A person that weighs 560 N stands on a scale in an elevator supported by a cable. The combined mass of the person and the elevator is 870 kg, and as the elevator starts moving the scale reads their weight as 450 N.
I want to find the acceleration of the elevator. I know the mass of the elevator, $m$, is 812.92 kg. Since the scales reading decreased then the elevator must be going down, so the downward force is greater in magnitude than the tension by the cable.
I know I'm supposed to apply Newton's second law, $\sum F_y=ma_y$. But the only vertical components I can think of on the elevator are the normal force and the weight of the elevator acting downward, both of which result in 0.
My "guess" has been $$(450/57.08)-9.81=-1.92$$ Which turned out to be the correct answer, but I basically got this by random guess and check, so I'm not sure if it only coincidentally gives the right answer.
You have used the right calculation for the answer.
By Newton's third law, the scale reading $450$ N means that the scale is pushing the person upward with a force of $450$ N.
So the person's upward-acceleration by the scale's force is :$$\frac{450 \text{ N}}{57.08\text{ kg}}$$
We also have the gravitational downward-acceleration, which is:
$$9.81\text{ m/s$^2$}$$
The elevator's acceleration is the same as that of the person, which would be the difference between those two.