The system of linear equations $x_1+2x_2+x_3=3,$ $2x_1+3x_2+x_3=3$and $3x_1+5x_2+2x_3=1$, has 

$\mathrm{\Delta }=\left|\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\end{array}\right|$
  $=\left|\begin{array}{r}1 \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em} \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 2 \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\\ 3 \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\end{array}\right|$ [Applying $C_2\to C_2-2C_1,C_3\to C_3-C_1$ ]
   $=0$

${\mathrm{\Delta }}_x=\left|\begin{array}{l}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\end{array}\right|=\left|\begin{array}{l}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\end{array}\right|$ [Applying $R_2\to R_2-R_1$]
      $=5\ne 0$
$\mathrm{\Delta }=0,{\mathrm{\Delta }}_x\ne 0$
Hence, the system has no solution.