Washer discharge line runs into my main drain line the washer discharge line is 11/2 line and the main drain line is 2 in line every time that the laundry is done the water backs up in the kitchen sink washer discharge line runs under my slab any suggestions?

Your house drain line is partially clogged, probably lint and hair. If you have a clean out, open it and rod. If no clean out remove the trap under the sink and rod into the horizontal line going into the wall. However, your discharge line may be connected to the main drain via a stand pipe and that would also be a good place to run a snake. Is this the only fixture in the home that gets the back up?

Any trap/cleanout you can open and check? Then drain cleaner, then snake.
Buy a filter 'sock' for the washer hose, especially if you wash a lot of linty clothes like hoodies and sweats.

NO no, do not put drain cleaner in this line before you snake. Since the line is not draining well now you can injure yourself when you go to run the snake. Most of us do not recommend drain cleaners, ever.

Recently got new sewer line put in front yard roots from tree destroyed the other sewer pipe I thought that my problem was solved I snaked out lines and did find lint in the line but still have same problem washing machine discharges water very fast.

Phil a newer machine may make your issue worse. Newer machines drain much faster and can , at times, almost overload a clean 2" pipe. Its your drain not your washer. Also Phil, don't type in the comments box, just use the ANSWER THIS QUESTION box