The input base variable in the Simplex method determines towards what new vertex is performed the displacement.
In this example, as P1 (corresponding to 'x') enters, the displacement is carried out by the OF-edge to reach the F-vertex, where the Z-function value is calculated.
The third column lists the solution of the basic variables in the same order as they are mentioned in the first column and the fourth column lists the Z value for the corresponding solution.
Please note that the values of the non-basic variables used to compute Z are freezed to zero.
Note that some of the rows are colored in a shade of red.
These rows corresponds to solutions that violate the sign restrictions.
What we did was, we found our the coordinates of the vertices of the feasible region, evaluated Z at each of these points and claimed with sufficient intuitive proof that the vertex that renders the best value for Z is indeed the optimum of the LP.
Hence, it only makes sense, to try and find the vertices of the feasible region in our algebraic method as well. Our knowledge of algebra allows us to solve equations and not inequalities to hopefully arrive at a singular point.
However, unfortunately one is never as lucky to find a real world solution that involves two or less decision variables.
As we saw in the article on formulation, there can indeed be quite a few decision variables in even a simple problem.