System of Equations Solver (2×2, 3×3, n×n)

A system of linear equations is a set of equations that must all be true at once, such as 2x + 3y − z = 5 together with two more. This solver eliminates variables in exact fractions, so 23/5 stays 23/5 instead of turning into a rounded decimal. It tells you whether the system has one solution, none, or infinitely many (with the general solution in terms of the free variables), and shows the determinant, Cramer's rule for 2×2 and 3×3 systems and the reduced row echelon form.

How to use this tool

1. Type one equation per line, with variables such as x, y, z or x1, x2 (terms may be on both sides).
2. Alternatively paste augmented-matrix rows: the coefficients, then the constant.
3. Read the solution type and each variable's value as a fraction and a decimal.
4. Use the determinant, Cramer's rule and the RREF table to check the working.

Formula

Gauss–Jordan elimination on [A | b] · Cramer: xᵢ = det(Aᵢ) ÷ det(A)

Elimination swaps rows, scales a row so its pivot is 1 and subtracts multiples of it from every other row. A row 0 = c (c ≠ 0) means no solution; fewer pivots than variables means free variables and infinitely many solutions. Aᵢ is the coefficient matrix with column i replaced by the constants; Cramer's rule only applies when det(A) ≠ 0.

Example

For 2x + 3y − z = 5, x − y + 2z = 3 and 3x + y + z = 10, the determinant is 5 and D_x = 23, D_y = −10, D_z = −9, so x = 23/5 = 4.6, y = −2 and z = −9/5 = −1.8. Check the second equation: 4.6 + 2 − 3.6 = 3.