https://file.aiquickdraw.com/mathbot/file/172747647208428zqw4fk.jpg

In the given problem, we are provided with three segment lengths:

- \( KL = 7x \)
- \( LM = 4x - 4 \)
- \( KM = 6x + 16 \)

Since \( K \), \( L \), and \( M \) are points on a straight line, the sum of \( KL \) and \( LM \) must equal \( KM \). Hence, we can set up the equation:

\[
KL + LM = KM
\]

Substitute the expressions for \( KL \), \( LM \), and \( KM \):

\[
7x + (4x - 4) = 6x + 16
\]

Now, solve for \( x \):

\[
7x + 4x - 4 = 6x + 16
\]

\[
11x - 4 = 6x + 16
\]

Subtract \( 6x \) from both sides:

\[
5x - 4 = 16
\]

Add 4 to both sides:

\[
5x = 20
\]

Now, divide by 5:

\[
x = 4
\]

Now that we have \( x = 4 \), substitute this value back into the expression for \( LM \):

\[
LM = 4x - 4 = 4(4) - 4 = 16 - 4 = 12
\]

Thus, the length of \( LM \) is **12 units**.

Would you like further details or explanations?

Here are some related questions:
1. What would happen if \( KM \) were shorter than the sum of \( KL \) and \( LM \)?
2. How do we know that \( K \), \( L \), and \( M \) are collinear?
3. If \( x = 3 \), what would the value of \( LM \) be?
4. Can you express \( KL \) and \( KM \) in terms of \( LM \) alone?
5. What is the geometric significance of solving this type of equation?

**Tip:** When solving equations, combining like terms as early as possible simplifies the process of finding the solution.

If KL = 7x, LM = 4x - 4, and KM = 6x + 16, what is LM?

Learn how to solve the algebraic equation with segment addition, where KL = 7x, LM = 4x - 4, and KM = 6x + 16. Step-by-step solution provided to find the length of LM using substitution and simplification.

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