Let $A = \begin{bmatrix} col_{1} & col_{2} & col_{3} \end{bmatrix}$ be any arbitrary $3 \times 3$ matrix.
Here, $col_{1}, col_{2}, col_{3}$ are the column vectors of A, each of dimension $3 \times 1$.
Also, let $x =\begin{bmatrix} x_{1}\\ x_{2}\\ x_{3} \end{bmatrix}$.
$ \therefore Ax = b \Rightarrow col_{1}.x_{1} + col_{2}.x_{2} + col_{3}.x_{3} = b$, i.e., a linear combination of the column vectors of $A$.
Therefore, finding the solution of the equation $Ax = b$ basically reduces to finding a linear combination of column vectors of matrix $A$ that result in vector $b$. Such a combination can only be found if vector $b$ lies in the column space of matrix $A$, otherwise not.