Math

System of Equations Solver

Solve Any System of Equations Instantly

Enter your system of linear equations and get the answer immediately. This solver auto-classifies the system (unique, infinite, or no solution), shows the determinant, and provides step-by-step working on demand. Supports 2x2 and 3x3 systems with three solving methods.

Quick Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Simple 2x2 System

Solve 2x + 3y = 7 and x - y = 1 using Gaussian elimination

Key values: 2×2 system · Unique solution · x=2, y=1

3x3 Engineering System

Solve a 3-variable system commonly found in circuit analysis

Key values: 3×3 system · Cramer's Rule · 3 unknowns

Parallel Lines (No Solution)

Explore what happens when two equations have no intersection

Key values: 2×2 system · Inconsistent · No solution

Linear System Solver

Enter your system of linear equations to find the solution. Supports 2×2 and 3×3 systems with step-by-step Gaussian elimination, Cramer's rule, and matrix inverse.

Documentation

This calculator is also known as System of Equations Solver.

Read the complete guide

How This Solver Works

Enter your coefficients, select a method, and the solver instantly computes the solution. It first classifies the system using rank analysis (Rouche-Capelli theorem), then applies your chosen method. The result includes exact fractional solutions when possible.

Examples

Basic 2x2 System

Solve 2x + 3y = 7 and x - y = 1.

Using Gaussian elimination: eliminate x from equation 2 by subtracting 0.5 times equation 1. Back-substitute to get y = 1, x = 2.

Key takeaway: The determinant det(A) = -5 confirms a unique solution exists.

Getting the Best Results

Tips for solving systems of equations:

Write each equation in standard form: ax + by (+ cz) = d
Check for obvious contradictions before entering (e.g., same LHS, different RHS)
Use Cramer's rule for 2x2 systems to verify by hand
For ill-conditioned systems, expect rounding warnings

Frequently Asked Questions about System of Equations Solver

What does "no solution" mean for a system of equations?

No solution (inconsistent system) means the equations are contradictory. In 2D, the lines are parallel. In 3D, the planes do not share a common intersection.

What are free variables in an infinite-solution system?

Free variables are parameters that can take any value. Other variables are expressed in terms of these free variables. For example, if y is free, x might be expressed as x = 3 - y.

Can I choose which method to use?

Yes. Select from Gaussian Elimination, Cramer's Rule, or Matrix Inverse in the method dropdown. If your chosen method fails (e.g., Cramer's rule with det(A)=0), the solver falls back to Gaussian elimination automatically.