Tip: Enter decimal or negative coefficients (for example: -2.5).
What this system of equations calculator does
This calculator solves linear systems with either 2 variables or 3 variables. You enter the coefficients for each equation, click Solve System, and the tool reports whether the system has:
- a unique solution (exactly one answer),
- no solution (inconsistent equations), or
- infinitely many solutions (dependent equations).
Behind the scenes, it uses Gaussian elimination with row reduction, which is the same method taught in algebra, precalculus, and linear algebra classes.
How to use the calculator
Step 1: Choose the number of variables
Select either 2 variables or 3 variables. The input fields update automatically.
Step 2: Enter equation coefficients
Each row is one equation. For example, if your equation is:
2x + 3y = 7
enter 2 for x, 3 for y, and 7 for the constant.
Step 3: Solve and interpret
The result panel explains the solution type and, when possible, gives the values of each variable. It also shows the reduced augmented matrix so you can check the algebraic process.
Example systems you can test
2-variable example
Try:
- 2x + y = 5
- x - y = 1
This has a unique solution: x = 2, y = 1.
3-variable example
Try:
- x + y + z = 6
- 2x - y + 3z = 14
- x + 2y - z = 2
The solver returns one exact triple for x, y, z.
Why solution type matters
In practical settings, systems of equations model budgets, engineering constraints, chemical balances, and more. Knowing whether a system has zero, one, or infinitely many solutions helps you decide if your model is valid.
- Unique solution: your constraints intersect at one point (or one state).
- No solution: constraints conflict; the model may contain an error or impossible assumptions.
- Infinite solutions: constraints overlap; you may need more independent equations.
Common input mistakes to avoid
- Forgetting a zero coefficient (example: write 0 for missing terms like 0z).
- Typing the constant on the wrong side (right side only in this form).
- Mixing signs (especially when moving terms across the equals sign).
- Leaving an input blank.
When to use this calculator
This is ideal for homework checking, exam prep, quick verification, and exploratory modeling. If you need symbolic parameters, nonlinear equations, or larger matrices (4x4 and up), you’ll want a full computer algebra system.