We have line $l\_1:\begin{cases} x=1+t\_1 \\ y=t\_1 \\ z=-1+t\_1\end{cases}$ and $l\_2:\begin{cases} x=10+5t\_2 \\ y=5+t\_2 \\ z=2+2t\_2\end{cases}$.

Find the line $l$ that goes through $P:(3, 2, −1)$ and intersects line $l\_1$ and $l\_2$.

I tried finding a point of intersection between line $l\_1$ and $l\_2$ but the equation I got is inconsistent.
$\begin{cases}
t\_1-5t\_2=9\\
t\_1-t\_2=5\\
t\_1-2t\_2=3
\end {cases}\iff \begin{pmatrix}1 & -5 & 9\\ 1 & -1 & 5\\ 1 & -2 & 3 
 \end{pmatrix} \iff \begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 
 \end{pmatrix} $. So there is no point of intersection. How do I find *the* line that goes through $P$ and intersects $l\_1$ and $l\_2$?