2150.62 – Lowest Terms?

Using a spread sheet perhaps, you'll find that some of these $x$ values are 5, 18, 31, 44, 57, and so on. Notice that these are in the form $x = 13k+5$. Plugging in, we can see what is happening:

$$\begin{aligned}\dfrac{9x+7}{6x + 9} &= \dfrac{9(13k+5) + 7}{6(13k+5) + 9} \\[1em]&= \dfrac{117k+ 52}{78k+ 39} \\[1em]&= \dfrac{13 \cdot 9k+ 13 \cdot 4}{13 \cdot 6k+ 13 \cdot 3} \\[1em]&= \dfrac{13 (9k+4)}{13 (6k+3)}\end{aligned}$$

My, my!