<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt


QUESTION Find all solutions (if any) of each of the following
systems of equations:

\begin{description}

\item[(a)]

\begin{eqnarray*}
3x+6y-6z&amp;=&amp;9\\ 2x-5y+4z&amp;=&amp;6\\ -x+16y-14z&amp;=&amp;-3
\end{eqnarray*}

\item[(b)]

\begin{eqnarray*}
x+y-z&amp;=&amp;7\\ 4x-y+5z&amp;=&amp;4\\ 6x+y+3z&amp;=&amp;20
\end{eqnarray*}

\item[(c)]

\begin{eqnarray*}
2w+x+2y-z&amp;=&amp;6\\ 6w+8x+12y-13z&amp;=&amp;-21\\ 10w+2x+2y+3z&amp;=&amp;59\\
-4w+y-3z&amp;=&amp;-30
\end{eqnarray*}

\end{description}



ANSWER
\begin{description}

\item[(a)]

Gaussian elimination leads to
$\left[\begin{array}{ccc|c}1&amp;2&amp;-2&amp;3\\0&amp;9&amp;-8&amp;0\\0&amp;0&amp;0&amp;0\end{array}\right]$

$\begin{array} {rl}\mathrm{So}\hspace{0.3in}
x=&amp;3+\frac{2z}{9}\\y=&amp;\frac{8z}{9}\\z=&amp;z\end{array}$
$\begin{array}{rl}\hspace{0.3in}\mathrm{or}\hspace{0.3in}x=&amp;3+\frac{y}{4}\\y=&amp;y\\z=&amp;\frac{9y}{8}\end{array}$
$\begin{array}{rl}\hspace{0.3in}\mathrm{or}\hspace{0.3in}x=&amp;3+2\lambda\\y=&amp;8\lambda\\z=&amp;9\lambda\end{array}$

\item[(b)]

Add twice row 1 to row 2, to get $6x+y+3z=18$, and this is clearly
inconsistent with row 3, so there are no solutions.

\item[(c)]
$w=4,x=1,y=1, z=5.$

\end{description}


\end{document}
</pre></body></html>