Gaussian Elimination

Solving systems of linear equations by row reduction to row echelon form

Golden Rule: Use row operations to produce zeros below each pivot. Back-substitute from the last row upward. Check for inconsistent systems (row of zeros = nonzero).

• Step 1: Write the augmented matrix [A|b] — Each row corresponds to one equation. Columns correspond to variables (x₁, x₂, …, xₙ), with the last column being the constants. Use 0 as a coefficient when a variable is absent.
• Step 2: Find the pivot in the first column — If the first entry is zero, swap rows (R1 ↔ Rᵢ where aᵢ₁ ≠ 0). Partial pivoting (swapping to get the largest entry) improves numerical stability.
• Step 3: Eliminate below the pivot — For each row i below the pivot row k: Rᵢ → Rᵢ − (aᵢₖ/aₖₖ)Rₖ. This sets the column-k entry in row i to zero. Repeat for each subsequent pivot column.
• Step 4: Back-substitute — The matrix is now in row echelon form. The last row gives xₙ directly. Substitute back into the second-to-last row to get xₙ₋₁, and continue upward.