Let $A = \left[{\begin{matrix}2 & 4 & 2 \cr 3 & 9 & 18 \cr 4 & 10 & 14 \cr \end{matrix}}\right]$.

We want to determine if the system $A{\bf x}={\bf b}$ has a solution for every ${\bf b} \in {\mathbb R}^3$.

To check this we add times the first row to the second.
We then add times the first row to the third.
We then add times the new second row to the new third row.
We conclude that

A. There is not a solution for every ${\bf b} \in {\mathbb R}^3$.
B. There is a solution for every ${\bf b} \in {\mathbb R}^3$.
C. We cannot tell if there is a solution for every ${\bf b} \in {\mathbb R}^3$.