https://file.aiquickdraw.com/mathbot/file/1728497465972jat37w17.jpg

The problem asks to find the value of \( TV \) given:

1. \( TU = 4x + 8 \)
2. \( UV = 9 \)
3. \( TV = 12x - 15 \)

From the diagram and the information provided:

\[
TV = TU + UV
\]

Using this relation, we can substitute the expressions for \( TU \) and \( UV \):

\[
TV = (4x + 8) + 9
\]

Now simplify the equation:

\[
TV = 4x + 17
\]

We are also given that \( TV = 12x - 15 \). Therefore, we can set the two expressions for \( TV \) equal to each other:

\[
4x + 17 = 12x - 15
\]

Now solve for \( x \):

\[
17 + 15 = 12x - 4x
\]
\[
32 = 8x
\]
\[
x = 4
\]

Now substitute \( x = 4 \) back into either equation for \( TV \) (we'll use \( TV = 12x - 15 \)):

\[
TV = 12(4) - 15 = 48 - 15 = 33
\]

So, \( TV = 33 \).

Would you like more details on the steps, or do you have any questions?

Here are 5 related questions:
1. What method could you use if both sides of the equation had decimals instead of integers?
2. How would you solve this if one segment's length had been given as a fraction?
3. Can you describe how the equation setup relates to real-life geometry problems?
4. What would change if one of the given lengths involved subtraction rather than addition?
5. How do you ensure the correct simplification when solving equations like these?

**Tip:** Always verify your solution by substituting the variable back into the original expressions to check consistency.

If TU = 4x + 8, UV = 9, and TV = 12x - 15, what is TV?

Learn how to solve for TV in the equation given TU = 4x + 8, UV = 9, and TV = 12x - 15. This problem involves key algebraic concepts, such as solving linear equations and using substitution. Suitable for students in Grades 8-10.

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