner point using the objective function $P = 9x + 8y$:

1. At $(0, 0)$: $P = 9(0) + 8(0) = 0$ So, $P = 0$.

2. At $(6, 0)$: $P = 9(6) + 8(0) = 54$ So, $P = 54$.

3. At $(0, 7)$: $P = 9(0) + 8(7) = 56$ So, $P = 56$.

4. At $(3, 4)$: $P = 9(3) + 8(4) = 27 + 32 = 59$ So, $P = 59$.

Step 2: Conclusion

The maximum value of $P$ is 59, which occurs at the corner point $(3, 4)$.

Let me know if you need further details or have any questions!

Here are some related questions:

1. How do you determine corner points in linear programming?
2. What is the significance of constraints in linear programming?
3. Can you explain how the feasible region is determined in this problem?
4. How would this change if the objective function had different coefficients?
5. What is the meaning of the term "feasible solution"?

Tip: When solving linear programming problems graphically, always check the objective function at each vertex of the feasible region to find the optimal value.