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Is there some reason why it needs to be 10" specifically? 10" is a pretty uncommon size, and may needlessly limit your options to ancient netbooks from 15 years ago.
If something like a used 11-13" business laptop would be acceptable for your use case then there are a whole slew of options in the $50-70 range on ebay, maybe less if you find a good deal. I think there is also something to be said for getting something with a halfway reasonable keyboard which will not be agonizing to type on.
I think an 11" laptop would probably be a better bet. Taking a quick look on Ebay, I am seeing things like Dell 3190s, 3180s and Lenovo Yoga 11e laptops for around $50 or so. I see some chromebooks with x86 processors for as little as $35, but I do not have much experience with installing Linux on chromebooks and so you may want to double-check how to do that before buying anything.