How does Gaussian elimination solve systems of linear equations?

Gaussian elimination is a systematic method for solving systems of linear equations. It involves a series of steps that convert the original system into an upper triangular matrix form. Once in this upper triangular form, it becomes straightforward to use back substitution to find the values of the variables.

The process begins with the formation of an augmented matrix, which combines the coefficients of the variables and the constants from the equations. The goal of Gaussian elimination is to manipulate this matrix using elementary row operations—such as swapping rows, multiplying a row by a non-zero scalar, or adding/subtracting rows—to drive the system towards an upper triangular format. In an upper triangular matrix, all elements below the main diagonal are zero, which allows for an easy sequential solution of the equations starting from the last row.

For instance, if the system of equations is arranged in such a way that each equation provides information about the variables in a descending manner, one can solve for the last variable first and substitute it back into the previous equations. This process continues until all variables are solved.

This method stands out because it provides a clear, structured approach to finding solutions, distinctly separating it from other techniques involving graphical methods or iterative approximations.