After some brainstorming, I realized that when I inserted the snake into the vent pipe from the roof, I did go past the trap. I put all 50 ft of the snake into the vent. I then measured the path of the vent to the drain, and the direction that the drain should go under the house. The assumed path went to a room that had a court yard/flower bed just outside the window. A tree is in that court yard. My snake ended just before this area.
My concern being that a root from this tree could have broken the pipe, I called the landlord to have a plumber come out. The plumber had an electric drill/snake device. He used that to trace the path of the pipe, which went in the opposite direction that I thought.
The washer drain pipe was built as a separate drain system from the grey water system. This pipe went over to the property line and ran down a set of trees. At the end of the pipe, holes had been drilled into the pipe to let the water drain into the ground. Keeping in mind that this pipe is underground, dirt and mud had gotten into the pipe and roots from the nearby trees had grown into the pipe.