Example 7. The equation $(x-a)(x-8)-1=0$ is marked with two $\uparrow$

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Solve the original equation, which is transformed into
$$
(x-8)^{2}+(3-a)(x-8)-1=0 \text {. }
$$

By Vieta's formulas, we have
$$
\begin{array}{l}
x_{1}-8+x_{2}-8=a-8, \\
\left(x_{1}-8\right)\left(x_{2}-8\right)=-1 . \\
\because x_{1}, x_{2} \in Z, \\
\therefore x_{1}-8, x_{2}-8 \in Z . \\
\therefore\left(x_{1}-8\right)+\left(x_{2}-8\right)=0 .
\end{array}
$$

Thus, $a=8$.