When trying to solve a Linear Program (I call it primary problem below) of the form
min $c^{T}x$
s.t. $ $ $Ax = b, x \geq 0, b \geq 0 $
The two phase simplex algorithm suggests to starts with the following auxiliary problem,
min $\sum y_{i}$
s.t. $Ax + y = b, x \geq 0, b \geq 0 , y \geq 0$
It is argued that a solution to the auxiliary problem which gives $\sum y_{i} = 0$, provides a basic solution to the primary optimization problem, from which we can initialize the simplex algorithm for the primary problem.
I understand that $\sum y_{i} = 0$ implies that we found a solution to $Ax = b$, and therefore a feasible solution to the primary problem, however I do not understand why this solution must be a basic feasible solution, which implies that at least $n - m$ of the original $n$ decision variables ($x_{i}, i \in \{1, ..., n\}$), are zero.
I would be very grateful if someone could illuminate this.