How does Gaussian elimination solve a system of linear equations?

Gaussian elimination is a systematic method used in linear algebra to solve systems of simultaneous linear equations. The process involves reducing the system by solving one equation for a single variable and substituting that expression into the remaining equations. This effectively creates a new, smaller system with one fewer equation and one fewer variable.1

This reduction continues iteratively until only one equation remains, containing a single unknown variable. Once this variable is determined, the process of back substitution is used: the known value is plugged into previous equations to solve for the remaining unknowns sequentially until all variables are evaluated.1

In practice, this procedure is most efficiently executed using matrix operations. By representing the system as an augmented matrix, the reduction steps can be performed systematically, a technique that is highly optimized for implementation by computers.1