# System of Equations Calculator

Solve 2×2 and 3×3 systems of linear equations instantly using Cramer's rule. Find unique solutions, detect no-solution or infinite-solution cases.

## What this calculates

Solve systems of linear equations with two or three unknowns. Enter the coefficients and constants, and this calculator applies Cramer's rule to find the solution, compute the determinant, and identify whether the system has a unique, infinite, or no solution.

## Inputs

- **System Size** — options: 2×2 System, 3×3 System — Choose whether to solve a 2-variable or 3-variable system.
- **a₁₁** — Coefficient of x in equation 1.
- **a₁₂** — Coefficient of y in equation 1.
- **a₁₃** — Coefficient of z in equation 1 (3×3 only).
- **b₁** — Right-hand side of equation 1.
- **a₂₁** — Coefficient of x in equation 2.
- **a₂₂** — Coefficient of y in equation 2.
- **a₂₃** — Coefficient of z in equation 2 (3×3 only).
- **b₂** — Right-hand side of equation 2.
- **a₃₁** — Coefficient of x in equation 3 (3×3 only).
- **a₃₂** — Coefficient of y in equation 3 (3×3 only).
- **a₃₃** — Coefficient of z in equation 3 (3×3 only).
- **b₃** — Right-hand side of equation 3 (3×3 only).

## Outputs

- **x** — The value of x.
- **y** — The value of y.
- **z** — The value of z (3×3 systems only).
- **Determinant** — The determinant of the coefficient matrix.
- **Solution Type** — formatted as text — Whether the system has a unique solution, infinitely many solutions, or no solution.

## Details

A system of linear equations consists of two or more linear equations with the same variables. For a 2×2 system (two equations, two unknowns), the solution is the point where two lines intersect. For a 3×3 system, it is the point where three planes meet.

This calculator uses Cramer's rule, which expresses each variable as a ratio of determinants. For the system Ax = b, the solution is x_i = det(A_i) / det(A), where A_i is the matrix formed by replacing column i of A with the constant vector b. The method works whenever det(A) is nonzero, indicating a unique solution.

When the determinant is zero, the system is either inconsistent (no solution, meaning the lines or planes are parallel) or dependent (infinitely many solutions, meaning the equations describe the same line or plane). The calculator detects both cases and reports the result accordingly.

## Frequently Asked Questions

**Q: What is Cramer's rule?**

A: Cramer's rule is a method for solving systems of linear equations using determinants. For a system Ax = b, each variable x_i equals det(A_i)/det(A), where A_i is formed by replacing the i-th column of A with the vector b. It works only when det(A) is not zero.

**Q: What does it mean when the determinant is zero?**

A: A zero determinant means the coefficient matrix is singular. The system either has no solution (the equations are contradictory, e.g. parallel lines) or infinitely many solutions (the equations are dependent, e.g. the same line expressed differently).

**Q: Can this solver handle larger systems?**

A: This calculator supports 2×2 and 3×3 systems. For larger systems, methods like Gaussian elimination or LU decomposition are more efficient than Cramer's rule, whose computational cost grows factorially with system size.

**Q: How do I set up a system of equations from a word problem?**

A: Identify the unknowns and assign variables (x, y, z). Translate each condition into an equation of the form a₁x + a₂y + a₃z = b. Enter the coefficients and constants into the calculator to find the solution.

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Source: https://vastcalc.com/calculators/math/system-of-equations
Category: Math
Last updated: 2026-04-21