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Linear Equation System Solver

Solve a bounded linear system and report a unique solution, inconsistency or an underdetermined numerical result.

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How this works

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One equation per line, with coefficients of x1, x2 and so on separated by commas. Up to 6 rows and 6 unknowns. Enter coefficients only, not expressions.

One number per equation, in the same row order. Plain decimals; magnitude at most 10^12.

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How to use Linear Equation System Solver

  1. Enter coefficient rows and a matching list of constants.
  2. Run the pivoted numerical solver.
  3. Read the classification, tolerance and residuals together.

Example: Linear Equation System Solver

Linear Equation System Solver: A unique numerical solution was found.

You add
Coefficient rows: 1,1 1,-1 Right-hand constants: 5 1
You get
A unique numerical solution was found. Classification: unique Coefficient rank: 2 Equations: 2 Unknowns: 2 x1: 3 x2: 2 Equation 1 residual (Ax − b): 0 Equation 2 residual (Ax − b): 0 Decimal arithmetic with row-scaled pivoting. Coefficients at or below 10^-12 after row scaling count as zero. Each row has a right-hand tolerance starting at 10^-12 times max(1, its scaled constant), propagated through the row operations. Nearly dependent equations may change classification under this rule. Residuals use the original equations; this is not symbolic or exact-rational proof. Classification | unique Coefficient rank | 2 Equations | 2

Options

Coefficient rows
Each row represents one equation and each column one unknown. Enter one matching right-hand constant for every row.
Numerical classification
The solver uses row scaling and a pivot tolerance to classify unique, inconsistent or underdetermined results.

Supported inputs and limits

1–6 coefficient rows, 1–6 unknowns, rectangular allowed; ≤6,000 coefficient chars and1,000 constant chars. Magnitude ≤10^12, 25 digits/12 fractional places. Row scaling and pivot threshold 10^-12; RHS consistency tolerance propagated per row. At most six unknowns and bounded finite coefficients. Row-scaled pivoting uses a stated numerical tolerance, so near-singular systems can change classification with scale or rounding. Results are numerical approximations with finite decimal precision rather than symbolic proofs.

Where your input is processed

This tool processes your input in this browser. Your text and files are not uploaded to UseFreeTools. Check this tool's limits for anything it may save on your device.

Near dependence deserves a precision check

An almost redundant equation can behave differently after rounding its coefficients. Keep the entered matrix and stated tolerance with the result. This is a bounded numerical calculation, so an underdetermined classification is not a symbolic description of every possible exact solution.

Questions about Linear Equation System Solver

What does underdetermined mean?

The numerical rank does not determine every unknown and no inconsistency was detected within the tolerance.

Why show residuals?

Substituting the computed solution back into Ax-b checks how closely it satisfies each entered equation.

Can tiny coefficients be treated as zero?

Yes, relative to the documented tolerance and row scale. Inspect near-singular results rather than treating their rank as exact.

Project manager: Tony Hines · Content updated 4 October 2026 · Report a problem

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