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Free system of equations calculator & linear system solver. Solve 2×2 and 3×3 systems using substitution, elimination, Cramer's rule, and Gaussian elimination with step-by-step solutions and matrix analysis.
Last updated: February 2, 2026
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unique
Substitution Method
2
x = 2.0000
y = 1.0000
Given: 2x + 3y = 7 and x - y = 1
Step 1: Solve equation 2 for x: x = y + 1
Step 2: Substitute into equation 1: 2(y + 1) + 3y = 7
Step 3: Simplify: 2y + 2 + 3y = 7
Step 4: Solve for y: 5y = 5, so y = 1
Step 5: Find x: x = 1 + 1 = 2
Solution: x = 2, y = 1
Main Determinant
-5
Dx
-10
Dy
-5
Best for
One variable isolated
Solve one equation for a variable, then substitute into other equations
Best for
Similar coefficients
Add or subtract equations to eliminate one variable at a time
Best for
2×2 and 3×3 systems
Uses determinants to find solutions when system has unique solution
Best for
Large systems
Systematic approach using elementary row operations
System type
Two equations, two variables
Most common system type in algebra and precalculus
System type
Three equations, three variables
Advanced system requiring matrix methods for efficient solution
For system: 2x + 3y = 7, x - y = 1
Solution
x = 2, y = 1
Method
Substitution
Our system of equations calculator uses multiple solution methods to solve linear systems efficiently. The calculator applies matrix theory and algebraic techniques to find solutions, analyze system properties, and provide step-by-step solutions for educational purposes.
Substitution: Solve one equation, substitute into othersElimination: Add/subtract equations to eliminate variablesCramer's Rule: Use determinants for 2×2 and 3×3 systemsGaussian Elimination: Row operations for systematic solutionEach method has advantages for different types of systems. The calculator automatically chooses the most appropriate method or allows manual selection for educational purposes.
Shows matrix representation and solution process
System of equations theory is based on linear algebra and matrix theory. The fundamental theorem states that a system has a unique solution if and only if the coefficient matrix has full rank (determinant ≠ 0). Other cases lead to no solution or infinitely many solutions.
Need help with other algebra topics? Check out our reduced echelon form calculator and linear interpolation calculator.
Get Custom Calculator for Your PlatformSolution: x = 2, y = 1
The system has a unique solution, indicating the lines intersect at point (2, 1).
Same system: 2x + 3y = 7, x - y = 1
D = -5, Dx = -10, Dy = -5
x = -10/-5 = 2, y = -5/-5 = 1
x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2
Solution: x = 1, y = 2, z = 3
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