To solve this linear programming problem using the simplex method, we first need to analyze the constraints and objective functions.
Given:
Maximize [tex]\( z = 2x_1 + 9x_2 \)[/tex]
Subject to:
[tex]\[
\begin{array}{l}
5x_1 + x_2 \leq 80 \\
9x_1 + 2x_2 \leq 100 \\
x_1 + x_2 \leq 90 \\
x_1, x_2 \geq 0
\end{array}
\][/tex]
We will introduce slack variables [tex]\( s_1 \)[/tex], [tex]\( s_2 \)[/tex], and [tex]\( s_3 \)[/tex] to convert the inequalities into equalities for the simplex method:
[tex]\[
\begin{array}{l}
5x_1 + x_2 + s_1 = 80 \\
9x_1 + 2x_2 + s_2 = 100 \\
x_1 + x_2 + s_3 = 90 \\
x_1, x_2, s_1, s_2, s_3 \geq 0
\end{array}
\][/tex]
Now, we proceed with the simplex method steps:
1. Identify the coefficients and constants in the constraints and the objective function.
2. Set up the initial simplex tableau.
3. Perform the pivot operations to optimize the objective function iteratively.
Eventually, from the given data, the optimal solution is:
[tex]\[
z_{max} = 450.0
\][/tex]
This optimal value of [tex]\( z \)[/tex] is achieved when:
[tex]\[
x_1 = 0.0,\quad x_2 = 50.0
\][/tex]
Substituting these values back into the constraints to find the values of the slack variables:
For the first constraint:
[tex]\[
5(0) + 50 + s_1 = 80 \implies s_1 = 80 - 50 = 30
\][/tex]
For the second constraint:
[tex]\[
9(0) + 2(50) + s_2 = 100 \implies s_2 = 100 - 100 = 0
\][/tex]
For the third constraint:
[tex]\[
0 + 50 + s_3 = 90 \implies s_3 = 90 - 50 = 40
\][/tex]
Thus, the values of the slack variables are:
[tex]\[
s_1 = 30,\quad s_2 = 0,\quad s_3 = 40
\][/tex]
So, the correct choice is:
A. Treating [tex]\( x_1 \)[/tex] as a nonbasic variable, the maximum is [tex]\( z = 450.0 \)[/tex] when [tex]\( x_1 = 0.0 \)[/tex], [tex]\( x_2 = 50.0 \)[/tex], [tex]\( s_1 = 30 \)[/tex], [tex]\( s_2 = 0 \)[/tex], and [tex]\( s_3 = 40 \)[/tex].
