Enter coefficients for each equation in the form ax + by (+ cz) = d. Empty fields are treated as 0.
What this system of equations calculator does
This calculator solves linear systems with either two variables or three variables. In practical terms, that means you can solve equations like 2x + 3y = 12 and x - y = 1, or a 3-variable system such as x + y + z = 6, 2x - y + z = 3, and 3x + 2y - z = 4. It is useful for students in algebra, anyone reviewing for exams, and professionals who need quick linear equation solutions.
How to use the calculator
- Select whether you want a 2x2 or 3x3 system.
- Type the coefficients and constants into the inputs.
- Click Solve System to compute the result.
- Use Load Example for a pre-filled problem, or Clear to start over.
The output includes both the final answer and a short list of elimination steps so you can understand how the solution is found.
Possible outcomes explained
1) Unique solution
A unique solution means there is exactly one value for each variable that satisfies every equation in the system. In geometric terms, lines (2D) or planes (3D) intersect at one point.
2) No solution (inconsistent system)
If equations conflict with one another, the system is inconsistent. Example: two parallel lines never intersect. The calculator reports this as No solution.
3) Infinitely many solutions (dependent system)
If at least one equation is a linear combination of others, there may be infinitely many solutions. The calculator reports this as Infinitely many solutions.
Method used: Gaussian elimination
Internally, this tool uses Gaussian elimination with partial pivoting. It converts your system into an augmented matrix, performs row operations, and then applies back substitution when a unique solution exists. This approach is stable and widely used in numerical linear algebra.
Worked examples
Example A (2x2)
System:
- 2x + 3y = 12
- x - y = 1
Solution: x = 3, y = 2.
Example B (3x3)
System:
- x + y + z = 6
- 2x - y + z = 3
- 3x + 2y - z = 4
Solution: x = 1, y = 2, z = 3.
Common mistakes to avoid
- Entering constants on the wrong side of the equation.
- Forgetting negative signs (for example, typing 2 instead of -2).
- Mixing equation order while entering coefficients.
- Assuming every system has exactly one solution.
Final notes
A reliable linear equations solver should do more than output numbers—it should help you reason about the structure of a system. Use this calculator for homework checks, quick analysis, and confidence building before tests. If you want, I can also help you build a version that supports 4x4 systems, fraction input/output, or full row-reduction tables.