Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations. Please refer to the below wikipedia page to learn more about it.

https://en.wikipedia.org/wiki/Gaussian_elimination

For methodological steps (and considerations such as partial pivoting and row scaling) of gaussian elimination approach for solving linear equations, please refer to

https://www.purplemath.com/modules/systlin6.htm

https://courses.engr.illinois.edu/cs357/su2013/lectures/lecture07.pdf

Gaussian_elm function

num_or_matrix – The function can either take in numbers or a matrix.

If the input is numbers: total_columns - (default is 0) must specify the total number of the columns (including the right hand side). DO NOT use total columns argument if matrix is your input.

Note: Total column argument requires total number of columns including the right hand side. The function recognizes the last column as the right hand side.

ByRow – (default is FALSE): If the input values (numbers or matrix) are needed to be considered rowwise (or to be transposed).

rowScaling – (default is FALSE): If rowscaling is required in addition to Gaussian elimination.

partialPivoting - (default is FALSE): If partial pivoting is required in addition to Gaussian elimination.

The input must either be in the form of a numbers or matrix. The right hand side values must also be provided along with left hand side. The function recognizes the last column of input values to be right hand side. ONLY if vector is an input, please specify the total number of columns (total columns including the right hand side).

The function prints the partial pivot matrix, if the partial pivoting is TRUE. It prints step-wise Gaussian elimination results. In the final stage it prints, the individual variable values. The upper triangular (or the final result) is returned by the function.