Enter coefficients and solve systems by elimination. Review augmented matrices, pivots, swaps, and determinant insights. Download results and examples for study, checking, and sharing.
Fill each equation card. Then solve the linear system with Gaussian elimination.
| Equation | x1 | x2 | x3 | Constant | Expected Note |
|---|---|---|---|---|---|
| Equation 1 | 2 | 1 | -1 | 8 | Unique system |
| Equation 2 | -3 | -1 | 2 | -11 | Unique system |
| Equation 3 | -2 | 1 | 2 | -3 | x1 = 2, x2 = 3, x3 = -1 |
Linear systems are written as A x = b.
The coefficient matrix is A. The unknown vector is x. The constants vector is b.
Gaussian elimination applies row operations to the augmented matrix [A | b].
For each pivot column, the elimination factor is:
factor = a(i,j) / a(j,j)
Then the row update is:
R(i) = R(i) - factor × R(j)
After the matrix reaches upper triangular form, back substitution is used:
x(n) = b(n) / a(n,n)
x(i) = (b(i) - Σ a(i,k)x(k)) / a(i,i)
The determinant is the product of diagonal entries after elimination, adjusted by row swaps. Rank compares the number of independent rows in the coefficient and augmented matrices.
The linear equations gaussian elimination calculator helps solve systems with speed and structure. It is useful for algebra practice, engineering work, data analysis, and numerical methods. You enter coefficients, constants, and the matrix size. The tool then applies row operations, tracks pivots, and reports the solution type.
Gaussian elimination changes an augmented matrix into an upper triangular form. Each step uses legal row operations. Rows can be swapped. A row can be reduced by subtracting a multiple of another row. These operations preserve the solution set. After elimination, back substitution finds the unknown values when a unique solution exists.
This calculator also checks important matrix properties. It measures the rank of the coefficient matrix and the augmented matrix. It estimates the determinant for square systems with full rank. These values help classify a system as unique, dependent, or inconsistent. That makes the result easier to trust and explain.
A clear step log is valuable for students and teachers. You can inspect every swap and elimination factor. That supports homework checking and classroom demonstrations. The example data table shows a ready-made system for quick testing. The export options also help. You can save results for reports, revision notes, or assignments.
Partial pivoting improves stability. The calculator searches for a stronger pivot in each column before elimination. This reduces division by very small numbers and makes the method more dependable with decimal inputs. While classroom examples often use clean integers, real problems may contain fractions and measured values. A pivot check helps those cases.
Linear systems appear in many fields. They model circuit currents, mixture balances, budgeting, traffic flow, and regression steps. Because the calculator presents both the process and the answer, it works as a learning aid and a verification tool. It saves time and reduces arithmetic mistakes.
The interface is simple and practical. Choose the number of variables. Enter each coefficient and constant. Submit the form. The result appears above the matrix form, so review is immediate. You can then compare the upper triangular matrix, determinant, ranks, and solution vector in one place.
Use this linear equations gaussian elimination calculator whenever manual elimination feels slow or error-prone. It supports learning and fast verification. It also strengthens understanding of augmented matrices, pivots, row reduction, consistency, and back substitution. For anyone solving simultaneous linear equations, this tool offers a reliable and readable workflow.
It solves square systems of simultaneous linear equations. You can work with 2 × 2 up to 6 × 6 systems and inspect the elimination process step by step.
Yes. It compares the rank of the coefficient matrix with the rank of the augmented matrix. If they differ, the system is inconsistent and has no solution.
Yes. When both ranks are equal but smaller than the number of variables, the system is dependent. That means free variables exist and there is no single unique answer.
Row swaps help place a stronger pivot in the working row. This improves numerical stability and prevents division by very small values during elimination.
The determinant helps describe whether the coefficient matrix is singular. A zero determinant usually means the system cannot have a unique solution.
An augmented matrix places the constants column beside the coefficient matrix. It lets you apply row operations to the full system in one organized structure.
After forward elimination, the matrix becomes upper triangular. Back substitution then solves the last variable first and moves upward until every variable is found.
Yes. Use the CSV button for spreadsheet-friendly output or the PDF button for a portable report. The export tools also include the example table.
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.