*FG*=5*x*+1, E, G, equals, 5, x, plus, 7, comma*EG*=5*x*+7, and E, F, equals, 2, x, minus, 2, comma*EF*=2*x*−2, determine the numerical length of start overline, E, F, end overline, .*EF*

Given the information:

- \( FG = 5x + 1 \)
- \( EG = 5x + 7 \)
- \( EF = 2x - 2 \)

Assuming that \( E \), \( F \), and \( G \) are collinear points and \( F \) is between \( E \) and \( G \), the relationship between these segments can be expressed as:

\[
EG = EF + FG
\]

Substituting the given expressions:

\[
5x + 7 = (2x - 2) + (5x + 1)
\]

Simplify the equation:

\[
5x + 7 = 7x - 1
\]

Now, solve for \( x \):

\[
5x - 7x = -1 - 7
\]

\[
-2x = -8
\]

\[
x = 4
\]

Now that we have the value of \( x \), substitute it back into the expression for \( EF \):

\[
EF = 2x - 2 = 2(4) - 2 = 8 - 2 = 6
\]

Thus, the numerical length of \( \overline{EF} \) is **6**.

Would you like further explanation on any of these steps?

Here are some related questions:
1. How can you verify that \( E \), \( F \), and \( G \) are collinear?
2. What happens to the equation setup if the points are not collinear?
3. How would this method change if \( F \) was not between \( E \) and \( G \)?
4. What are the geometric interpretations of different segment lengths?
5. How does the choice of which segment to calculate affect the solution process?

**Tip:** Always ensure the relationships between segments are clearly understood before solving; whether they add up or subtract depends on the point's arrangement.

Learn how to determine the numerical length of segment EF using the segment addition postulate in geometry. This problem involves collinear points and segment relationships, exploring algebraic expressions and their application in geometric contexts. Suitable for high school students (Grades 9-12).

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