back-substitution
After you have wrestled a system down into a neat staircase, the hard part is over and back-substitution is the easy walk back up. The idea: the bottom equation has been reduced to a single unknown, so solve it first; then carry that known value up into the equation above to find the next unknown, and keep climbing until every variable is known.
It is the second half of methods like Gaussian elimination. Suppose elimination leaves you with z = 3 at the bottom, x + 2z = 9 in the middle, and a full equation on top. You substitute z = 3 to get x = 3, then push both values into the top equation to solve for the last variable. Each step uses values already found.
Back-substitution works precisely because the system has been put in triangular (upper-echelon) form, so each equation, read from the bottom up, introduces exactly one new unknown. The name captures the motion: you substitute backward, from the last equation toward the first. It is reliable and quick — just guard against carrying forward an arithmetic error from the elimination stage, since one slip propagates upward.
After reduction a system reads z = 3, y + z = 5, x + y + z = 9. From the bottom: z = 3, then y = 5 − 3 = 2, then x = 9 − 2 − 3 = 4. Solution: (4, 2, 3).
Solve the bottom, then climb upward.