Solving the upper triangular system

The upper triangular system

is equivalent to the initial system

in the sense that they admit the same solution .

The upper triangular system is

From the last equation . We substitute this value in equation 2

from which we infer that .

Finally, we substitute the known values of , in equation 1

and we obtain .

In short, we start with the last equation and find . Then, we move backwards through the equations; at each step we substitute the known values in the current equation , and compute the value of . This process is called backward substitution.