Does anyone know a CHEAP way to cut slate rock without costly expensive saws?

You can split slate with a hammer and or chisel but it is not very accurate and sometimes you ruin the stone.

It depends upon how mechanical you are... My grandfather built a rock saw by purchasing a (then) $60 diamond saw blade then building the rest of the saw from "junk" parts using a washing machine motor, a system of pulleys and belts to change speeds, a rolling feed table on rails motivated by using gravity a cable and weights, I don't remember what he used to construct the pump for the coolant but he used the same soluble oil used in automotive valve grinding machines for coolant

The saw he built was used for many years to cut stone for making jewelry, cutting slabs for display, and cutting Thunder Eggs (geodes) also for display.

This was back in the 1960's and I believe the cost was under $150 but I believe all the parts could be scrounged up for just a few of todays dollars also.

The least expensive rock saw available on the market at that time was $600.