Computing initial feasible basis in the simplex algorithm

Question

My textbook introduces the following method to compute initial feasible basis in the simplex algorithm:

What is the implication for the original LP if the auxiliary LP's objective function can't attain the value $0$, or if it is infeasible?

Answer 1

If the auxiliary problem can’t attain the value zero, then the original problem is infeasible.

The auxiliary problem can’t be infeasible. For a general LP with constraints $Ax=b$ with $x\geq0$, the auxiliary problem has constraints $Ax+s=b$, $x\geq0$, $s$ free**. There is always the feasible solution $x=0$ and $s=b$.

**If you require $s\geq0$, then you can write the slack variables $s$ as $s^+-s^-$ for $s^+,s^-\geq0$.