All echelon forms of a linear system have the same free variables

Definition: The three equation operations (elementary row operations) on a linear system are:

1. Interchange — Swap two equations: $E_i \leftrightarrow E_j$
2. Scaling — Multiply an equation by a nonzero constant $c$: $E_i \leftarrow c \cdot E_i$
3. Replacement — Replace equation $E_i$ by $E_i + c \cdot E_j$ (add a multiple of one equation to another)

These operations correspond directly to row operations on the augmented matrix. They are used in Gaussian elimination to simplify the system while preserving its solution set.

Example: Starting with:
$$\begin{cases} x + 2y = 4 \\ 3x + y = 5 \end{cases}$$

Apply Replacement ($E_2 \leftarrow E_2 - 3E_1$):
$$\begin{cases} x + 2y = 4 \\ -5y = -7 \end{cases}$$