Fill content

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

Put content here.**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 ($E_2 \leftarrow E_2 - 3E_1$):
$$\begin{cases} x + 2y = 4 \\ -5y = -7 \end{cases}$$

# Parents

* Linear systems of equations