When equity is zero, the check you write is determined by her motivation, not by arithmetic. If she is behind on payments, facing a divorce, relocating on a deadline, or simply done being a landlord, the value of a clean exit without a realtor commission, without a closing timeline measured in months, and without her credit taking a foreclosure hit can easily be worth two to five thousand dollars even when the spreadsheet says she has nothing coming. That number is not a purchase price, it is a moving-cost payment or a goodwill close, and framing it that way tends to land better than offering her equity she does not have.
The assumption doing the most work in your question is that her pain is financial. If it is logistical, the dollar figure matters less than certainty of close and speed. The risk you have not mentioned is the due-on-sale clause sitting inside that 3.1 percent loan. Every conventional loan has one, and while lenders have historically been slow to call it, a missed payment or an insurance claim that surfaces the new owner's name can accelerate it. You need to know the loan servicer's track record on this and whether you are wrapping the deed in a land trust before the seller signs anything. What is the actual reason she is selling, and how much runway does she have before her situation forces her hand?