🔢System of Equations Solver
Solve 2×2 and 3×3 systems of linear equations with full step-by-step Gaussian elimination working, exact fractions, and special-case handling.
Step-by-step working
Free System of Equations Solver with Steps
Stuck on a simultaneous equations problem? Our free system of equations solver handles 2×2 and 3×3 linear systems and shows every step of the working, so you understand the method — not just the answer. Enter the coefficients in the clean grid, and get the values of each variable clearly displayed, followed by a full elimination walkthrough.
It uses Gaussian elimination with partial pivoting under the hood, so it handles awkward systems correctly — including those with fractional answers, which are shown as proper fractions rather than rounded decimals. Special cases are explained too: if the system has no solution or infinitely many, you'll see exactly why, with the contradiction or parametric form shown. A verification step substitutes the answer back into each equation to confirm it's correct.
Simply enter your coefficients to solve instantly. No sign-up, no downloads, no limits — just clear, step-by-step solutions.
About the System of Equations Solver
Features
2×2 and 3×3 support
Switch system size with a single tab; the coefficient grid updates instantly.
Gaussian elimination with pivoting
Partial pivoting keeps the elimination numerically well-conditioned.
Exact fraction answers
Solutions are shown as simplified fractions, not rounded decimals.
Special-case handling
No-solution and infinite-solution systems are detected and explained, with a parametric form where possible.
Verification step
Each solution is substituted back into every original equation to confirm it.
Fully client-side
All computation runs in your browser — no data ever leaves your device.
Where systems of equations appear
Systems of linear equations show up whenever multiple related quantities need to be found at once — balancing chemical equations, solving circuit networks in engineering, optimizing costs in economics, or simply solving two related word problems in an algebra class. Gaussian elimination is the standard, reliable method for solving them by hand or by computer.
Frequently Asked Questions
What is a system of equations?
Two or more equations sharing the same variables; solving the system means finding the values that satisfy all equations simultaneously.
Which systems can this solve?
2×2 (two equations, two unknowns) and 3×3 (three equations, three unknowns) linear systems, entered via a coefficient grid.
Does it show working?
Yes. Every elimination and back-substitution step is shown in full so you can follow — and learn — the method, not just copy the answer.
What if the system has no solution?
It detects the inconsistency and explains it clearly — showing the contradiction row (e.g. 0 = 5) that makes the system impossible.
What if there are infinite solutions?
It detects the dependent system and displays the parametric form of the solution rather than a single answer.
Are fractional answers shown as fractions?
Yes. Results like x = 2/3 are displayed as fractions rather than rounded decimals, keeping the answer exact.
How does the solver work?
It uses Gaussian elimination with partial pivoting — a reliable method that handles near-singular systems correctly and scales to 3×3 without the instability of simple Cramer's rule.
Is there a verification step?
Yes. After solving, it substitutes the answer back into each original equation and confirms both sides match.
Is my input sent anywhere?
No. All computation runs locally in your browser; your equations never leave your device.
Does it work on mobile?
Yes, fully responsive; the coefficient grid, working, and solution display fit any screen without overflow.