How can I repair my worn carpets?

Fiber-Lok Non-Skid Rug Backing, or any latex backing should work. Look under rug hooking supplies.

I agree with Non Skid Rug Backing. Normally its placed under the rugs so they don't slip. But you could hot glue the backing to the backs of the carpets. If the carpets are large you can use canvas drop cloths or heavy fabric, and glue it to the back with spray adhesive. Spray the adhesive on the canvas and carpet back. Once it is tacky to the touch lay the canvas on the carpet back. Use a rolling pin, piece of 2X4 lumber, hard cover book to press the canvas onto the carpet.