This has little-to-nothing to do with the garbage disposal or anything in front of where the washing machine ties in to the drain line in the back.

You have a restricted drain line that will probably only be cleared by snaking the drain from that cleanout. The problem is likely a combination of lint from the washing machine and grease and such from the kitchen sink.

I can't really tell from the photos, but I think that the drain is also undersized. Laundry drains are supposed to be 2". Again, I'm guessing because I can't tell from the photo, but that looks like standard 1-1/2" drain for the sink. I'm going to assume that the washing machine was added later.

Washing machines, particularly standard top loaders, can move a lot of water when they are draining. That's why they need a larger drain line.

You can't fix the drain line size, but you can snake the drain.

Jake & Joseph,

Thanks for trying to help me. I will get you better photos. I think there's a bit of the pipe, that comes from the washing machine, that's partly visible. If you see this pipe will you be able to tell whether it's the right size? I would be very surprised if the pipe is the wrong size because the plumber who installed it works according to code.

If there wasn't a washing machine there originally, then there wasn't 2" pipe put into place. It's not really possible to do that with 2" pipe without opening up a lot of the wall and replacing the pipe.

The solution is to snake the drain.

When the washing machine was installed, the carpenter opened up the wall for the plumber to install the pipe work.

Here are two photos of the washer pipe. The second photo is taken from the top of the kitchen sink cabinet. It's a view of the back of the washer. You will notice that the baseboard is missing (that's why some insulation is dropping off onto the floor). I'm about to install the baseboard..

And this is a photo of my washer: