Point U is on line segment start overline, T, V, end overline
TV
. Given U, V, equals, 2, commaUV=2, T, U, equals, 5, x, commaTU=5x, and T, V, equals, 3, x, plus, 6, commaTV=3x+6, determine the numerical length of start overline, T, V, end overline, .
TV

We are given a geometric problem involving a line segment \( \overline{TV} \), with the point \( U \) located on the line between \( T \) and \( V \). The lengths of different parts of the segment are given in terms of \( x \).

Given:
- \( UV = 2 \)
- \( TU = 5x \)
- \( TV = 3x + 6 \)

Since \( U \) is on the line segment \( \overline{TV} \), the total length \( TV \) is the sum of the lengths of \( TU \) and \( UV \):
\[
TU + UV = TV
\]
Substituting the given values:
\[
5x + 2 = 3x + 6
\]
Now, solve for \( x \).

1. Subtract \( 3x \) from both sides:
\[
5x - 3x + 2 = 6
\]
This simplifies to:
\[
2x + 2 = 6
\]

2. Subtract 2 from both sides:
\[
2x = 4
\]

3. Divide both sides by 2:
\[
x = 2
\]

Now, substitute \( x = 2 \) into the expression for \( TV \):
\[
TV = 3x + 6 = 3(2) + 6 = 6 + 6 = 12
\]

Thus, the length of \( \overline{TV} \) is \( 12 \).

Would you like further details or have any questions?

Here are some related questions to think about:
1. What is the length of \( TU \) when \( x = 2 \)?
2. Can you find the midpoint of \( \overline{TV} \)?
3. If \( U \) is the midpoint of \( \overline{TV} \), how does that change the values of \( TU \) and \( UV \)?
4. What would happen if \( TV = 4x + 5 \) instead of \( 3x + 6 \)?
5. How would you solve the problem if the lengths of \( TU \) and \( UV \) were given in decimals?

**Tip**: When solving equations involving geometric segments, always check if the sum of the parts equals the whole.

Point U is on line segment \( \overline{TV} \). Given \( UV = 2 \), \( TU = 5x \), and \( TV = 3x + 6 \), determine the numerical length of \( \overline{TV} \).

Learn how to solve for the length of line segment \( \overline{TV} \) using algebraic equations. The problem involves segment addition, where \( TU + UV = TV \). Suitable for students in Grades 7-9.

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