A triangular system of equations is efficiently solved by a series of variable substitutions. By definition, there will always be at least one equation with a single unknown, one equation with only two unknowns (one of which will correspond to the unknown in the single variable equation), etc. This procedure can be formalized into two simple algorithms.
A lower triangular system is solved by forward substitution.
An upper triangular system is solved by backward substitution.
These operations are described with reference to the solution of systems of linear equations whose LU decomposition is known. Equation 25 (Ly = b) and Equation 24 (Ux = y) pose the problem in this context.