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### Theory:

The number of repeating decimals in the decimal expansion of the rational number is called the period of the decimal.
Example:
Verify $7=6.\overline{9}$

Let $x=6.\overline{9}=6.9999...$

As the period of the circle for $$x$$ is one(only the number $$9$$ repeating), multiply the number by $$10$$.

We have $\mathit{10x}=69.9999...$

Subtracting $$x$$ from $$10x$$,

$\begin{array}{l}\mathit{10x}-x=69.9999...-6.999...\\ \\ 9x\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}63\\ \\ x\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{63}{9}\\ \\ x=7\end{array}$

Thus, we have $x=6.\overline{9}$ and $x=7$. This implies $7=6.\overline{9}$.