Read the result
The last row gives the last unknown first. Each row above then has one fewer unknown left to find.
Linear systems
Back Substitution Calculator
Start from an upper triangular system and solve it from the bottom upward.
Formula
x_i = (y_i - sum from j=i+1 to n of u_ij*x_j) / u_ii
Result
Worked substitution
Back substitution works because each lower row contains fewer unknowns than the one above it.
Where it helps
Use this after Gaussian elimination, or whenever a problem already gives triangular equations.
Common slip
A zero diagonal entry stops ordinary back substitution. That row needs classification before continuing.
Try it
Change u33 and watch how the final variable affects every row above it.