Gaussian Elimination: Solving Linear Systems Step-by-Step
Gaussian Elimination is a fundamental algorithm in linear algebra for solving systems of linear equations. Named after Carl Friedrich Gauss, this method transforms a system into row echelon form through systematic row operations, making it straightforward to find solutions through back-substitution. Our calculator solves 2×2 systems instantly while showing the complete solution process.
What is Gaussian Elimination?
The method uses three types of row operations: swapping rows, multiplying a row by a nonzero constant, and adding/subtracting multiples of rows. These operations transform the coefficient matrix into upper triangular form (row echelon form), where solutions can be easily extracted.
How to Use This Calculator
Enter the coefficients for your system of equations in the format: a1·x + b1·y = c1 and
a2·x + b2·y = c2. Click Calculate to get the solution for x and y, along with
step-by-step explanation of the elimination process.
Applications
Gaussian Elimination is essential in engineering, computer science, economics, and physics for solving simultaneous equations in circuit analysis, structural engineering, economic modeling, and computational physics. It's also the foundation for more advanced techniques like LU decomposition and matrix inversion.