A unique pair
The rows 2 1 5 and 1 −1 1 represent 2x + y = 5 and x − y = 1. The solution is x = 2 and y = 1.
Calculations run locally in your browser. This tool does not upload the numbers you enter.
How to use the Linear Equation Solver
- 01
Choose two or three variables. Write each equation in x, y, z order with the constant on the right.
- 02
Enter one equation per line as its coefficients followed by the right-hand constant. Use zero for a missing variable.
- 03
Calculate, inspect the classification and reduced matrix, then copy or download the result.
Examples
A contradiction
Rows 1 1 2 and 2 2 5 cannot both hold. Doubling the first equation would give 2x + 2y = 4, so the solver reports no solution.
From equations to rows
For a three-variable system, the row 2 0 −3 7 means 2x + 0y − 3z = 7. The zero is essential: omitting it shifts the remaining entries into the wrong columns. The tool expects exactly as many equations as variables.
What rank tells you
Elimination identifies pivot columns. Full coefficient rank gives a unique numerical solution. A missing pivot without a contradictory row indicates free variables and infinitely many solutions. A row whose coefficients are effectively zero but whose constant is nonzero indicates an inconsistent system.
Checking the answer
For a unique solution, the residual is the largest absolute difference between an original left-hand side evaluated at the returned solution and its right-hand constant. A small residual supports the calculation for your entered equations, although an ill-conditioned system can still have sensitive solution values.
What to know
- Only square 2 × 2 and 3 × 3 systems are supported.
- Coefficient magnitude is limited to 1e12; singularity decisions are numerical, using a 1e−12 scaled pivot tolerance.
Common questions
Can I type x + y = 2 directly?
No. Enter coefficient rows such as 1 1 2. This avoids ambiguous equation parsing and makes missing terms explicit.
Why use a pivot tolerance?
The calculation uses row-scaled floating-point elimination. Scaled pivots smaller than 1e−12 are treated as zero, so near-dependent systems may be classified as singular.
Does an infinite-solution result list a parameterization?
No. It identifies free-variable behavior through rank and the reduced augmented matrix, but does not return a symbolic parametric family.
Are answers exact fractions?
No. Coefficients and results are numerical approximations. Use exact rational algebra when a proof or an exact singularity decision is required.