Systems of Equations Solver
Solve systems of linear equations with step-by-step solutions. Supports 2x2, 3x3, and larger systems using Gaussian elimination.
Formula:Ax = b → x = A⁻¹b
Enter Equations
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Systems of Equations Practice
Test your skills with practice problems
Practice with 5 problems to test your understanding.
Solving Methods
Substitution
Solve one equation for a variable and substitute into others
Elimination
Add/subtract equations to eliminate variables
Gaussian Elimination
Convert to row echelon form and back-substitute
Cramer's Rule
Use determinants to find each variable (square systems only)
What is a System of Equations?
A system of linear equations is a collection of two or more linear equations involving the same set of variables. Solving the system means finding values for all variables that satisfy every equation simultaneously. Methods include substitution, elimination, and matrix methods like Gaussian elimination.
A system of equations is a set of two or more equations with the same variables. Use our free calculator to solve systems of linear equations instantly with step-by-step solutions. Simply enter equations like "2x + 3y = 7" and "x - y = 1" and get the solution with detailed Gaussian elimination steps.
Key Facts About Systems of Equations
- *A system has a unique solution when the equations intersect at exactly one point
- *Inconsistent systems have no solution (parallel lines that never meet)
- *Dependent systems have infinitely many solutions (same line)
- *Gaussian elimination converts the system to row echelon form
- *Back substitution finds values from the bottom row up
- *The number of equations should match the number of unknowns for a unique solution
- *Partial pivoting improves numerical stability
- *Systems can be represented as augmented matrices [A|b]
Frequently Asked Questions
How do I solve a 2x2 system of equations?
For a 2x2 system like x + y = 5 and x - y = 1, you can use substitution (solve one equation for a variable, substitute into the other), elimination (add/subtract equations to eliminate a variable), or Gaussian elimination (convert to matrix form and reduce).
What does "no solution" mean for a system?
A system has no solution when the equations are inconsistent - they represent parallel lines (2D) or parallel planes (3D) that never intersect. For example, x + y = 1 and x + y = 3 have no solution because x + y cannot equal both 1 and 3.
What does "infinite solutions" mean?
Infinite solutions occur when the equations are dependent - they represent the same line or plane. For example, x + y = 2 and 2x + 2y = 4 are the same equation, so any point on the line x + y = 2 is a solution.
What is Gaussian elimination?
Gaussian elimination is a systematic method for solving systems by converting the augmented matrix to row echelon form through elementary row operations: swapping rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another. Then back substitution finds the solution.
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).