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The problem you're working on asks for the length of segment \( KL \), with the given expressions for the lengths of three segments on a line.

The setup provided includes:
- \( JK = 7x \)
- \( KL = 10x - 3 \)
- \( JL = 10x + 16 \)

These segments correspond to the points \( J \), \( K \), and \( L \) on the number line, and the entire segment \( JL \) equals the sum of segments \( JK \) and \( KL \):
\[
JL = JK + KL
\]

Given this, we can set up the equation:
\[
10x + 16 = 7x + (10x - 3)
\]

Now, let's solve for \( x \). I'll calculate this step by step.The value of \( x \) is \( \frac{19}{7} \).

Now that we have \( x = \frac{19}{7} \), we can find the length of \( KL \) by substituting this value into the expression for \( KL = 10x - 3 \). I'll calculate this next.The length of \( KL \) is \( \frac{169}{7} \), which is approximately 24.14 when expressed as a decimal.

Would you like more details or clarifications on this solution?

Here are some related questions to explore:
1. How do you derive the equation for \( JL = JK + KL \)?
2. Can the value of \( x \) affect other segment lengths?
3. What is the relationship between the expressions for each segment?
4. How do you simplify expressions involving fractions in geometry problems?
5. How do you convert a proper fraction into a mixed number?
6. How would the problem change if \( JL \) were expressed differently?
7. How can such problems be applied in real-world contexts like construction or design?
8. How do different methods (e.g., graphing) help solve geometry problems?

**Tip:** When solving algebraic equations involving geometry, always ensure all parts of the geometric figure are properly represented in the equation.

Learn how to calculate the length of segment KL on a number line using linear equations and geometric concepts. This problem involves solving for the length of segment KL given expressions for segments JK and JL. Suitable for high school students studying geometry and linear equations.

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