https://file.aiquickdraw.com/mathbot/file/1727703959778ans8vg1u.jpg

Let's break down the information provided to solve the problem step by step:

### Given:
1. \( RS = 8y + 2 \)
2. \( ST = 3y + 9 \)
3. \( RT = 77 \)

From the number line, we know that \( R \), \( S \), and \( T \) are points on a straight line, so:

\[
RT = RS + ST
\]

### Step 1: Set up the equation
We can express \( RT \) in terms of \( RS \) and \( ST \):

\[
RT = (8y + 2) + (3y + 9)
\]

Now simplify:

\[
RT = 8y + 2 + 3y + 9 = 11y + 11
\]

Since we know \( RT = 77 \), substitute this into the equation:

\[
11y + 11 = 77
\]

### Step 2: Solve for \( y \)
Now solve the equation for \( y \):

\[
11y = 77 - 11
\]
\[
11y = 66
\]
\[
y = \frac{66}{11} = 6
\]

So, the value of \( y \) is:

\[
y = 6
\]

### Step 3: Find \( RS \) and \( ST \)
Now that we know \( y = 6 \), substitute it into the expressions for \( RS \) and \( ST \):

1. \( RS = 8y + 2 = 8(6) + 2 = 48 + 2 = 50 \)
2. \( ST = 3y + 9 = 3(6) + 9 = 18 + 9 = 27 \)

### Final answers:
- \( y = 6 \)
- \( RS = 50 \)
- \( ST = 27 \)

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What happens to the value of \( RS \) and \( ST \) if \( y \) increases?
2. How would the total length \( RT \) change if the expression for \( RS \) had a different constant?
3. Can we solve the problem using a graphical method? How would it differ?
4. What if \( RT \) was given in a different form (e.g., an inequality)? How would the solution change?
5. How do you verify the correctness of the solution?

**Tip**: Always substitute the variable value back into all expressions to check if your solution satisfies all conditions.

Use the number line below, where RS = 8y + 2, ST = 3y + 9, and RT = 77. a. What is the value of y? b. Find RS and ST.

Learn how to solve a linear algebra problem involving segments on a number line with RS = 8y + 2, ST = 3y + 9, and RT = 77. This solution involves step-by-step calculations, making it suitable for Grades 6-8.

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