System of Equations Solver - Solve 2x2, 3x3 and 4x4 Linear Systems Exactly
System of Equations Solver
Solve linear systems exactly, with fractions and the determinant
Exact Fractions
Works in exact arithmetic, so 3x = 1 gives x = 1/3, not a rounded 0.33333.
Three Outcomes
Tells you clearly whether the system has one solution, none (contradictory equations) or infinitely many.
Determinant Included
The determinant of the coefficient matrix is shown; a value of zero means there is no unique solution.
Full Privacy
Everything is calculated in your browser.
How a System of Equations Is Solved
This solver uses Gauss-Jordan elimination: it combines equations to eliminate one unknown at a time until each variable stands alone. Geometrically, two equations in two unknowns are lines, and the solution is where they cross. Parallel lines never meet (no solution) and identical lines overlap everywhere (infinitely many solutions).
The determinant is a quick test: when it is not zero, there is exactly one solution. When it is zero the equations are dependent or contradictory, and the solver tells you which. For example, x + y = 2 and 2x + 2y = 4 are the same line, while x + y = 2 and 2x + 2y = 5 are parallel.
Key Takeaways
- Flexible entries: Integers, decimals and fractions like -7/2 all work; blank counts as 0.
- No rounding: Answers are exact fractions, with a decimal shown alongside.
- Up to 4 unknowns: Choose 2, 3 or 4 equations; the grid adjusts.