Theorem (Back Substitution).  Suppose that is an upper-triangular system with the form given above.  
If   for then there exists a unique solution to the linear system.

Constructive Proof.

The solution is easy to find. The last equation involves only , so we solve it first  .  Now is known and it can be used in the next-to-last equation  .  Now are known and they can be used in the second-to-last equation to find :

        .  

Once the values are known, the general step is

          
the summation form is  
           for  .  

    The uniqueness of the solution is easy to see.  The n-th equation implies that is the only possible value of .  Then finite induction is used to establish that    are unique.

Q. E. D.

(c) John H. Mathews 2003