Final answer:
Gaussian elimination reduces a system of equations to a row-echelon form followed by back substitution for solutions, whereas Gauss-Jordan elimination reduces it further to a reduced row-echelon form providing direct solutions without back substitution.
Step-by-step explanation:
Differences Between Gaussian Elimination and Gauss-Jordan Elimination
Gaussian elimination and Gauss-Jordan elimination are both methods used to solve systems of linear equations. The primary difference between them is the form in which the solutions are presented. Gaussian elimination involves a series of steps that reduce the system to its row-echelon form, where each leading entry is to the right of the leading entry in the row above. This method results in a triangular matrix, from which one proceeds with back substitution to find the solutions to the unknowns.
On the other hand, Gauss-Jordan elimination takes the process a step further by not only reducing the system to row-echelon form but also converting it to reduced row-echelon form (RREF). This is achieved by making the leading entry of each row a 1 (if it is not already) and then making all other entries in the column containing this leading entry zeros. The benefit of this method is that it directly provides the solutions to the unknowns, without the need for back substitution, as each variable corresponds to one of the rows in the matrix.
Therefore, while both Gaussian and Gauss-Jordan elimination can be used to solve linear systems, the Gauss-Jordan method is often considered more direct since it results in the RREF, which eliminates the need for the additional step of back substitution required in Gaussian elimination.