Back Substitution

Linear Algebra for Machine Learning · Axiom Academy

Solving for unknowns starting from the last equation After reducing a system to row echelon form, we solve it from the bottom up, substituting known values into equations above. Starting with the row echelon form: Step 1: Solve for z from equation 3 Step 2: Substitute z = 1 into equation 2 Step 3: Substitute y = 15/4 and z = 1 into equation 1 Bottom-up approach: Always start with the last equation where only one variable is unknown Substitution chain: Each solved variable feeds into the equations above No ambiguity: Row echelon form guarantees a clear solving path Computational efficiency: Back substitution is much faster than solving the original system directly Verification: Always substitute back into original equations to check correctness

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