I apologize as this is probably a very elementary question. I would like to know whether you can find other solutions to $Ax=b$, that aren't the least squares solution, a.k.a. a general solution.
Suppose I have found a unique solution to $Ax=b$ from either:
$ \begin{align} \bar{x} = (A^TA)^{-1}A^Tb \end{align}$ or $\begin{align} \tilde{x} = A^{+}b \end{align} $
Would I be able to use these solutions to find more general solutions? If I take a multiple of one solution, would it still be considered a solution?