MATH · LINEAR EQUATION

Linear Equation Calculator

Solve systems of 2 or 3 linear equations with step-by-step solutions. Detects unique solutions, no solution (parallel lines), and infinitely many solutions.

System Size
Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
2×2 Solution
Solution
x = 3
y = 2
↳ Steps
System: (1) 1x + 1y = 5
(2) 1x + -1y = 1
Determinant D = 1·-1 − 1·1 = -2
Dx = 5·-1 − 1·1 = -6
Dy = 1·1 − 1·5 = -4
x = Dx / D = -6 / -2 = 3
y = Dy / D = -4 / -2 = 2

About This Calculator

Enter the coefficients of a 2×2 or 3×3 system of linear equations to find the solution instantly. The calculator shows step-by-step working and correctly identifies when no unique solution exists (parallel lines or dependent equations).

How It Works

For 2×2 systems, Cramer's rule computes the determinant D and the sub-determinants Dx and Dy. For 3×3 systems, Gaussian elimination with partial pivoting reduces the augmented matrix to row echelon form, then back-substitution finds x, y, z. Both methods detect singular cases (det = 0) and report whether the system is inconsistent or has infinitely many solutions.

The Formula

x = Dx/D, y = Dy/D (2×2 — Cramer's rule)

D
determinant of the coefficient matrix
Dx, Dy
determinants with the RHS column substituted in

Frequently Asked Questions

What does "no solution" mean?
The system has no solution when the two (or three) equations represent parallel lines (or planes) that never intersect. Algebraically, the coefficient determinant is zero but the equations are inconsistent — they cannot all be true simultaneously.
What does "infinitely many solutions" mean?
This occurs when the equations are dependent — one equation is a multiple of another, so they represent the same line (2×2) or the same plane (3×3). Any point on that line or plane satisfies both equations.
Can I use negative or decimal coefficients?
Yes. All coefficient fields accept negative numbers and decimals. Use the − sign before a number to enter a negative coefficient.
What is the format for entering the system?
For a 2×2 system enter a₁, b₁, c₁ (first equation: a₁x + b₁y = c₁) and a₂, b₂, c₂ (second equation). For a 3×3 system the right-hand side is d₁, d₂, d₃ and the first three coefficient columns are a, b, c for x, y, z respectively.