Questions: Use the simplex method to solve the linear programming problem. Maximize z=7x1+5x2+x3 subject to 2x1+2x2+x3 ≤ 10 x1+4x2+5x3 ≤ 12 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The maximum is when x1= , x2= x3= ,s1= , and s2= . B. There is no maximum.

To solve this linear programming problem using the simplex method, we first convert the inequalities into equalities by introducing slack variables. Then, we set up the initial simplex tableau and perform pivot operations to find the optimal solution that maximizes the objective function. The solution will be found when there are no more negative indicators in the bottom row of the tableau.

Se busca maximizar la función objetivo \( z = 7x_1 + 5x_2 + x_3 \) sujeta a las restricciones: \[ \begin{align_} 2x_1 + 2x_2 + x_3 & \leq 10 \\ x_1 + 4x_2 + 5x_3 & \leq 12 \\ x_1, x_2, x_3 & \geq 0 \end{align_} \]

Al aplicar el método simplex, se obtiene que el valor máximo de la función objetivo es \( z = 35 \) cuando: \[ \begin{align_} x_1 & = 5 \\ x_2 & = 0 \\ x_3 & = 0 \end{align_} \]

Las variables de holgura se calculan como sigue: \[ \begin{align_} s_1 & = 10 - (2 \cdot 5 + 2 \cdot 0 + 0) = 0 \\ s_2 & = 12 - (5 + 4 \cdot 0 + 5 \cdot 0) = 7 \end{align_} \]

El máximo es \( z = 35 \) cuando \( x_1 = 5 \), \( x_2 = 0 \), \( x_3 = 0 \), \( s_1 = 0 \) y \( s_2 = 7 \). Por lo tanto, la respuesta es A.

\[ \boxed{z = 35, \, x_1 = 5, \, x_2 = 0, \, x_3 = 0, \, s_1 = 0, \, s_2 = 7} \]