https://file.aiquickdraw.com/mathbot/file/1724112975493i8enhp7a.jpg

To solve this problem, let's break it down step by step.

### Problem Analysis
The diagram shows a line segment \(HK\) with points \(G\) and \(F\) lying on it. The lengths of the segments are given as follows:

- \(HG = 13x - 993\)
- \(GF = 3x - 129\)
- \(FK = 3x - 183\)
- \(HK = 4\)

We need to find the length of segment \(GE\).

### Steps to Solve
1. **Expression for \(HK\)**: 
   Since \(HK\) is the entire line segment, it is the sum of the segments \(HG\), \(GF\), and \(FK\):
   \[
   HK = HG + GF + FK
   \]
   Substitute the given expressions:
   \[
   4 = (13x - 993) + (3x - 129) + (3x - 183)
   \]

2. **Simplify the Equation**:
   Combine the terms involving \(x\) and the constant terms:
   \[
   4 = 13x + 3x + 3x - 993 - 129 - 183
   \]
   \[
   4 = 19x - 1305
   \]

3. **Solve for \(x\)**:
   Add 1305 to both sides:
   \[
   1309 = 19x
   \]
   Divide by 19:
   \[
   x = \frac{1309}{19} = 69
   \]

4. **Find \(GE\)**:
   \(GE\) is simply the length of \(GF\) since \(E\) is located at point \(F\):
   \[
   GE = GF = 3x - 129
   \]
   Substitute \(x = 69\) into the equation:
   \[
   GE = 3(69) - 129 = 207 - 129 = 78
   \]

### Final Answer
The length of segment \(GE\) is \(78\) units.

Would you like any further details or have any questions?

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Here are some related questions to consider:

1. How would you solve for \(x\) if \(HK\) had a different value?
2. What if the segments were not collinear—how would that affect the solution?
3. Could you determine the length of segment \(HE\) using similar methods?
4. What role does the equation balancing play in solving for \(x\)?
5. How could you check your answer for accuracy?

**Tip:** Always double-check your algebraic manipulations to ensure accuracy, especially when combining like terms.

Explore how to solve a geometry and algebra problem involving segment lengths step-by-step. This problem includes simplifying equations and applying geometric concepts. Suitable for high school students.

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