Solve simultaneous linear equations with 2, 3 or 4 unknowns. Type the coefficients and this free system of equations solver returns exact answers (fractions, not just decimals), shows Cramer’s rule determinants and the row-reduced matrix, and tells you when a system has no solution or infinitely many.
How the solver works
A system of linear equations can be written as a matrix equation A·x = b. The solver uses two standard methods and shows both:
- Cramer’s rule: each variable equals a determinant ratio, x = Dₓ / D, where D is the determinant of the coefficient matrix and Dₓ replaces the x-column with the constants. It works only when D ≠ 0.
- Gaussian (Gauss–Jordan) elimination: row operations reduce the augmented matrix [A | b] to reduced row echelon form, which also reveals systems with no solution or infinitely many.
No solution or infinitely many?
If the determinant is 0, the equations are not independent. When row reduction produces a row like 0 = 5, the system is inconsistent (the lines or planes never meet). When it produces 0 = 0 instead, there are infinitely many solutions and the calculator lists the free variables.
Frequently asked questions
How do I solve a system of equations with this calculator?
Choose 2, 3 or 4 variables, enter the coefficient of each variable and the constant on the right of each equation, then press Solve system. You get exact answers plus the determinant and elimination working.
What is Cramer’s rule?
Cramer’s rule solves a square linear system using determinants: each variable equals the determinant of the matrix with that variable’s column replaced by the constants, divided by the main determinant.
Why does it say the system has no unique solution?
The coefficient determinant is zero, so the equations are dependent. Row reduction then shows whether there is no solution (a contradiction such as 0 = 3) or infinitely many solutions.
Can I enter fractions or decimals?
Yes. Type values such as 1/2, −3 or 0.25. Answers are converted back to simple fractions where possible.
Is this the same as a simultaneous equations calculator?
Yes. Simultaneous equations and systems of linear equations are the same thing, so you can use it for GCSE, A-level, SAT and college algebra homework.
Related tools
Free educational tool by HawkInc. Results are calculated in your browser and rounded for display; check critical work by hand or with a second method.