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\begin{abstract_online}{Gaussian Elimination with Parameters}{%
        \underline{Aharon Naiman}$^{1}$}{[naiman@jct.ac.il]
        }{%
        $^1$ Department of Applied Mathematics, Jerusalem College of
    Technology, Jerusalem, Israel}
    
Basic mathematics courses, at all levels, involve \textit{many}
opportunities to include CAS packages like \emph{Mathematica} [2],
Maple [1], REDUCE [3], Sage [4], amongst many others.  Computer
algebra packages assist with the preparation of:
%
\begin{itemize}
    \item classroom slides/notes,
    \item individualized homework assignments,
    \item in-class, randomized quizzes,
    \item class projects,
    \item extra-credit, further reading,
    \item final examinations,
    \item etc.
\end{itemize}
%
In this paper we discuss an aspect which affects all of the areas
above, i.e., that of solving Gaussian elimination with parameters, in
particular for the teaching of basic, first-semester linear algebra.

Right from the beginning of the semester, students are shown how to
perform row reduction.  As we know, they need to show that there are
either no solutions, one unique solution (and what it is), or an
infinite number of solutions (and what they are).  Are the standard
functions of the available packages prepared to show these three
possibilities?

The linear systems are then ``complicated'' by including input
parameters.  The students need to continue to solve these systems, and
specify, based on the input parameters, the same three questions
above.  Again, do the standard, available functions supply all of the
necessary solutions?  As we shall show, not all solutions and special
cases are covered.

Three approaches are presented (using \emph{Mathematica}), including
one which gets back to basic, row reduction.  We compare these,
demonstrating that some have more satisfying results than the others,
handling all special cases, and \emph{not} unnecessary ones.

We end off with applying a final approach to most of the exercises
posed in the remainder of the linear algebra course.

\textbf{Keywords} \newline{} linear algebra; education; automated
Gaussian elimination with parameters; \emph{Mathematica}

\textbf{References} \newline{}
\emph{Maple} at \texttt{www.maplesoft.com/products/Maple} \newline{}
\emph{Mathematica} at \texttt{www.wolfram.com/mathematica} \newline{}
\emph{REDUCE} at \texttt{reduce-algebra.com} \newline{}
\emph{Sage} at \texttt{www.sagemath.org} \newline{}
\end{abstract_online}
    
