| # Backwards substitution for solving triangular linear systems Consider a triangular system of the form , where the vector is given, and is upper-triangular. Let us first consider the case when , and is invertible. Thus, has the form with each , non-zero. The backwards substitution first solves for the last component of using the last equation: and then proceeds with the following recursion, for : Example: Solving a triangular system by backwards substitution