System of Linear Equations Calculator (SLE)
Solve 2x2 and 3x3 linear systems with clear step-by-step working using elimination, substitution, or Gaussian elimination. Includes 2D intersection plotting and an interactive step-by-step trainer.
📚Schritt-für-Schritt-Lösungsweg
Wähle dein bevorzugtes mathematisches Lösungsverfahren:
Multipliziere Gleichung I mit -1 und Gleichung II mit -3, damit die Koeffizienten von $y$ entgegengesetzt gleich sind:
Durch Addition der beiden Gleichungen hebt sich die Variable $y$ vollständig auf:
Setze $x = 3$ in Gleichung I ein und löse nach $y$ auf:
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Multi-Variable Crop Accounting & Geodetic Surveys
Over 2,000 years ago, Chinese grain administrators needed to determine yields of individual crop grades from aggregated wagon loads. Later, Gauss needed to determine Earth's ellipsoidal shape from hundreds of noisy survey angles.
2. Gaussian Elimination & Row Reduction
Gauss systematized row operations to reduce linear systems into upper triangular echelon form, enabling straightforward backward substitution to find exact solutions.
3. CT Scanners, Computer Graphics & Logistics Optimization
Medical CT scanners solve linear systems with millions of variables in milliseconds to reconstruct 3D cross-sectional organ scans from X-ray data.
Praxisnutzen & Reale Anwendungsfälle
Warum lernen wir das? So wird diese Mathematik in Wirtschaft, Technik, Wissenschaft und Alltag praktisch angewendet:
Kirchhoff's Circuit Laws
Calculating electrical networks with multiple loops and resistors requires systems of linear equations for branch currents and voltages.
A circuit with 3 interconnected loops generates a 3x3 linear system for currents I1, I2, and I3.
Production & Resource Allocation
Businesses determine optimal product mixes and cost allocations subject to linear capacity and material constraints.
Calculating production outputs of products A and B given machine runtime and raw material limits.
3D Transformation & Kinematics
Rendering engines and robotic arms continuously solve linear systems for ray-triangle intersection and inverse kinematics.
Ray tracing calculations detecting light intersections with 3D polygon surfaces.
Mathematical Foundations: Systems of Linear Equations
A system of linear equations consists of multiple linear equations with several variables. Geometrically, each equation in 2 variables represents a line on the Cartesian plane. The intersection point of the lines gives the unique solution (x, y).
Standard solving methods for 2x2 systems include elimination (addition/subtraction of equations to cancel a variable), substitution (solving for one variable and plugging into the other), and Cramer's rule using determinants.
For larger systems (e.g. 3x3), Gaussian elimination is used to transform the augmented matrix into row-echelon form. A linear system can have exactly one unique solution, infinitely many solutions (coincident lines/planes), or no solution (parallel lines/planes).
Frequently Asked Questions (FAQ) about Linear Systems
When does a linear system have no solution or infinite solutions?
If eliminating variables leads to a contradiction such as 0 = 5, the system has no solution (parallel lines). If it results in a true identity like 0 = 0, there are infinitely many solutions (identical lines).
Which method is the fastest to solve linear systems?
For 2x2 systems, elimination is usually the fastest when coefficients can easily match. For 3x3 and higher dimensions, Gaussian elimination is standard.
What does the graphical representation show?
The 2D plot draws both linear equations in a coordinate plane. The highlighted intersection point represents the solution set (x, y).