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Gaussian elimination is the classic two-stage method for solving a system of linear equations: eliminate downward to reach a triangular form, then solve upward one variable at a time by back-substitution — distinct from Gauss-Jordan, which instead keeps eliminating until the answer appears directly with no separate back-substitution step.
This calculator solves a square system of equations using forward elimination and back-substitution.
How it works
Enter the coefficient matrix and constants for your system. The calculator performs forward elimination with partial pivoting to reach row echelon form (upper triangular), then solves for each variable starting from the last row and substituting upward.
- Click Calculate to see your results.
Examples
2x + y = 8, x + 3y = 13
This system reduces to an upper triangular form, and back-substitution gives x = 2.2 and y = 3.6.
Who should use it
- Learning or checking the Gaussian elimination method for a linear algebra course.
- Solving a small system of linear equations step by step.
Industry applications
- Linear algebra and mathematics education
- Engineering and applied mathematics coursework
Advantages
- Demonstrates the classic two-stage elimination-then-back-substitution method directly.
- Uses partial pivoting for numerical stability during elimination.
Limitations
- Limited to square, non-singular systems — see Gauss-Jordan Elimination for a more general solver.
Common mistakes to avoid
- Expecting a non-square system to work here — this tool is specifically for square systems with exactly one solution.
- Confusing row echelon form (this tool's intermediate step) with reduced row echelon form (Gauss-Jordan's result).
Best practices
- If you specifically need to see the back-substitution process taught in an introductory linear algebra course, use this tool; if you just need the solution (or need to handle a non-square/rank-deficient system), use Gauss-Jordan Elimination instead.
Tips
- Gaussian elimination and Gauss-Jordan elimination always produce the same final solution for a system with a unique answer — they differ only in method and how much of the matrix gets reduced along the way.