System of Linear Equations Calculator
Enter coefficients for a square system (same number of equations and variables), then solve instantly using Gaussian elimination.
Tip: You can enter integers, decimals, or fractions like 3/4.
What is a system of linear equations?
A system of linear equations is a collection of equations that all describe straight-line (or flat-plane) relationships between variables. Your goal is to find values of the variables that satisfy every equation at the same time.
For example, in two variables:
x - y = 1
The solution is the point where both lines intersect. In higher dimensions (3 variables, 4 variables, and beyond), the same idea applies: you're looking for the shared intersection that satisfies all equations simultaneously.
How to use this calculator
Step-by-step
- Select the size of the system (2×2 up to 5×5).
- Fill in the coefficient matrix (left side) and constants column (right side).
- Click Solve System.
- Read whether the system has a unique solution, infinitely many solutions, or no solution.
If you make a typo, click Clear Inputs and start over.
What the results mean
1) Unique solution
There is exactly one set of values for all variables. This usually means the equations are independent and intersect at one point.
2) Infinitely many solutions
At least one equation is dependent on the others. Geometrically, equations overlap in a way that leaves a full line, plane, or subspace of answers.
3) No solution
The system is inconsistent. In 2D this is like parallel lines: they never meet.
Quick worked example (3×3)
2x - y + z = 3
x + 2y - z = 3
Enter these rows in the calculator. You should get a unique solution: x = 1, y = 2, z = 3.
Method used: Gaussian elimination
This calculator uses Gaussian elimination with pivoting, a standard numerical method taught in algebra, engineering, data science, and numerical analysis.
- Build an augmented matrix from your coefficients and constants.
- Eliminate lower entries column by column to create an upper-triangular form.
- Check consistency and rank conditions.
- Use back substitution when a unique solution exists.
Pivoting improves stability by reducing division by very small numbers when possible.
Common input mistakes to avoid
- Leaving a coefficient blank (use 0 if a variable is missing in an equation).
- Mixing rows accidentally (double-check each equation before solving).
- Typing invalid fractions (use format like -5/2).
- Forgetting signs in front of negative constants.
Why this tool is useful
Solving systems quickly helps in budgeting models, physics constraints, optimization setups, balancing equations, and machine learning preprocessing. Even if you know the manual steps, a reliable calculator saves time and catches arithmetic errors.