 Solution

The equation is solvable exactly when $b\in \mathrm{im}A$, the image of a linear map, equivalently when the rank of a matrix satisfies $\mathrm{rank}A=\mathrm{rank}[A\mid b]$. If it is solvable and $x_0$ is one solution, every solution is preciselyIndeed differences of solutions lie in the kernel of a linear map, and adding a kernel vector preserves the equation. Hence there are no solutions when $b\notin \mathrm{im}A$; one solution when $A$ is invertible; and infinitely many solutions when $b\in \mathrm{im}A$ and $A$ is singular. In the last case the rank-nullity theorem gives a nonzero kernel vector $v$, and $x_0+tv$ gives distinct solutions for every real $t$. This proves that a finite number greater than one is impossible.