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A matrix equation in the form <math>\mathbf{L}\mathbf{x} = \mathbf{b}</math> or <math>\mathbf{U}\mathbf{x} = \mathbf{b}</math> is very easy to solve by an iterative process called '''forward substitution''' for lower triangular matrices and analogously '''back substitution''' for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes <math>x_1</math>, then substitutes that ''forward'' into the ''next'' equation to solve for <math>x_2</math>, and repeats through to <math>x_n</math>. In an upper triangular matrix, one works ''backwards,'' first computing <math>x_n</math>, then substituting that ''back'' into the ''previous'' equation to solve for <math>x_{n-1}</math>, and repeating through <math>x_1</math>.
The process is so called because for lower triangular matrices, one first computes <math>x_1</math>, then substitutes that ''forward'' into the ''next'' equation to solve for <math>x_2</math>, and repeats through to <math>x_n</math>. In an upper triangular matrix, one works ''backwards,'' first computing <math>x_n</math>, then substituting that ''back'' into the ''previous'' equation to solve for <math>x_{n-1}</math>, and repeating through <math>x_1</math>.
Notice that this does not require inverting the matrix.
===Forward substitution===
The matrix equation '''L'''''x''' = '''b''' can be written as a system of linear equations
