Questions: Solve the linear programming problem using the simplex method. Maximize P = -x1 + 2x2 subject to -x1 + x2 ≤ 2 -x1 + 3x2 ≤ 14 x1 - 4x2 ≤ 6 x1, x2 ≥ 0 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The maximum value of P is P= when x1= and x2= . (Simplify your answers.) B. There is no optimal solution.

Solution Steps

We are tasked with maximizing the objective function \( P = -x_1 + 2x_2 \) subject to the following constraints: \[ \begin{aligned} -x_1 + x_2 & \leq 2 \\ -x_1 + 3x_2 & \leq 14 \\ x_1 - 4x_2 & \leq 6 \\ x_1, x_2 & \geq 0 \end{aligned} \]

The constraints can be rewritten as:

1. \( x_2 \leq x_1 + 2 \)
2. \( x_2 \leq \frac{14 + x_1}{3} \)
3. \( x_2 \geq \frac{x_1 - 6}{4} \)

Upon applying the simplex method, it was determined that the problem is unbounded. This indicates that there is no maximum value for the objective function \( P \) within the given constraints.

Since the optimization process indicates that the problem is unbounded, we conclude that there is no optimal solution.

Final Answer