augmented matrix

For a linear system $Ax=b$, the augmented matrix appends $b$ as an extra column:

$$(A|b):=\left(\begin{array}{cccccc}{a}_{11}& a_{12}& \cdots & a_{1n}& |& b_1\\ a_{21}& a_{22}& \cdots & a_{2n}& |& b_2\\ ⋮& ⋮& \ddots & ⋮& |& ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}& |& b_m\end{array}\right)$$

This lets us perform gaussian elimination on $A$ and $b$ simultaneously — every row operation on $A$ is mirrored on $b$.

Row operations preserve the solution set

If $(C|b^{\prime })$ is obtained from $(A|b)$ by row operations (swapping, scaling, adding rows), then $L(A,b)=L(C,b^{\prime })$.
This is why we can solve $Ax=b$ by transforming $(A|b)$ to row echelon form and reading off solutions from $(C|b^{\prime })$.