Systems Equation Solver Calculator

Systems equation solver form

Set the matrix size, enter coefficients and constants, then solve the system using automatic selection or a chosen method.

Configuration

System size

Primary method

Display precision

Coefficient matrix and constants

Enter values row by row. The rightmost input in each row is the constant term.

Quick actions

Use the example loader for a ready-made matrix, then adjust any cell as needed.

Example data table

This sample shows a consistent 3 × 3 system with one exact solution.

Equation	Coefficients	Constant	Expected result
1	2, 1, -1	8	x₁ = 2, x₂ = 3, x₃ = -1
2	-3, -1, 2	-11
3	-2, 1, 2	-3

Formula used

Matrix form: A x = b

The coefficient matrix is A, the variable vector is x, and the constants form b.

Gaussian elimination: Apply row operations until the augmented matrix becomes row-echelon or reduced row-echelon form.

Back substitution or direct reading then provides the variable values.

Cramer's rule: x_i = det(A_i) / det(A)

Replace column i with b, compute determinants, and divide by det(A) when det(A) is nonzero.

Inverse method: x = A^-1b

This method only works when A is invertible, which means det(A) is not zero.

How to use this calculator

Select the system size from 2 × 2 through 5 × 5.
Choose a solving method or keep automatic selection enabled.
Enter every coefficient and constant into the matrix grid.
Press Solve system to place the result above the form.
Review the determinant, ranks, solution vector, and verification table.
Inspect the elimination steps and use the CSV or PDF export buttons.
Use the Plotly graph to study intersections or solution magnitude patterns.

FAQs

1. What kinds of systems can this calculator solve?

It solves square linear systems from 2 × 2 up to 5 × 5. The page can report unique, infinite, or impossible cases by comparing matrix ranks after row reduction.

2. Why does the calculator show no solution?

No solution appears when the equations contradict each other. In matrix language, rank(A) becomes smaller than rank([A|b]), proving the system is inconsistent.

3. Why does it report infinitely many solutions?

That result means at least one equation is dependent on the others. The ranks match, but they stay below the number of variables, leaving free parameters.

4. When should I use Cramer's rule?

Use it for smaller non-singular systems when you want determinant-based insight. It becomes computationally heavier as size grows, so elimination is usually more practical.

5. What does the determinant tell me?

A nonzero determinant confirms the coefficient matrix is invertible and the system has one unique solution. A zero determinant signals dependence or singularity.

6. Why is there a verification table?

It multiplies A by the computed solution vector and compares the result with b. Small residuals confirm the calculation is internally consistent.

7. What does the Plotly graph show?

For 2-variable systems, it plots both equations as lines and marks the intersection when one exists. For larger systems, it charts solution values by variable.

8. Can I export the result for reports?

Yes. Use the CSV button for spreadsheet-friendly output and the PDF button for a compact report containing the main metrics and solved values.