Back substitution

Summary

Back substitution is a method to solve a system $Ux = b$, where $U$ is an $n \times n$ upper triangular matrix with nonzero diagonal entries. The last equation gives $x_n = b_n / u_{nn}$. Then, for $i = n-1, \dots, 1$ in this order, equation $i$ gives $$x_i = \frac{1}{u_{ii}} \left( b_i - \sum_{j = i+1}^{n} u_{ij} x_j \right),$$ which uses only the unknowns that are already known.