To write the initial simplex tableau, we first introduce slack variables [tex]\( s_1, s_2, \)[/tex] and [tex]\( s_3 \)[/tex] to convert the inequalities into equalities. The given constraints transform as follows:
1. [tex]\( x_1 + 2x_2 + s_1 = 13 \)[/tex]
2. [tex]\( 5x_1 + x_2 + s_2 = 10 \)[/tex]
3. [tex]\( 4x_1 + 2x_2 + s_3 = 9 \)[/tex]
Next, the objective function [tex]\( z = x_1 + 4x_2 \)[/tex] is written in the form [tex]\( -z + x_1 + 4x_2 = 0 \)[/tex].
Now, we set up the initial simplex tableau with these equations. It will include coefficients of [tex]\( x_1, x_2, s_1, s_2, s_3 \)[/tex], and the constant term on the right-hand side. Additionally, the objective function row is included with a [tex]\( z \)[/tex] column.
The initial simplex tableau looks like:
[tex]\[
\left[\begin{array}{rrrrrrr}
x_1 & x_2 & s_1 & s_2 & s_3 & z & \text{RHS} \\
\hline
1 & 2 & 1 & 0 & 0 & 0 & 13 \\
5 & 1 & 0 & 1 & 0 & 0 & 10 \\
4 & 2 & 0 & 0 & 1 & 0 & 9 \\
\hline
-1 & -4 & 0 & 0 & 0 & 1 & 0
\end{array}\right]
\][/tex]
Plugging in the given data, the complete initial simplex tableau is:
[tex]\[
\left[\begin{array}{rrrrrrr}
x_1 & x_2 & s_1 & s_2 & s_3 & z & \text{RHS} \\
1 & 2 & 1 & 0 & 0 & 0 & 13 \\
5 & 1 & 0 & 1 & 0 & 0 & 10 \\
4 & 2 & 0 & 0 & 1 & 0 & 9 \\
\hline
-1 & -4 & 0 & 0 & 0 & 1 & 0
\end{array}\right]
\][/tex]
This tableau will be our starting point for the Simplex Method to solve the given linear programming problem.
