Point T is between points S and U. SU=40n-40, ST=20n-2, and TU=5n+7. Find TU.

To solve the problem, let's first understand the relationship between the points. Since point \( T \) is between points \( S \) and \( U \), the sum of the segments \( ST \) and \( TU \) should equal the entire segment \( SU \):

\[
SU = ST + TU
\]

We are given:

\[
SU = 40n - 40
\]

\[
ST = 20n - 2
\]

\[
TU = 5n + 7
\]

Now, we can set up the equation:

\[
40n - 40 = (20n - 2) + (5n + 7)
\]

**Step 1: Simplify the right side**

First, combine like terms on the right side:

\[
40n - 40 = 20n + 5n - 2 + 7
\]

\[
40n - 40 = 25n + 5
\]

**Step 2: Solve for \( n \)**

Subtract \( 25n \) from both sides to isolate the variable \( n \):

\[
40n - 25n - 40 = 5
\]

\[
15n - 40 = 5
\]

Next, add 40 to both sides:

\[
15n = 45
\]

Finally, divide both sides by 15:

\[
n = 3
\]

**Step 3: Find \( TU \)**

Now that we know \( n = 3 \), substitute this value back into the expression for \( TU \):

\[
TU = 5n + 7 = 5(3) + 7 = 15 + 7 = 22
\]

**Final Answer:**

\[
TU = 22
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the problem change if point \( T \) were not between \( S \) and \( U \)?
2. Can you find the lengths of \( SU \) and \( ST \) when \( n = 3 \)?
3. What if \( n \) were negative? Would the segment lengths still make sense?
4. How do you verify if the solution is consistent with the condition \( T \) is between \( S \) and \( U \)?
5. What happens to the segment lengths if you double the value of \( n \)?

**Tip:** When solving problems involving points on a line, always remember to set up an equation that reflects the sum of the segments if the points are collinear.

Learn how to find the length of segment TU when point T is between points S and U in a geometric context. This problem involves using equations to determine segment lengths based on given relationships. Suitable for high school students.

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