How old ? as in how old was the human use of the slate piece in question.
Slate was used by the Egyptians for their 'palettes', while in the UK from Bronze Age at least, we know slate was used for cooking upon. Sometimes slate was used for whetstones, one was found at Verulamium, iirc. apart from roofs and floors.
Why do you want to go to the expense of petrographic analysis anyway ? that's the basis I suspect of your answer. If it's for source then I know it can be done but it needs a background record of slate quarries to compare.
I'd want to know what had been done to the slate. What was it used for ? Were there any evidences of residue for instance….where was it found though ? what context? in all it's ramifications. Work wear analysis perhaps….mind slate can be mde into an ulu type knife by rubbing and polishing it, too.
Very widespread use of slates for roofing is 'relatively' recent. The Romans here
mostly used tiles….and the slates they did use were small, and often pointed. It's heavy to transport and it's not always easy to split evenly, tiles came standard sized and were large enough to cover the roofs on purlins without sarking boards beneath.
Slates were sometimes used as floor tiling. If it was local it'd be cheaper than making those fancy tiles, and it contrasted well with other stones that split, like sandstone, colour wise, for designs in 'posh' buildings.
English Heritage has this pdf on stone.
https://archaeologydataservice.ac.u...ch-1416-1/dissemination/pdf/9781848021334.pdf
Best of luck with it
M