Point U is on line segment start overline, T, V, end overline*TV*. Given T, U, equals, 4, comma*TU*=4, U, V, equals, 5, x, comma*UV*=5*x*, and T, V, equals, 3, x, plus, 6, comma*TV*=3*x*+6, determine the numerical length of start overline, T, V, end overline, .*TV*.

To determine the numerical length of line segment \( \overline{TV} \), we can use the information given:

- \( TU = 4 \)
- \( UV = 5x \)
- \( TV = 3x + 6 \)

Since \( U \) is a point on the line segment \( \overline{TV} \), we know that:

\[
TV = TU + UV
\]

Substituting the given values:

\[
3x + 6 = 4 + 5x
\]

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

\[
3x + 6 = 4 + 5x
\]

Subtract \( 3x \) from both sides:

\[
6 = 4 + 2x
\]

Subtract 4 from both sides:

\[
2 = 2x
\]

Divide both sides by 2:

\[
x = 1
\]

Now that we know \( x = 1 \), substitute it back into the expression for \( TV \):

\[
TV = 3x + 6 = 3(1) + 6 = 3 + 6 = 9
\]

Therefore, the numerical length of \( \overline{TV} \) is \( \boxed{9} \).

Would you like to see more details or have any questions about this? Here are five related questions:

1. How would the problem change if \( U \) were the midpoint of \( \overline{TV} \)?
2. What is the length of \( UV \) given that \( x = 1 \)?
3. How would you set up the problem if \( TU = 3x + 2 \) and \( UV = x + 4 \)?
4. Can you verify the solution by substituting back into the original expressions?
5. What happens to the length of \( \overline{TV} \) if \( x \) were to double?

**Tip:** Always double-check your solution by substituting the found value back into the original equation to ensure it satisfies all given conditions.

Learn how to calculate the length of line segment TV using points T, U, V and variables given in the problem. This problem involves coordinate geometry and requires solving equations to find the length of the segment. Suitable for Grades 9-12 students and anyone interested in coordinate geometry.

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