The bathtub will not drain the water out. When the sink in the bathroom is used, the water in the tub will bubble and sludge will com up through the drain. Please help

The tub is the lowest point to your drain lines that service the bathroom. The back up is in the bathroom main drain line and not any one individual fixture. If there is no close clean out you may need to take up the toilet and rod down that drain. You can try to rod via the sink but first remove the trap and rod staight into the wall. It has a slim chance of working but its worth a try before pulling the toilet.