# Dictionary:Back substitution

When simultaneous equations can be expressed as ${\displaystyle A\cdot X=B}$, where ${\displaystyle A}$ is a triangular matrix and ${\displaystyle X}$ and ${\displaystyle B}$ are vectors, the solution for the last (or first) value becomes trivial. Back-substitution then involves successively replacing the last (or first) value in the equations for the other elements so that their solutions also become trivial.