Backhoe ? in someones yard ? think about this tire tracks in the yard, minimum of a 12" wide trench. torn up sod , a whole lot of hand raking doing back fill and filling in the tire tracks. I think a backhoe to be extreme in this case. I would have added the cost of rental of trencher into the job. whats a trencher cost to rent? how much more then $100.
Take no offense these are just my thoughts based on alot of commercial trench work.