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In this problem, we are given that point \( S \) is the midpoint of line segment \( RT \). The expressions for the distances \( RS \) and \( ST \) are provided as \( 7y - 4 \) and \( 2y + 6 \), respectively. Since \( S \) is the midpoint, we know that:

\[
RS = ST
\]

This allows us to set the expressions equal to each other:

\[
7y - 4 = 2y + 6
\]

### Let's solve for \( y \):

1. Subtract \( 2y \) from both sides:

\[
7y - 2y - 4 = 6
\]

\[
5y - 4 = 6
\]

2. Add 4 to both sides:

\[
5y = 10
\]

3. Divide by 5:

\[
y = 2
\]

### Now that we know \( y = 2 \), we can find the length of \( RT \). Since \( RS = ST \), we calculate \( RS \) using \( y = 2 \):

\[
RS = 7(2) - 4 = 14 - 4 = 10
\]

Thus, \( ST \) is also 10. Therefore, the total length of \( RT \) is:

\[
RT = RS + ST = 10 + 10 = 20
\]

### Final Answer:
The length of \( RT \) is 20 units.

Would you like further details on any step, or do you have any additional questions?

Here are 5 related questions for further practice:
1. What would the length of \( RT \) be if \( RS = 3y + 5 \) and \( ST = 6y - 7 \)?
2. How do you find the midpoint if the coordinates of \( R \) and \( T \) are given instead of lengths?
3. If the midpoint formula is used with coordinates, how would you calculate \( S \)'s coordinates if \( R(2, 3) \) and \( T(6, 7) \)?
4. What if the distances were in a ratio of 2:1 instead of being equal? How would you solve for \( y \)?
5. Can you solve for \( y \) if the equation given was \( RS = 4y - 3 \) and \( ST = 2y + 5 \)?

**Tip:** Always double-check that you've correctly set up equations based on given relationships, especially when dealing with midpoints or ratios.

Point S is the midpoint of RT. Find RT. Given RS = 7y - 4 and ST = 2y + 6.

Learn how to find the total length of line RT using the midpoint theorem. This problem involves solving algebraic equations with expressions for RS and ST and is suitable for students in Grades 6-8.

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