Question

Consider the Linear Programming Problem \( P \): \[ \text{Minimize } 2x_1 - 5x_2 \]  subject to  \[ 2x_1 + 3x_2 + s_1 = 12, \] \[ -x_1 + x_2 + s_2 = 1, \] \[ -x_1 + 2x_2 + s_3 = 3, \] \[ x_1 \geq 0, x_2 \geq 0, s_1 \geq 0, s_2 \geq 0, \text{ and } s_3 \geq 0. \] If \[ \left[ \begin{array}{c} x_1 \\ s_1 \\ s_2 \\ s_3 \end{array} \right] \]  is a basic feasible solution of \( P \), then \( x_1 + s_1 + s_2 + s_3 = \underline{\hspace{1cm}}. \)

Hint:

For basic feasible solutions, solve the system of constraints to find the value of the variables at equilibrium.

Answer 1

Correct Answer: 5

Substitute the given basic feasible solution into the system of equations. Solving for the values of \( x_1, s_1, s_2, s_3 \) gives: \[ x_1 + s_1 + s_2 + s_3 = 5. \] Thus, \( \boxed{5} \).