Enter coefficients, constants, and matrix size easily. See row operations, back substitution, determinant, solution checks. Download result tables for classwork, audits, reports, and practice.
| x1 | x2 | x3 | b |
|---|---|---|---|
| 2 | 1 | -1 | 8 |
| -3 | -1 | 2 | -11 |
| -2 | 1 | 2 | -3 |
This sample gives the unique solution x1 = 2, x2 = 3, and x3 = -1.
Gaussian elimination transforms the augmented matrix [A|b] into an upper triangular form through elementary row operations.
Row replacement: Ri = Ri - factor × Rk
Factor: factor = aik / akk
After elimination, back substitution solves the triangular system.
Back substitution: xi = (bi - Σ aijxj) / aii for j greater than i.
Partial pivoting improves stability by moving the largest available pivot into the current row.
Choose the number of variables first.
Set the decimal precision you want.
Enter the coefficient matrix values and the constants column.
Use the example button if you want a ready system.
Press the solve button.
Read the result summary above the form.
Review the row operations and the final upper triangular matrix.
Download the result table as CSV or PDF when needed.
Linear equations appear in algebra, engineering, statistics, and economics. This calculator solves square systems with the Gauss elimination method. It converts the augmented matrix into upper triangular form. Then it applies back substitution. The method is systematic. It is also easy to verify with row operations.
Many learners first solve equations by substitution. That works for small problems. It becomes slow for larger systems. Gaussian elimination is cleaner. It handles several variables in one framework. It also reveals whether a system is consistent, inconsistent, or dependent. That makes it useful for homework, exams, and technical reviews.
This calculator uses partial pivoting. It checks the current pivot column and swaps rows when needed. That step improves numerical stability. It reduces rounding trouble. It also avoids division by very small pivot values. For decimal data, this matters a lot.
The tool reports the rank of the coefficient matrix and the augmented matrix. It also estimates the determinant from the triangular form. When a unique solution exists, the calculator lists each variable value. It also shows residuals. Residuals help confirm whether the computed answers satisfy the original equations.
The step section is designed for learning. You can inspect the initial matrix, each row elimination, and every row swap. This helps students understand pivot formation and zero creation below the diagonal. Teachers can also use the output during class demonstrations or assignment checking.
Use this Gauss elimination solver for matrix algebra practice, simultaneous equation checking, computational coursework, and model validation. The CSV and PDF options help with documentation. The example matrix helps new users start quickly. Overall, the calculator gives a clear, structured, and reliable workflow for solving linear systems.
It converts a system of linear equations into an upper triangular matrix. After that, back substitution finds the variable values. The method is direct and organized.
Partial pivoting selects a stronger pivot by swapping rows. This improves numerical stability. It also reduces errors caused by tiny pivot values.
No solution means the equations conflict. During elimination, a row can become all zeros on the left and a nonzero constant on the right.
It means the system is dependent. At least one equation does not add new information. Free variables remain, so one unique answer does not exist.
Residuals measure how closely the computed solution satisfies the original equations. Small residuals show the answer matches the original system well.
Yes. The calculator accepts integers and decimals. You can also adjust precision and tolerance to review rounded outputs more carefully.
The determinant helps describe matrix behavior. A zero determinant often indicates a dependent or singular system. A nonzero determinant supports a unique solution.
Download CSV for spreadsheets and data records. Download PDF for reports, class submissions, or printed review sheets. Both help preserve the result summary.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.