Gaussian Elimination

Algebra 2 · Axiom Academy

Reduce a system's augmented matrix to a triangular row-echelon staircase with row operations, then back-substitute for . Consider a system of three linear equations in three unknowns. All of the information lives in the coefficients and the constants — the letters x , y , z are just placeholders that mark the columns. Its augmented matrix — coefficients, then constants To simplify the matrix we are allowed exactly three moves — the elementary row operations . Each one is reversible, and none of them changes the solution of the system. Swap two rows — reorder the equations Scale a row — multiply it by a nonzero constant Add a multiple of one row to another row Our goal is row-echelon form : a staircase where each row's leading entry sits to the right of the one above it and every entry below a pivot is zero. We build it top-down, using each pivot to clear the entries beneath it. All zero rows (if any) sit at the bottom Each row's leading entry is to the right of the one in the row above Every entry below a leading entry is zero Making each leading entry a 1 is optional, but handy Once the matrix is in row-echelon form, we solve from the bottom up . Each row hands its variable's value to the row above it, until every unknown is pinned down. Read the bottom row — it has only one unknown — and solve for it Substitute that value into the row above Solve that row for the next unknown Keep climbing until every variable is known

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