Point S is on line segment start overline, R, T, end overline*RT*. Given S, T, equals, 2, x, minus, 10, comma*ST*=2*x*−10, R, S, equals, 2, x, minus, 5, comma*RS*=2*x*−5, and R, T, equals, 3, x, plus, 1, comma*RT*=3*x*+1, determine the numerical length of start overline, R, T, end overline, .*RT*.

We are given that point \( S \) lies on the line segment \( RT \), with the following information:

- \( ST = 2x - 10 \)
- \( RS = 2x - 5 \)
- \( RT = 3x + 1 \)

Since point \( S \) lies on line segment \( RT \), the sum of \( RS \) and \( ST \) should be equal to \( RT \). Therefore, we can set up the following equation:

\[
RS + ST = RT
\]

Substituting the given values:

\[
(2x - 5) + (2x - 10) = 3x + 1
\]

Now simplify the left side:

\[
2x - 5 + 2x - 10 = 3x + 1
\]
\[
4x - 15 = 3x + 1
\]

Next, subtract \( 3x \) from both sides:

\[
x - 15 = 1
\]

Add 15 to both sides:

\[
x = 16
\]

Now that we have the value of \( x \), substitute it back into the expression for \( RT \) to find the numerical length of \( RT \):

\[
RT = 3x + 1 = 3(16) + 1 = 48 + 1 = 49
\]

Thus, the length of \( RT \) is 49 units.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this problem:
1. What would happen if \( S \) were the midpoint of \( RT \)?
2. How would the solution change if one of the segments \( RS \) or \( ST \) had a different expression?
3. Can you determine the lengths of \( RS \) and \( ST \) now that \( x \) is known?
4. How would you calculate the ratio of the segments \( RS \) to \( ST \)?
5. How can this concept be extended to three-dimensional geometry?

**Tip:** When solving for variables in equations, always double-check your substitutions to avoid small mistakes.

Point S is on line segment RT. Given ST = 2x − 10, RS = 2x − 5, and RT = 3x + 1, determine the numerical length of RT.

Learn how to find the numerical length of RT when given ST = 2x − 10, RS = 2x − 5, and RT = 3x + 1. This problem focuses on key algebraic concepts and the Segment Addition Postulate, suitable for students in Grades 8-10.

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