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In this problem, we need to convert \( 0.\overline{7} \) (a repeating decimal) into a fraction by solving an equation.

### Step-by-Step Solution:

1. Let \( x = 0.\overline{7} \). This repeating decimal is the same as writing \( 0.77777... \) with \( 7 \) repeating indefinitely.

2. Multiply both sides of the equation \( x = 0.\overline{7} \) by 10 to shift the decimal point:
   \[
   10x = 7.\overline{7}
   \]

3. Now we have two equations:
   \[
   x = 0.\overline{7}
   \]
   \[
   10x = 7.\overline{7}
   \]

4. Subtract the first equation from the second:
   \[
   10x - x = 7.\overline{7} - 0.\overline{7}
   \]
   \[
   9x = 7
   \]

5. Divide both sides by 9 to isolate \( x \):
   \[
   x = \frac{7}{9}
   \]

Therefore, \( 0.\overline{7} \) can be written as the fraction \( \frac{7}{9} \).

Would you like further details on any of the steps or have any questions?

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### Additional Questions for Further Exploration

1. How can repeating decimals other than \( 0.\overline{7} \) be converted to fractions?
2. What is the difference in approach for converting non-repeating decimals to fractions?
3. Can you use this method to convert \( 0.\overline{3} \) to a fraction?
4. Why does subtracting the equations help in isolating \( x \)?
5. How does this approach apply to repeating decimals with more than one digit, like \( 0.\overline{23} \)?

#### Tip
When converting a repeating decimal to a fraction, multiply by a power of 10 that matches the number of repeating digits. This helps align the decimal parts for subtraction.

You can write 0.7 (repeating) as a fraction using equations. Simplify the left side of the equation 10x - x = 7.7 - 0.7.

Learn how to convert the repeating decimal 0.7 to a fraction using equations. This solution simplifies the left side of the equation by solving 10x - x = 7.7 - 0.7. Suitable for students in Grades 7-9.

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