Gaussian elimination is a systematic procedure for solving a linear system by using row operations to create zeros below successive pivots.

Forward elimination

Starting from an augmented matrix, choose a nonzero pivot in the leftmost available column. Use row operations to eliminate entries below it, then move to the next row and the next pivot column.

For example,

$$\left[\begin{array}{cc|c} 1&1&5\ 2&-1&1 \end{array}\right]$$

becomes

$$\left[\begin{array}{cc|c} 1&1&5\ 0&-3&-9 \end{array}\right]$$

after $R_2\leftarrow R_2-2R_1$.

Back substitution

The second row gives

$$-3y=-9,$$

so $y=3$. Substituting into the first row gives

$$x+3=5,$$

so $x=2$.

Thus

$$(x,y)=(2,3).$$

Gauss-Jordan elimination

Continuing past row-echelon form to reduced row-echelon form eliminates entries above the pivots as well. This variation, often called Gauss-Jordan elimination, lets the solution be read directly from the final matrix.

The important invariant is that every row operation preserves the solution set. Elimination changes the representation of the system, not the system's solutions.