So if you switch from 35mm to 10" X 8" with a 50mm lens the focal length stays unchanged. I think not.
No change in focal length. And for the same subject distance, you would need to focus by having the film the same distance from the nodal point of the lens. That would be difficult - even a bag bellows with a recessed lens board would probably not let you have the film that close.
And as that lens is designed to cover a 35mm frame, it would only fill a small central part of the 8"x10" film with image.
If somehow you had a 50mm lens designed to fill the 8"x10" film - an extreme wide angle lens - it would be huge and difficult to mount on your 35mm camera, and the film in the 35mm camera would only use a very small centre portion of the image formed by the lens.
A more reasonable thought experiment would be to take a 90mm lens designed for 35mm film and mount it on the 8"x10" camera. Again, you would need to focus by having the film the same distance from the nodal point of the lens, although you could probably achieve that using a bag bellows and/or a recessed lens board.
And as that lens is also designed to cover a 35mm frame, it would also only fill a small central part of the 8"x10" film with image.