GAUSS AND CRAMER’S METHOD FOR SOLVING LINEAR EQUATIONS

Аннотация

     Solving systems of linear equations is a fundamental problem in mathematics, engineering, and computational sciences. Among the most important classical techniques for finding the solutions to such systems are Gauss’s method (commonly known as Gaussian elimination) and Cramer’s method (based on Cramer’s rule). Both methods provide systematic procedures for determining unknown variables in a set of simultaneous linear equations, but they differ significantly in their computational approach, efficiency, and practical applicability.

     Gaussian elimination is an algorithmic method that transforms a system of linear equations into an equivalent upper triangular form through a series of elementary row operations: row swapping, scaling, and row addition. Once the system is in triangular form, the unknown variables can be found using the process of back substitution. This method is particularly advantageous for large systems because it can be easily implemented in computer algorithms and extended to matrix decomposition techniques such as LU decomposition. Gaussian elimination serves as the foundation for many numerical linear algebra methods and is widely used in scientific computing, data analysis, and engineering modeling.