A tangle of linear equations — three unknowns, three equations — looks hopeless until you systematically subtract rows to knock out one variable at a time, melting the system into a TRIANGLE. Then you read the answers off from the bottom up. That’s Gaussian elimination, the bedrock of solving linear systems, and it’s inside every simulation, circuit, and regression. Slide to eliminate, column by column.
Gaussian elimination solves Ax=b by forward elimination: for each pivot column, subtract multiples of the pivot row from the rows below to zero out that column, reducing A to upper-triangular form; then back-substitute from the last equation up. PARTIAL PIVOTING (swap in the largest-magnitude pivot) keeps it numerically stable. The cost is O(n³), and storing the multipliers gives the LU factorization for reusing the same A with new b’s. It underlies linear solvers, matrix inversion, determinants, and least-squares. A fail-loud self-check throws unless it solves a known 3×3 system to [2,3,−1]. ◆ real numerical methods, node-verified.
Without pivoting a tiny pivot wrecks accuracy; genuinely singular/ill-conditioned systems amplify error regardless. Dense O(n³) is costly for huge sparse systems (iterative solvers win there). The elimination + back-substitution is exact.