From AMS Glossary
A systematic procedure for solving systems of linear algebraic equations of the form
where the xj represent unknown quantities and where the ai,j and bi are prescribed constants. Multiples of one equation are combined with another to eliminate variables successively until the system has the upper triangular form
where the constants in all but the first equation have been altered by the elimination process. The last equation is solved for xn and the resulting value substituted in the preceding equation which then can be solved for xn-1. The back-substitution process is continued until all the remaining unknowns are determined.