If LM = 14, MN = x + 18, and LN = 3x − 6, what is LN?

Step-by-step solution

1. 1

   Write the segment addition relation. With $M$ between $L$ and $N$, the two adjacent pieces sum to the whole segment:

   $LM + MN = LN$

   The whole segment $LN$ is the one whose endpoints are the two outer points, so it belongs on its own side.

2. 2

   Substitute the given expressions.

   $14 + (x + 18) = 3x - 6$

   One length is a plain number and two are expressions in $x$; that is fine, the equation still has a single unknown.

3. 3

   Simplify the left side.

   $x + 32 = 3x - 6$

4. 4

   Solve for x. Gather the $x$ terms on the right and the constants on the left:

   $32 + 6 = 3x - x \ \Longrightarrow \ 38 = 2x \ \Longrightarrow \ x = 19$

5. 5

   Substitute into the requested length. The question asks for $LN$, so use its expression:

   $LN = 3(19) - 6 = 57 - 6 = 51$

6. 6

   Verify the whole picture. With $x = 19$: $LM = 14$, $MN = 19 + 18 = 37$, and $14 + 37 = 51 = LN$. Every length is positive and the sum checks, so $LN = 51$.