1. **State the problem:** We are given six basic solutions to the system: $$\begin{cases} 2x_1 + 3x_2 + s_1 = 9 \\ 4x_1 + 3x_2 + s_2 = 12 \end{cases}$$ We want to maximize the objective function: $$P = 4x_1 + 17x_2$$ using the given basic feasible solutions. 2. **List the basic feasible solutions:** $ (x_1, x_2, s_1, s_2) $ - (A) (0, 0, 9, 12) - (B) (0, 3, 0, 3) - (C) (0, 4, -3, 0) \text{ (not feasible since } s_1 = -3 < 0)\) - (D) (4.5, 0, 0, -6) \text{ (not feasible since } s_2 = -6 < 0)\) - (E) (3, 0, 3, 0) - (F) (1.5, 2, 0, 0) 3. **Identify feasible solutions:** Only those with nonnegative slack variables are feasible: (A), (B), (E), and (F). 4. **Calculate the objective function value for each feasible solution:** - For (A): $$P = 4(0) + 17(0) = 0$$ - For (B): $$P = 4(0) + 17(3) = 51$$ - For (E): $$P = 4(3) + 17(0) = 12$$ - For (F): $$P = 4(1.5) + 17(2) = 6 + 34 = 40$$ 5. **Determine the maximum value:** Among feasible solutions, the maximum value of $P$ is $51$ at solution (B). **Final answer:** The maximum value of $P$ is **51** at $x_1=0$, $x_2=3$.