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FG = 6x + 5, GH = 7x, and FH = 18, what is GH?

We are given that the line segment is made up of three points, F, G, and H. The segments between these points are labeled as:

- \( FG = 6x + 5 \)
- \( GH = 7x \)
- \( FH = 18 \)

Since \( F \), \( G \), and \( H \) are collinear, the total distance \( FH \) must equal the sum of the distances \( FG + GH \):

\[
FH = FG + GH
\]

Substituting the given expressions:

\[
18 = (6x + 5) + 7x
\]

Now, let's solve for \( x \):

\[
18 = 6x + 5 + 7x
\]
\[
18 = 13x + 5
\]
\[
18 - 5 = 13x
\]
\[
13 = 13x
\]
\[
x = 1
\]

Now that we know \( x = 1 \), we can substitute it into the expression for \( GH \):

\[
GH = 7x = 7(1) = 7
\]

Thus, \( GH = 7 \).

Would you like more details or have any further questions?

### Related questions:
1. What is the length of \( FG \) when \( x = 1 \)?
2. How do we verify the total length \( FH \) after finding \( FG \) and \( GH \)?
3. What method can be used to solve for \( x \) in similar problems?
4. What happens if \( x \) results in a negative distance? Is that physically meaningful?
5. Can this process be extended to cases where there are more segments along a line?

**Tip:** When solving linear equations involving segments, always check that the total length agrees with the sum of individual parts!

This page provides a step-by-step guide to solving for GH when FG = 6x + 5, GH = 7x, and FH = 18. Learn key algebra concepts, including how to solve linear equations and work with distances between collinear points. Suitable for students in Grades 8-10.

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