9 Let the coordinates of moving points $P$ and $P^{\prime}$ be $(x, y)$ and $\left(x^{\prime}, y^{\prime}\right)$, respectively, and they satisfy
$$
\left\{\begin{array}{l}
x^{\prime}=3 x+2 y+1, \\
y^{\prime}=x+4 y-3 .
\end{array}\right.
$$

If $P$ and $P^{\prime}$ move on the same straight line, does such a line exist? If it exists, find its equation.
9 Let $P$ and $P^{\prime}$ be on the line $A x+B y+C=0$, then we should have
$$
A x^{\prime}+B y^{\prime}+C=0,
$$

which means
$$
(3 A+B) x+(2 A+4 B) y+(C+A-B)=0 .
$$

From $\frac{3 A+B}{A}=\frac{2 A+4 B}{B}=\frac{C+A-3 B}{C}=k$ eliminating $A, B, C$ gives $k=2$ or $k=5$.

When $k=2$, $A: B: C=1:(-1): 4$, the equation is $x-y+4=0$.
When $k=5$, $A: B: C=4: 8:(-5)$, the equation is $4 x-8 y-5=0$.