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

Click to show tips

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

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.