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

\begin{description}

\item[(a)]

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

\item[(b)]

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

\item[(c)]

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

\end{description}



ANSWER
\begin{description}

\item[(a)]

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

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

\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}


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