Example 2 Let the coordinates of moving points $P$ and $P^{\prime}$ be $(x, y)$ and $\left(x^{\prime}, y^{\prime}\right)$, respectively, and they satisfy $\left\{\begin{array}{l}x^{\prime}=3 x+2 y+1 \\ y^{\prime}=x+4 y-3\end{array}\right.$, can $P$ and $P^{\prime}$ move on the same straight line?
Assume $P, P^{\prime}$ both move on the line $A x+B y+C=0$, then we have $A x^{\prime}+B y^{\prime}+C=0$.
Substituting $x^{\prime}=3 x+2 y+1, y^{\prime}=x+4 y-3$ into (2) yields
$$
(3 A+B) x+(2 A+4 B) y+(A-3 B+C)=0 \text {. }
$$

Since equations (1) and (3) represent the same line, and obviously $A, B, C$ are not zero, we have
$$
\frac{3 A+B}{A}=\frac{2 A+4 B}{B}=\frac{A-3 B+C}{C} \text {. }
$$

From the first equality in (4), we get $B=2 A$ or $B=-A$.
When $B=2 A$, from the last equality in (4), we get $C=-\frac{5}{4} A$;
When $B=-A$, from the last equality in (4), we get $C=4 A$.
Therefore, $A: B: C=4: 8:(-5)$ or $A: B: C=1:(-1): 4$, which means points $P, P^{\prime}$ can move on the line $x-y+4=0$ together, or they can move on the line $4 x+8 y-5=0$ together.