Point K is on line segment start overline, J, L, end overline 
JL
 . Given J, L, equals, x, plus, 10, commaJL=x+10, K, L, equals, 4, x, commaKL=4x, and J, K, equals, 2, x, commaJK=2x, determine the numerical length of start overline, K, L, end overline, . 
KL
 .

We are given that:

- \( JL = x + 10 \)
- \( KL = 4x \)
- \( JK = 2x \)

Since \( K \) is a point on the line segment \( JL \), we can use the fact that the sum of the lengths of the segments \( JK \) and \( KL \) equals the total length of \( JL \). Therefore, we can set up the equation:

\[
JK + KL = JL
\]

Substituting the given expressions:

\[
2x + 4x = x + 10
\]

Simplifying the left-hand side:

\[
6x = x + 10
\]

Now, subtract \( x \) from both sides:

\[
5x = 10
\]

Solving for \( x \):

\[
x = 2
\]

Now that we have \( x = 2 \), we can find the numerical length of \( KL \):

\[
KL = 4x = 4(2) = 8
\]

Thus, the length of \( KL \) is \( \boxed{8} \).

Would you like further details or have any questions? Here are five related questions:

1. What is the length of the entire segment \( JL \) when \( x = 2 \)?
2. What is the length of segment \( JK \) when \( x = 2 \)?
3. How do you verify that the total length of \( JL \) is consistent with the sum of \( JK \) and \( KL \)?
4. If point \( K \) was shifted, how would it affect the lengths of \( JK \) and \( KL \)?
5. Can we generalize the method for any point \( K \) on a line segment \( JL \)?

**Tip:** Always check your solution by substituting the values back into the original equation to ensure consistency.

Point K is on line segment JL. Given JL = x + 10, KL = 4x, and JK = 2x, determine the numerical length of KL.

Solve for the length of segment KL given JL = x + 10, KL = 4x, and JK = 2x. This problem uses algebraic manipulation and the Segment Addition Postulate to find the solution. Suitable for Grades 7-8 students studying algebra.

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