I've just finished the first lecture of MIT 18.06 Linear Algebra with Gilbert Strang. The professor briefly discusses how one can find out whether $A x = b$ as a solution.
I don't understand this example that he gave:
Imagine that there are $9$ components in a $9$-dimensional space and the eighth component is the same as the ninth component; then you will not be able to find a solution.
Think about every equation in the system as describing an affine subspace of the domain. If a solution exists, that vector must satisfy all the equations simultaneously, therefore it must lie in the intersection of the subspaces described by the equations in the system. Now it's easy to construct example where:
A unique solution exists, for example take two lines in $\mathbb{R}^2$ instersecting in exactly one point: $$\ell_1 = \{x-y=0\}, \quad \ell_2 =\{x+y =1\},$$ then the linear system $$\begin{pmatrix}1 & -1\\ 1&1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix} = \begin{pmatrix}0\\ 1\end{pmatrix}$$ has a unique solution (the intersection of the lines).
A solution doesn't exists: take two parallel line, there is no intersection.
Infinite solution exists: take $\ell_1 = \ell_2$.
In general the information you need is encoded in the ranks of the matrices $A$ and $(A|b)$.