https://file.aiquickdraw.com/mathbot/file/1727064583217gd32p6v9.jpg

Let's analyze the given problem step by step:

We are provided with the following information from the number line:

- \( RS = 4y + 3 \)
- \( ST = 3y + 7 \)
- \( RT = 12y - 30 \)

Since \( R \), \( S \), and \( T \) are points on the number line, the total distance from \( R \) to \( T \) should be the sum of \( RS \) and \( ST \). Therefore, we can write the equation:

\[
RT = RS + ST
\]

Substituting the expressions for \( RS \), \( ST \), and \( RT \) from the problem:

\[
12y - 30 = (4y + 3) + (3y + 7)
\]

Now let's solve for \( y \).

\[
12y - 30 = 4y + 3 + 3y + 7
\]
\[
12y - 30 = 7y + 10
\]
\[
12y - 7y = 10 + 30
\]
\[
5y = 40
\]
\[
y = \frac{40}{5} = 8
\]

### a. The value of \( y \) is \( \boxed{8} \).

### b. Now, let's find \( RS \), \( ST \), and \( RT \).

- \( RS = 4y + 3 = 4(8) + 3 = 32 + 3 = 35 \)
- \( ST = 3y + 7 = 3(8) + 7 = 24 + 7 = 31 \)
- \( RT = 12y - 30 = 12(8) - 30 = 96 - 30 = 66 \)

Thus:
- \( RS = 35 \)
- \( ST = 31 \)
- \( RT = 66 \)

Let me know if you want further details, or if you have any questions!

Here are 5 related questions for practice:
1. If \( RS = 4y - 1 \) and \( ST = 2y + 5 \), find \( RT \) in terms of \( y \).
2. What is the value of \( y \) if \( RS = 3y + 2 \), \( ST = 2y + 5 \), and \( RT = 60 \)?
3. If the total distance \( RT = 50 \) and \( ST = 2x + 4 \), find \( x \) when \( RS = 3x + 10 \).
4. How would the equation change if \( RS = 2y \), \( ST = 3y \), and \( RT = 5y - 4 \)?
5. If \( RT = 45 \) and \( ST = 30 \), how long is \( RS \)?

**Tip:** When solving for unknown variables, always combine like terms and move constants to one side for clarity!

Use the number line where RS = 4y + 3, ST = 3y + 7, and RT = 12y - 30. Find the value of y, and determine the lengths of RS, ST, and RT.

Learn how to solve for the variable y and determine the lengths of RS, ST, and RT on a number line. This algebraic problem involves solving a system of equations with linear expressions. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation