In simplex method, to get the initial feasible solution. We often use Big-M method. My question is can we simply use elementary row transformation to get basis fesible solution instead introduce extra variables?
In the primal simplex method, you want an initial basic feasible solution that is feasible with respect to both the equation constraints $Ax=b$ and to lower/upper bound constraints $l \le x \le u$. Conventional row operations can easily help you find a nonsingular subset of the columns of $A$ (a basis matrix, $B$), but this wouldn't help you in finding a basic solution that is feasible with respect to the bounds constraints $l \le x \le u$.