System of Equations Solver - Gaussian Elimination with Visual Steps
System Size
Equation 1: ax + by = c
Equation 2: ax + by = c
This solver uses Gaussian Elimination with step-by-step matrix visualization.
Solution & Process
Enter coefficients and click Solve System
Understanding Gaussian Elimination
What is a System of Equations?
A system of equations is a collection of two or more equations with the same set of variables. Solving a system means finding values for the variables that satisfy all equations simultaneously.
2x + 3y = 8
x - y = 1
Solution: x = 2.2, y = 1.2
The Gaussian Elimination Method
- Form Augmented Matrix: Write equations as a matrix
- Forward Elimination: Create zeros below diagonal
- Back Substitution: Solve from bottom to top
- Verify: Check solution in original equations
Row Operations
- • Swap rows: Exchange two rows
- • Multiply row: Multiply all elements by a constant
- • Add rows: Add multiple of one row to another
These operations don't change the solution!
Real-World Applications
- Economics: Supply-demand equilibrium, market analysis
- Engineering: Electrical circuits, structural analysis
- Chemistry: Balancing chemical equations
- Computer Graphics: 3D transformations, rendering
- Operations Research: Resource allocation, optimization
Solver Features
- • Visual Matrix Display: See each transformation step
- • Color Highlighting: Track pivot elements and changes
- • Detailed Steps: Understand every operation
- • Automatic Verification: Solution checked in original equations
- • Error Detection: Identifies systems with no unique solution