If ST = 6x + 4, TU = 8x − 16, and SU = 13x − 5, what is TU?

Step-by-step solution

1. 1

   Turn the betweenness into an equation. The segment addition postulate says that when $T$ is between $S$ and $U$, the two short pieces add to the whole:

   $ST + TU = SU$

   This is the only geometry in the problem — everything after it is solving a linear equation.

2. 2

   Substitute the three expressions.

   $(6x + 4) + (8x - 16) = 13x - 5$

   Note which expression goes where: $SU$ is the whole segment, so it must sit alone on the right.

3. 3

   Combine like terms on the left.

   $14x - 12 = 13x - 5$

   Here $6x + 8x = 14x$ and $4 - 16 = -12$.

4. 4

   Solve for x. Move the $x$ terms to one side and the constants to the other:

   $14x - 13x = -5 + 12 \ \Longrightarrow \ x = 7$

5. 5

   Substitute back into TU — not into x. The question asks for a length, so finish the substitution:

   $TU = 8(7) - 16 = 56 - 16 = 40$

   Stopping at $x = 7$ is the most common way to lose the mark on this type of question.

6. 6

   Check that all three lengths are consistent. With $x = 7$: $ST = 46$, $TU = 40$, $SU = 86$, and indeed $46 + 40 = 86$. All three are positive, so the configuration is geometrically valid.