In this simplex algorithm tableau, what are the basic variables?

Question

At some point while running the simplex algorithm, we find this tableau:

Would I be correct in saying that at this stage, our basic variables are $x_4,x_3,x_6$ as they are the only ones not equal to zero, or have I missed something?

Answer 1

In order for a variable to become a basic variable in the tableau, there needs to exist a column in the tableau in which the value for that variable in one row is one and all other values in all other rows in that column is zero. Below, highlighted green, are all columns that satisfy that property for all the basic variables.

Since this tableau is not optimal, we are able to pick a column to pivot to find a new extreme point (the $x_2$ column since it has a $C^\pi_j=-1$ that is negative). After doing the minimum ratio test, and entering the pivoting phase, notice how we're establishing that mentioned column basic variable property to the $x_2$ column, and after doing all the math for the pivoting, we will notice that $x_4$ is now a non-basic variable, and $x_2$ is now a basic variable. Every time we do a pivot, we are exchanging a basic variable for another basic variable as shown in the set of images below.