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.