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We are given the following equations based on the line segment \( SU \) with point \( T \) on it:

- \( ST = 4x - 4 \)
- \( TU = 4x - 10 \)
- \( SU = 5x + 1 \)

Since \( T \) is on the line segment \( SU \), we know that the total length of \( SU \) must be the sum of the lengths of \( ST \) and \( TU \). Therefore:

\[
SU = ST + TU
\]

Substitute the given expressions for \( ST \), \( TU \), and \( SU \):

\[
5x + 1 = (4x - 4) + (4x - 10)
\]

Now, solve this equation for \( x \).

### Step 1: Simplify the right-hand side
\[
5x + 1 = 4x - 4 + 4x - 10
\]
\[
5x + 1 = 8x - 14
\]

### Step 2: Rearrange the equation to isolate \( x \)
\[
5x + 1 = 8x - 14
\]
\[
1 + 14 = 8x - 5x
\]
\[
15 = 3x
\]
\[
x = 5
\]

### Step 3: Calculate \( TU \)
Now that we know \( x = 5 \), substitute this value into the expression for \( TU \):

\[
TU = 4x - 10 = 4(5) - 10 = 20 - 10 = 10
\]

Thus, the numerical length of \( TU \) is **10**.

Would you like more details or have any questions? 

Here are 5 related questions you might find interesting:
1. How do you verify if a point lies on a line segment?
2. Can line segments have negative lengths in geometric problems?
3. How can we interpret the relationship between variables and geometric distances?
4. If \( ST \) and \( TU \) had different forms, how would we adjust our approach?
5. How can we generalize this problem for other types of segments?

**Tip:** Always double-check your solution by substituting the value of \( x \) back into the original conditions to confirm consistency.

Point T is on line segment SU. Given ST = 4x - 4, TU = 4x - 10, and SU = 5x + 1, determine the numerical length of TU.

Learn how to solve a geometric algebra problem where ST = 4x - 4, TU = 4x - 10, and SU = 5x + 1. This step-by-step solution demonstrates how to find the length of TU by solving for x. Suitable for students in Grades 9-10.

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