LGS Rechner (2x2 & 3x3)
Solve lineare Gleichungssysteme quickly with Gaussian elimination. Enter coefficients for equations of the form ax + by + cz = d.
Equation 1
Equation 2
Equation 3
What is an LGS Rechner?
An LGS Rechner is a calculator for solving a lineares Gleichungssystem, which means a system of linear equations. Instead of solving by hand with substitution or elimination, you enter coefficients and let the calculator do the matrix operations. This is useful for students, engineers, developers, and anyone working with constrained numeric models.
How this calculator works
This tool applies Gaussian elimination (in Gauss-Jordan form) to the augmented matrix of your system. It can detect three outcomes:
- Unique solution: exactly one point satisfies all equations.
- Infinitely many solutions: equations are dependent.
- No solution: equations are inconsistent.
Supported formats
You can solve:
- 2x2 systems with variables
x, y - 3x3 systems with variables
x, y, z
Decimal values are fully supported, so you can use integer and non-integer coefficients.
Quick interpretation of results
Unique solution
If the output gives fixed values for all variables, the system intersects at exactly one point (or one line crossing point for 2D).
Infinitely many solutions
If one equation is a linear combination of others, you can get free variables and therefore infinitely many valid points.
No solution
If row reduction produces a contradiction (for example 0x + 0y + 0z = 5), there is no possible solution.
When to use an LGS calculator
- Checking homework and exam preparation in algebra.
- Solving balance and mixture equations in chemistry and economics.
- Parameter estimation in simple modeling problems.
- Validating equations before coding them into software workflows.
Manual methods vs. calculator
It is still important to understand substitution, elimination, and matrix notation. But once the concept is clear, a calculator saves time and reduces arithmetic mistakes. A practical strategy is:
- Solve one example by hand.
- Use the calculator to verify the answer.
- Analyze why mismatches happen (usually sign or arithmetic errors).
Final note
A good LGS Rechner is more than a “number machine.” It helps you test ideas fast, understand structure in equations, and build confidence when working with linear systems in school, university, and applied projects.