Gaussian Elimination Calculator
Reducing a System Row by Row
Gaussian elimination solves linear systems the way most people first learn to by hand — systematically combining equations to eliminate variables one at a time — but does it with matrix rows and partial pivoting for numerical stability. Forward elimination reduces the augmented matrix to a triangular form; back substitution then unwinds it into the final answer.
The Method
Back substitution: solve the last variable first, then substitute upward
Partial pivoting swaps in the row with the largest available value in each column before eliminating, which keeps the arithmetic numerically stable and avoids dividing by a small or zero pivot.
Worked Example
Solving 2x + 3y = 8 and x − 2y = −3 — the same system as the Cramer's rule example, for comparison:
| Variable | Value |
|---|---|
| x | 1 |
| y | 2 |
Gaussian elimination and Cramer's rule always agree on a unique solution — they're two different routes to the same answer.
Where This Matters
- Larger linear systems — elimination scales far better than Cramer's rule as the number of equations grows, which is why it's the standard method in numerical software.
- Engineering simulations — finite element and circuit solvers reduce their underlying equations with this same row-elimination approach.
- Balancing chemical equations — systems of linear equations describing atom balance can be solved this way.
How to Use This Calculator
- Choose the system size: 2 Equations, 2 Unknowns or 3 Equations, 3 Unknowns.
- Enter each equation's coefficients and constant term.
- Select Calculate to see the row-reduction steps and the final solved values.
Related Calculations
Compare against the determinant-based Cramer's Rule Calculator, or invert the coefficient matrix directly with the Matrix Inverse Calculator.