We are given three equations that are essentially the same expression with different divisors, all equating to \( x \):

\[
\frac{n+5}{6} = x, \quad \frac{n+5}{8} = x, \quad \frac{n+5}{10} = x
\]

Since each expression equals \( x \), we can set them equal to each other:

\[
\frac{n+5}{6} = \frac{n+5}{8} = \frac{n+5}{10}
\]

Let's solve this step by step.

### Step 1: Set the expressions equal to each other

First, set \(\frac{n+5}{6}\) equal to \(\frac{n+5}{8}\):

\[
\frac{n+5}{6} = \frac{n+5}{8}
\]

Cross-multiply to eliminate the denominators:

\[
8(n+5) = 6(n+5)
\]

Simplifying:

\[
8(n+5) = 6(n+5) \implies 8 = 6 \quad (\text{This is clearly a contradiction.})
\]

### Step 2: Conclusion
Since the equation leads to a contradiction, there is no possible value of \( n \) or \( x \) that satisfies all three conditions simultaneously. Thus, there is no solution to the system of equations as presented.

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Do you want further details or have any questions?

Here are five related questions you might find useful:
1. How do you solve equations with multiple variables?
2. What is the method for cross-multiplying fractions?
3. Can systems of equations sometimes have no solutions?
4. How do you recognize contradictions in algebraic expressions?
5. What are the conditions for a system of equations to be consistent?

**Tip:** When you encounter equations with multiple fractions, cross-multiplying is a helpful technique to simplify and compare expressions!

(n+5) ÷ 6 = x, (n+5) ÷ 8 = x, (n+5) ÷ 10 = x. What is x?

Learn how to approach a system of equations involving fractions, where no solution exists. The problem explores algebraic concepts such as cross-multiplication and detecting inconsistencies in equations. Suitable for students in Grades 8-10.

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