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You do not need a stud to replace a small piece of sheetrock. My husband repairs holes all the time, too often from the kids, but he is never cutting holes to go to the stud.
Why the heck are you replacing an entire piece of sheet rock for a 3"x12" hole. Get a patch kit from Lowes with the wire mesh. Easy and fast. Even saw a cooler patch kit on shark tank. Will cost less than $10. Slap a little paint up and then hang a pic there.