Find the ordered triple $(p,q,r)$ that satisfies the following system:
\begin{align*}
p - 2q &= 3 - 4p + 7q, \\
q - 2r &= -2 - 5q, \\
p + r &= 9 + 10q.
\end{align*}
To begin, we can solve the first equation for $p$: \begin{align*}
p - 2q &= 3 - 4p + 7q \\
5p &= 2 + 9q \\
p &= \frac{2 + 9q}{5}.
\end{align*}We can also solve the second equation for $r$: \begin{align*}
q - 2r &= -2 - 5q \\
2r &= -2 - 6q \\
r &= -1 - 3q.
\end{align*}Finally, we substitute for $p$ and $r$ above into the third equation: \begin{align*}
p + r &= 9 + 10q \\
\frac{2 + 9q}{5} - 1 - 3q &= 9 + 10q \\
2 + 9q - 5 - 15q &= 45 + 50q \\
4q &= 42 \\
q &= 10.5.
\end{align*}Now substitute $q = 10.5$ into $p = \frac{2 + 9q}{5}$ and $r = -1 - 3q$ to find that $p = \frac{2 + 9 \times 10.5}{5} = 6$ and $r = -1 - 3 \times 10.5 = -32.5$. Therefore, the ordered triple $(p,q,r)$ that satisfies the system is $\boxed{(6, 10.5, -32.5)}$.