How much slope is required in drain lines from bath room to main house drain? We need to go about 15' before dropping 4' to connect with existing 3" line for another 12 feet to mmain drain from house.

Code requires 1/4" pitch per foot of pipe, in most cases. You can have a maximum of 1/2" per foot, but try to stick with the 1/4" per foot so the solids will stay with the liquid waste, OK?

Don't forget to add cleanouts at every change in direction and over at the main drain right after the drain heads over to the bathroom.

Mark