Gaussian elimination as a method to solve a linear system

Definition: The echelon form (or row echelon form) of a linear system is a form obtained by applying Gaussian elimination, satisfying these properties:

1. All nonzero rows are above any rows of all zeros
2. The leading entry (first nonzero entry) of each nonzero row is to the right of the leading entry of the row above it
3. All entries below a leading entry are zero

A matrix is in reduced row echelon form (RREF) if additionally:
4. Each leading entry is $1$
5. Each leading $1$ is the only nonzero entry in its column

Example of echelon form:
$$\begin{bmatrix} 2 & 3 & 1 & | & 5 \\ 0 & 1 & -2 & | & 3 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}$$

The echelon form reveals which variables are basic (corresponding to pivot columns) and which are free.