Gaussian elimination method

For $3\times 3$ systems or larger, Cramer becomes cumbersome: it requires $n+1$ determinants of size $n\times n$. The Gauss method does better: it transforms the system into a triangular one using three elementary operations that do not change the set of solutions.

1. Swap two rows.
2. Multiply a row by a scalar $\ne 0$.
3. Add to a row a multiple of another.

The triangular system is then solved “backwards” (back-substitution): from the last equation you obtain the last unknown, which you substitute into the second-to-last, and so on.

Links

Topics: Linear systems
Concepts: Gauss’s method · Elementary row operations · Triangular system
Skills: Solve systems
People: Carl Friedrich Gauss