<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}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

Find the solutions for the following systems of equations.

\begin{description}
\item[(i)] $\ds \left( \begin{array}{ccc}2 &amp; 4 &amp; 1 \\ 1 &amp; 1 &amp; 1 \\ 2 &amp; 3 &amp; 1
 \end{array} \right)
\left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right) =
\left(
\begin{array}{c} 5 \\ 6 \\ 6\end{array} \right)$
\item[(ii)]$\ds \left( \begin{array}{ccc} 5 &amp; 15 &amp; -10 \\ -2 &amp; -2 &amp; -4 \\ 3 &amp; 4 &amp; 1
\end{array} \right)
\left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right) =
\left( \begin{array}{c} 0 \\ 4 \\ -5 \end{array} \right)$
\item[(iii)]$\ds \left( \begin{array}{cccc} 1 &amp; -2 &amp; 1 &amp; -5 \\ 3 &amp; -6 &amp; 4 &amp; -14 \\
-2 &amp; 4 &amp; -4 &amp; 9 \\ 4 &amp; -7 &amp; 5 &amp; -18 \end{array} \right) \left(
\begin{array}{c} x_ 1 \\ x_2 \\ x_3 \\x_4 \end{array} \right) = \left(
\begin{array}{c} 0 \\ 4 \\ 5 \\ 6\end{array} \right)$
\item[(iv)]$\ds \left( \begin{array}{ccc} 2 &amp; 1 &amp; -1 \\ 1 &amp; -2 &amp; 1 \\ 4 &amp; -3 &amp; 1
\end{array} \right)
\left( \begin{array}{c}x_1 \\ x_2 \\ x_3 \end{array} \right) =
\left(
\begin{array}{c} 1 \\ 3 \\ 5\end{array} \right)$
\item[(v)]$\ds \left( \begin{array}{ccc} 2 &amp; -1 &amp; 3 \\ 1 &amp; 1 &amp; -3 \\ 5 &amp; -1 &amp; 3
\end{array} \right)
\left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right) =
\left(
\begin{array}{c} 4 \\ -1 \\ 7 \end{array} \right)$
\item[(vi)]$\ds \left( \begin{array}{ccc} 5 &amp; -2 &amp; 7 \\ -3 &amp; 5 &amp; -1 \\ 1 &amp; 1 &amp; -3
 \end{array} \right)
\left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right) =
\left(
\begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right)$
\item[(vii)] $\left(\begin{array}{cccc} 1 &amp; -2 &amp; 3 &amp; -4 \\ 2 &amp; 1 &amp;
-1 &amp; 1 \\ 3 &amp; -1 &amp; 2 &amp; -1 \end{array} \right)\left(
\begin{array}{c} x_ 1
\\ x_2 \\ x_3 \\x_4 \end{array} \right) = \left(
\begin{array}{c} 0 \\ 0 \\ 0\end{array} \right)$
\end{description}

\vspace{.25in}

{\bf Answer}

Solutions
\begin{description}
\item[(i)] $\left( \begin{array}{c} 2\\ -1 \\ 5 \end{array} \right)$
\item[(ii)] $\left( \begin{array}{c} -3 \\1\\0\end{array} \right)$
\item[(iii)] $\left( \begin{array}{c} 52\\-11\\-9\\13\end{array} \right)$
\item[(iv)] no solutions
\item[(v)] $\left( \begin{array}{c} 1\\3t-1\\t\end{array} \right)$
\item[(vi)] $\left( \begin{array}{c} 0\\0\\0\end{array} \right)$
\item[(vii)] $\left( \begin{array}{c} -t\\7t\\5t\\0\end{array} \right)$

\end{description}


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