How to check consistency of matrix-vector product by carrying it to REF

If we have the product $A\vec{x}=\vec{b}$, this is basically same as:

$$\begin{array}{r}\vec{b}=x_1{\vec{a}}_1+x_2{\vec{a}}_2+x_3{\vec{a}}_3\end{array}$$

Now “converting” this expression into an augmented matrix:

$$\begin{array}{r}\left[\begin{array}{cccc}.& .& .& b_1\\ .& .& .& b_2\\ .& .& .& b_3\end{array}\right]\end{array}$$

• Where the first column is ${\vec{a}}_1$, second is ${\vec{a}}_2$, third is ${\vec{a}}_3$.
• And the first column’s coefficient is $x_1$, second is $x_2$ and so on.