7. Given an arithmetic sequence $\left\{a_{n}\right\}$, where $S_{n}$ is the sum of its first $n$ terms, if $S_{p}=q, S_{q}=p(p \neq q)$, try to find the value of $S_{p+q}$.
7. Generally, list the system of linear equations in two variables about the first term $a_{1}$ and the common difference $d$ according to the problem, solve for $a_{1}$ and $d$, and then use the sum formula to get $S_{p+q}$. Since it involves symbolic computation, there is a large amount of computation, and it is easy to make mistakes. Also, since
$$
S_{p+q}=(p+q) a_{1}+\frac{(p+q)(p+q-1)}{2} d=(p+q)\left[a_{1}+\frac{(p+q-1)}{2} d\right],
$$

if we can find the whole expression $a_{1}+\frac{(p+q-1)}{2} d$, things will be much easier.
Thus, according to the problem, we have
$$
\left\{\begin{array}{l}
S_{p}=p a_{1}+\frac{p(p-1)}{2} d=q, \\
S_{q}=q a_{1}+\frac{q(q-1)}{2} d=p .
\end{array}\right.
$$
(1)-(2) gives
$$
(p-q) a_{1}+\frac{(p-q)(p+q-1)}{2} d=q-p \text {. }
$$

Since $p \neq q$,
from $a_{1}+\frac{(p+q-1)}{2} d=-1$,
we have $\quad S_{p+q}=(p+q)\left[a_{1}+\frac{(p+q-1)}{2} d\right]=-(p+q)$.