Gaussian elimination is the systematic recipe for solving any linear system. Use row operations to carve the augmented matrix into a staircase (row-echelon) form — zeros below a diagonal of leading entries — and then keep going until each leading entry is a lone 1 with zeros above and below it. At that point the answer is just sitting in the last column.

It always works, it never guesses, and it scales to systems of any size — which is why it's the algorithm running quietly inside almost every piece of software that solves equations.

Step it through

Watch the system 2x + y = 5, x - y = 1 get solved, one row operation at a time. By the final step the matrix reads x = 2, y = 1 directly — no algebra left to do.

Reading the staircase

The final reduced row-echelon form — an identity block beside the answer column — is the goal whenever a unique solution exists. If a row collapses to 0 = 0 you've found infinitely many solutions; if it collapses to 0 = 5 (something impossible) there are none. The shape of the staircase tells you everything — the idea behind rank.