I remember reading an article in a fitness magazine a few years ago which said the best value equipment anyone could buy to keep fit at home is a set of dumbbells, due to their low cost and versatility, and dumbbells can be used to strengthen your back as well.
For example, the exercise pictured above is called a 'Bent Over Row'. If you start in the position as seen, reaching down with a dumbbell in both hand, you then raise the dumbbells up, lowering them down to the original position once the upper arms are level with your body, pausing a few seconds first, so you can feel the tension in your upper arms and back.
Try two sets of ten at first and see how you get on. Take it easy, don't rush and don't force it. Of course, this is just one exercise, there are lots of others you can do with dumbbells to work on your back.
I use dumbbells in a workout twice a week, and find them quite useful and not just for my arms. If you get two sets of dumbbells, one heavy and one light, you can use the lighter ones in your warm-up session.
YouTube has plenty of workouts to follow, and this one is for women getting started on using dumbbells for back strengthening: