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Solution of linear equations system by using the row-echelon form.

This m-file gives the solution of a system of n linear equations with x unknowns by using the row-echelon form.

A linear system of equations is a system of equations in which each equation is linear. For any linear system, exactly one of the following will be true: There is only one solution, there are infinitely may solutions (underdetermined), or there are no solutions (overdetermined or inconsistent).

There are many algebraic procedures to solve a given linear system of equations. One procedure is to reduce the augmented matrix in echelon form, and solve by back substitution.

An augmented matrix is a matrix form of a linear system of equations obtained from the coefficient matrix created by adding an additional column for the constants on the right of the equal signs. The new column is set apart by a vertical line.

A matrix is said to be in row-echelon form if:
1. All rows consisting entirely of zeros are at the bottom.
2. In each row, the first non-zero entry form the left is a 1, called the leading 1.
3. The leading 1 in each row is to the right of all leading 1's in the rows above it.
If, in addition, each leading 1 is the only non-zero entry in its column, then the matrix is in reduced row-echelon form.

Gaussian elimination is one algorithm that reduces matrices to row-echelon form.

The function's name is giving by the acronym of Facultad de Ciencias Marinas-Universidad Autonoma de Baja California

Input:
A - augmented matrix of the system.
Output:
x - vector of the unknowns. It can give only one solution, infinitely many solutions (one can input a vector of arbritary values), or no solutions.