Thanks Alexey,
The dissolve succeeded in combining the intersections and roads into single objects. But how do I execute this same function so that the two different attributes combine?
If you look at the images, I want the brown intersections and the blue roads to be combined into the same object resulting in the symbolization in the second image.
Is the eliminate tool what I'm looking for? If so, is there another option? (I currently don't have the spatial analyst license)