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Free system of linear equations calculator & linear system calculator. Solve 2x2 and 3x3 systems using Cramer's rule, Gaussian elimination & matrix methods. Our calculator uses determinant analysis to provide accurate solutions for algebra, linear algebra, and mathematics education.
Last updated: February 2, 2026
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System has a unique solution: x = 2.00, y = 1.00
Method
Determinant-based
Uses determinants to find unique solutions for 2x2 and 3x3 systems
Method
Row Operations
Transforms augmented matrix to row-echelon form for systematic solution
Method
Matrix Inversion
Uses matrix inversion to solve systems: X = A⁻¹B
Method
Algebraic Substitution
Solves one equation for a variable and substitutes into others
Analysis
Solution Classification
Determines if system has unique, no, or infinite solutions
Capabilities
Multi-variable Systems
Handles systems with 2, 3, or more variables and equations
For system: 2x + 3y = 7, x - y = 1
Solution Type
Unique
x =
2.00
y =
1.00
Our system of linear equations calculator uses fundamental algebraic and matrix methods to solve systems of linear equations. The calculation applies determinant analysis and elimination techniques to provide accurate solutions for algebra, linear algebra, and mathematical problem-solving.
Cramer's Rule: x = det(Ax)/det(A), y = det(Ay)/det(A)Gaussian Elimination: Row operations → Row-echelon formMatrix Method: X = A⁻¹BEach method has advantages: Cramer's Rule is direct for small systems, Gaussian elimination is systematic for larger systems, and matrix methods are computationally efficient.
Shows different solution methods and their applications
Systems of linear equations are fundamental in linear algebra and have applications across mathematics, science, and engineering. The solution methods are based on the properties of matrices and determinants. A system has a unique solution when the coefficient matrix is invertible (determinant ≠ 0), no solution when the system is inconsistent, or infinitely many solutions when the system is dependent.
Need help with other algebraic calculations? Check out our matrix calculator and determinant calculator.
Get Custom Calculator for Your PlatformResult: x = 2, y = 1 (Unique Solution)
The system has exactly one solution since the determinant is non-zero.
x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2
Solution: x = 1, y = 2, z = 3
x + y = 3, x + y = 5
No solution (parallel lines)
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