If LM = x + 10, MN = 4, and LN = 3x, what is LM?

Step-by-step solution

1. 1

   Apply the Segment Addition Postulate. Point $M$ lies between $L$ and $N$ — that is what naming the three segments $LM$, $MN$, $LN$ signals — so the two short pieces add to the whole:

   $LM+MN=LN$

   Every problem of this type is this one equation with different labels; identifying which segment is the whole is the only real decision.

2. 2

   Substitute the given expressions.

   $(x+10)+4=3x$

   Combining the constants on the left gives

   $x+14=3x$

3. 3

   Solve the linear equation. Collect the $x$ terms on one side:

   $14=3x-x=2x\;\Longrightarrow\;x=7$

4. 4

   Substitute back — the question asks for $LM$, not for $x$. This is where most answers go wrong: stopping at $x=7$ answers a question nobody asked.

   $LM=x+10=7+10=17$

5. 5

   Check all three lengths for consistency. With $x=7$: $LM=17$, $MN=4$, $LN=3(7)=21$, and indeed $17+4=21\;\checkmark$. All three lengths are positive, so the configuration is geometrically possible — worth checking, since a negative length would mean the setup had no valid solution.