T
 . Given R, S, equals, 4, x, minus, 10, commaRS=4x−10, S, T, equals, 2, x, minus, 10, commaST=2x−10, and R, T, equals, 4, x, minus, 4, commaRT=4x−4, determine the numerical length of start overline, R, S, end overline, . 
RS


Given the following equations for the lengths of line segments:

1. \( RS = 4x - 10 \)
2. \( ST = 2x - 10 \)
3. \( RT = 4x - 4 \)

We are asked to determine the numerical length of \( RS \).

### Step 1: Determine the value of \( x \)
To find \( x \), we can use the fact that:

\[
RS + ST = RT
\]

Substitute the given expressions for \( RS \), \( ST \), and \( RT \):

\[
(4x - 10) + (2x - 10) = 4x - 4
\]

### Step 2: Simplify the equation
Combine like terms on the left side:

\[
6x - 20 = 4x - 4
\]

### Step 3: Solve for \( x \)
Subtract \( 4x \) from both sides:

\[
2x - 20 = -4
\]

Add 20 to both sides:

\[
2x = 16
\]

Divide by 2:

\[
x = 8
\]

### Step 4: Calculate the length of \( RS \)
Now substitute \( x = 8 \) into the equation for \( RS \):

\[
RS = 4x - 10 = 4(8) - 10 = 32 - 10 = 22
\]

Thus, the numerical length of \( RS \) is \( \boxed{22} \).

Would you like further details or clarifications? 

Here are 5 related questions to consider:
1. How can we check the values of \( ST \) and \( RT \) for consistency?
2. How would the result change if \( RS \) had a different equation?
3. Can you find the length of \( ST \) using the same approach?
4. How would you express the total length \( RT \) in terms of \( RS \) and \( ST \)?
5. How does solving systems of linear equations help in geometry?

**Tip:** When solving for multiple variables, always check for consistency by substituting the value back into all original equations.

Given RS = 4x − 10, ST = 2x − 10, and RT = 4x − 4, determine the numerical length of RS.

Discover how to solve for the length of the line segment RS using linear equations. The problem involves solving a system of equations with RS = 4x − 10, ST = 2x − 10, and RT = 4x − 4. Suitable for students in Grades 9-10, this solution covers important concepts like solving for x and applying equations to geometry.

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