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# Definition of equation operations on a linear system

Put content here**Theorem:** Applying any of the three equation operations to a linear system produces a new system that is **equivalent** to the original — i.e., both systems have exactly the same solution set.
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**Proof sketch:** Each operation is reversible:
- Swapping equations can be undone by swapping them back
- Scaling by $c \neq 0$ can be undone by scaling by $1/c$
- Replacing $E_i$ with $E_i + cE_j$ can be undone by replacing with $E_i - cE_j$
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Since each operation is invertible, no solutions are lost or gained.
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**Why this matters:** This theorem justifies Gaussian elimination. We can transform a system into a simpler equivalent form (echelon form) and solve that instead of the original system.

# Parents

* Linear systems of equations