System of Equations Calculator | 2×2 & 3×3 Solver
Start typing to discover tools…
 View all results
Linear Equation Solver

System of Equations Calculator

Solve 2×2 and 3×3 linear equation systems, identify unique, infinite or missing solutions, and verify each calculated result instantly.

Instant solutions 2×2 and 3×3 systems Result verification

Linear System Solver

Enter each coefficient and equation constant

Ready

Enter Your Equations

2 Equations · 2 Variables
x
+
y
=
b₁
x
+
y
=
b₂

Equation Preview

Equation 1 will appear here
Equation 2 will appear here

Unique Solution

The system classification will appear here.

Determinant 0
Coefficient Rank 0
Augmented Rank 0

Entered Equations

Try an example

What Is a System of Equations Calculator?

A system of equations calculator finds values that satisfy several linear equations at the same time. The shared values represent the point where the equations intersect.

This calculator solves systems containing two equations and two variables or three equations and three variables. It also identifies systems with no solution or infinitely many solutions.

2×2 and 3×3 support Switch between two-variable and three-variable linear equation systems.
Solution classification Identify unique, infinite and inconsistent equation systems.
Automatic verification Substitute unique solutions back into every original equation.
Simple Process

How the System of Equations Calculator Works

Solve and classify a linear equation system in three straightforward steps.

01

Select the System Size

Choose a 2×2 system for two equations or a 3×3 system for three equations.

02

Enter the Coefficients

Add every x, y and optional z coefficient followed by each equation constant.

03

Review the Solution

View the variable values, determinant, matrix ranks and verification results.

Methods and Rules

How Linear Equation Systems Are Solved

The calculator uses matrix reduction and rank comparison to solve and classify each system.

Matrix Representation

The variable coefficients form matrix A, the unknown variables form vector X and the constants form vector B.

AX = B

Gaussian Elimination

Row operations transform the augmented matrix into a simpler form from which the variables can be read.

[A | B] → Reduced Row-Echelon Form

Unique Solution

A square system has one unique solution when its coefficient matrix has full rank.

rank(A) = rank([A|B]) = n

Infinite Solutions

A consistent system has infinitely many solutions when its rank is smaller than the number of variables.

rank(A) = rank([A|B]) < n

No Solution

A system is inconsistent when the augmented matrix rank is greater than the coefficient matrix rank.

rank(A) < rank([A|B])

Determinant Test

A nonzero determinant guarantees a unique solution for a square linear system.

det(A) ≠ 0 → Unique Solution
Common Outcomes

System of Equations Examples

Compare systems that produce unique, infinite and missing solutions.

Equation System Classification Result
2x + 3y = 7
x − y = 1 		Unique Solution x = 2, y = 1
x + 2y = 5
2x + 4y = 10 		Infinite Solutions Dependent equations
x + 2y = 5
2x + 4y = 12 		No Solution Inconsistent equations
x + y + z = 6
2x − y + z = 3
x + 2y − z = 2 		Unique 3×3 Solution x = 1, y = 2, z = 3
Helpful Answers

System of Equations Calculator FAQs

Find answers about linear systems, determinants, matrix ranks and solution types.

What is a system of equations?

A system of equations is a group of equations that use the same variables. A solution must satisfy all equations simultaneously.

Can this calculator solve 3×3 systems?

Yes. Select the 3×3 option and enter the x, y and z coefficients and constant for all three equations.

What method does the calculator use?

It uses Gaussian elimination with partial pivoting. It also compares coefficient and augmented matrix ranks to classify the result.

What does a zero determinant mean?

A zero determinant means the system does not have one unique solution. It may have infinitely many solutions or no solution.

What are infinitely many solutions?

Infinitely many solutions occur when the equations are dependent and describe the same line, plane or mathematical relationship.

Why would a system have no solution?

A system has no solution when its equations contradict each other. In two dimensions, this often means two parallel lines that never intersect.

Can coefficients be negative or decimal values?

Yes. The calculator accepts positive numbers, negative numbers, zeros and decimal coefficients.

How are solutions verified?

Each calculated variable value is substituted into the original equations. The resulting left side is compared with the equation constant.

Solve Linear Equation Systems Instantly

Enter your coefficients above to solve and verify 2×2 or 3×3 linear equation systems.

Open Calculator
Result copied successfully.