https://file.aiquickdraw.com/mathbot/file/1727367057533eus4uxjq.jpg

The problem provided is to find the length of \( EG \) on a line segment involving several sub-segments. Here's what we have from the image:

- \( E \) to \( F \) has a length of \( x + 23 \),
- \( F \) to \( G \) has a length of \( 14 + x \),
- \( G \) to \( H \) has a length of 12,
- The total length from \( E \) to \( H \) is 26.

We want to find \( EG \), which is the length from \( E \) to \( G \). 

### Step 1: Express the total length of \( EH \)
The total length \( EH \) is:
\[
EH = EF + FG + GH
\]
Substituting the given values:
\[
26 = (x + 23) + (14 + x) + 12
\]

### Step 2: Simplify and solve for \( x \)
Simplify the equation:
\[
26 = (x + 23) + (14 + x) + 12
\]
\[
26 = 2x + 23 + 14 + 12
\]
\[
26 = 2x + 49
\]
Now solve for \( x \):
\[
26 - 49 = 2x
\]
\[
-23 = 2x
\]
\[
x = -\frac{23}{2} = -11.5
\]

### Step 3: Find \( EG \)
Now, we can find \( EG \). The length \( EG \) is the sum of \( EF \) and \( FG \), so:
\[
EG = (x + 23) + (14 + x)
\]
Substitute \( x = -11.5 \):
\[
EG = (-11.5 + 23) + (14 - 11.5)
\]
\[
EG = 11.5 + 2.5 = 14
\]

Thus, the length of \( EG \) is \( 14 \).

---

Let me know if you need further details or have any questions!

### Related Questions:
1. How would the problem change if the total length of \( EH \) was 30 instead of 26?
2. What would be the value of \( x \) if the segment \( GH \) had a different length?
3. How can we use algebraic expressions to represent line segments in geometry problems?
4. Why is it necessary to solve for \( x \) before calculating the length of \( EG \)?
5. How can this type of problem help in understanding coordinate geometry?

### Tip:
Always organize your information clearly when solving geometry problems involving multiple line segments!

Find the length of EG using the provided segment lengths.

Solve the problem to find the length of EG using algebraic expressions and linear equations. This problem involves calculating unknown values from segment lengths and solving for variables in an equation. Suitable for students in Grade 7-8.

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