What is Gaussian elimination?

Gaussian elimination is an algorithm that transforms a matrix into row echelon form using three elementary row operations: swapping two rows, scaling a row by a non-zero constant, and adding a multiple of one row to another. Back-substitution then extracts the variable values.

What does "no unique solution" mean?

If the system is inconsistent (no solution exists) or underdetermined (infinitely many solutions), the calculator will report this. A unique solution requires the coefficient matrix to be full rank (non-singular determinant).

Why does the calculator use partial pivoting?

Partial pivoting selects the row with the largest absolute value in the pivot column before each elimination step. This reduces floating-point rounding errors that accumulate when dividing by very small numbers.

How many equations can this tool solve?

The tool supports systems from 2 equations with 2 unknowns up to 5 equations with 5 unknowns.

Can I use decimal coefficients?

Yes. The input accepts any real number including decimals and negative values. Results are rounded to 6 significant figures to avoid floating-point display noise.

Gaussian Elimination Calculator

Enter a system of linear equations as an augmented matrix and get the solution with full step-by-step row reduction.

Augmented Matrix Input

Number of variables:

Solution

Enter a matrix and click Solve.

Step-by-Step Row Operations

Steps will appear here after solving.

Summary

The Gaussian Elimination Calculator solves systems of linear equations by reducing an augmented matrix to row echelon form, then applying back-substitution to find the values of each variable. It shows every row operation performed, making it ideal for checking homework or learning the algorithm. Supports 2×2 up to 5×5 systems.

How it works

Select the number of variables (2 to 5) to set the matrix size.
Enter the coefficients of each equation into the augmented matrix grid.
Click "Solve" to run Gaussian elimination with partial pivoting.
The step panel shows each row operation (swap, scale, eliminate) in order.
The solution panel displays the value of each variable, or reports no unique solution.

Use cases

Solve simultaneous equations for algebra homework.
Check hand-computed Gaussian elimination row operations.
Visualize how partial pivoting improves numerical stability.
Find solutions to 3×3 or 4×4 systems in engineering problems.
Understand back-substitution by tracing each elimination step.
Verify linear system answers before submitting coursework.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu