About the System of Equations Solver
A free tool for solving 2×2 and 3×3 linear systems, with full working shown.
What it is
The System of Equations Solver takes a 2×2 or 3×3 system of linear equations entered as a coefficient grid and solves it using Gaussian elimination with partial pivoting. Every row operation is shown, results are displayed as exact fractions, and special cases — no solution or infinitely many solutions — are explained clearly rather than left as an error.
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.