To solve a system we simplify it without changing its answer. Three moves — the elementary row operations — are allowed, because each one rewrites the equations into an equivalent set with exactly the same solution:

• Swap two rows (reorder the equations).
• Scale a row by a non-zero number (multiply an equation through).
• Combine — add a multiple of one row to another (add equations).

We usually work on the augmented matrix — the coefficients with the right-hand side tacked on as a final column — so the bookkeeping is just numbers in a grid.

The three moves

Pick an operation and watch it act on the augmented matrix. Each leaves the underlying solution untouched — that's the whole reason these (and only these) moves are legal.

Why only these three?

Each operation is reversible — you can always undo it with another operation of the same kind — so no information is lost and no false solutions sneak in. Strung together cleverly, they drive a system toward a form you can read the answer straight off of. That deliberate sequence is Gaussian elimination, the workhorse algorithm we build next.