To simplify the expression \( 12 \frac{5}{6}x - 14x + \frac{1}{6}x \), let's break it down:

1. **Convert the mixed fraction** \( 12 \frac{5}{6} \) into an improper fraction:
   \[
   12 \frac{5}{6} = \frac{72}{6} + \frac{5}{6} = \frac{77}{6}
   \]
   So, the term becomes \( \frac{77}{6}x \).

2. Now, rewrite the expression:
   \[
   \frac{77}{6}x - 14x + \frac{1}{6}x
   \]

3. Combine the like terms. To do that, first express \( 14x \) as a fraction with a denominator of 6:
   \[
   14x = \frac{84}{6}x
   \]
   Now the expression is:
   \[
   \frac{77}{6}x - \frac{84}{6}x + \frac{1}{6}x
   \]

4. Combine the terms with a common denominator:
   \[
   \frac{77 - 84 + 1}{6}x = \frac{-6}{6}x = -x
   \]

Thus, the simplified expression is:
\[
-x
\]

Would you like more details on any of the steps?

### Related Questions:
1. How do you simplify algebraic expressions involving fractions?
2. What is the process of converting mixed fractions to improper fractions?
3. How can you factor common terms in expressions like this?
4. How do you solve equations that involve combining like terms?
5. What are the rules for handling negative signs in algebra?

**Tip:** Always ensure that you express all terms in a consistent form (like common denominators) before combining them.

Learn how to simplify the algebraic expression 12 5/6x - 14x + 1/6x by converting mixed fractions to improper fractions and combining like terms. This detailed step-by-step solution is suitable for students in Grades 7-9 and covers key algebraic concepts involving fractions.

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