Solve 2x2 and 3x3 linear systems instantly with step-by-step working
System Size
Enter Coefficients
Each row: a·x + b·y = c
Eq 1: 2x + 3y = 8
Eq 2: 5x - y = 3
Solution
Determinants
D = -17
Dx = -17
Dy = -34
x = Dx / D = -17 / -17 = 1
y = Dy / D = -34 / -17 = 2
Verification
How It Works
Solve a system of linear equations instantly with this free online calculator. Enter coefficients for a 2×2 or 3×3 system and get exact solutions using Cramer's Rule — complete with determinant working and automatic back-substitution verification. No software to install, no account required. Ideal for students, engineers, and anyone checking algebra homework.
A system of equations is a set of two or more equations that share the same variables. Solving the system means finding values for each variable that satisfy all equations simultaneously.
The tool uses Cramer's Rule, which expresses each variable as a ratio of determinants. For a 2×2 system, x = Dx/D and y = Dy/D, where D is the coefficient matrix determinant and Dx, Dy are determinants formed by replacing the respective column with the constants.
When the main determinant D equals zero, the system either has no solution (inconsistent — the lines or planes are parallel) or infinitely many solutions (dependent — the equations describe the same line or plane). The solver detects and reports both cases.
Results are calculated using IEEE 754 double-precision floating-point arithmetic and rounded to 10 significant figures. The solver also substitutes the answers back into each original equation so you can confirm correctness directly on screen.