System of Linear Equations Calculator - Linear System Calculator & Algebra Calculator
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: October 19, 2025
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System has a unique solution: x = 2.00, y = 1.00
How to Use:
- • Select the system type (2x2, 3x3, or custom)
- • Choose your preferred solution method
- • The calculator will solve the system automatically
- • Results show solution type, values, and step-by-step process
- • Matrix form and determinant are calculated for analysis
System of Linear Equations Calculator Types & Solution Methods
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
Quick Example Result
For system: 2x + 3y = 7, x - y = 1
Solution Type
Unique
x =
2.00
y =
1.00
How Our System of Linear Equations Calculator Works
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.
The Solution Methods
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
Mathematical Foundation
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.
- Unique solution: det(A) ≠ 0 (system is consistent and independent)
- No solution: det(A) = 0 and system is inconsistent
- Infinitely many solutions: det(A) = 0 and system is dependent
- Cramer's Rule works only for systems with unique solutions
- Gaussian elimination reveals the solution structure
- Matrix methods are computationally efficient for large systems
Sources & References
- Linear Algebra and Its Applications - Lay, Lay, McDonaldComprehensive coverage of linear systems and matrix methods
- Elementary Linear Algebra - Larson, EdwardsDetailed explanation of Cramer's Rule and elimination methods
- Khan Academy - Systems of Linear EquationsEducational resources for understanding linear systems
Need help with other algebraic calculations? Check out our matrix calculator and determinant calculator.
Get Custom Calculator for Your PlatformSystem of Linear Equations Calculator Examples
Given System:
- Equation 1: 2x + 3y = 7
- Equation 2: x - y = 1
- Method: Cramer's Rule
- Matrix: [[2, 3], [1, -1]]
Solution Steps:
- Calculate determinant: det = 2(-1) - 3(1) = -5
- Apply Cramer's rule: x = (7(-1) - 3(1))/(-5) = 2
- Apply Cramer's rule: y = (2(1) - 7(1))/(-5) = 1
- Verify: 2(2) + 3(1) = 7 ✓, 2 - 1 = 1 ✓
Result: x = 2, y = 1 (Unique Solution)
The system has exactly one solution since the determinant is non-zero.
3x3 System Example
x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2
Solution: x = 1, y = 2, z = 3
Inconsistent System Example
x + y = 3, x + y = 5
No solution (parallel lines)
Frequently Asked Questions
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