Gauss elimination

A systematic procedure for solving systems of linear algebraic equations of the form

where the x_j represent unknown quantities and where the a_i,j and b_i 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 x_n and the resulting value substituted in the preceding equation which then can be solved for x_n−1. The back-substitution process is continued until all the remaining unknowns are determined.

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