I take back my previous statement. The problem is indeed solvable once you know that one tension is twice the other. After doing some algebra I see that we can find the accelerations based on only just the ratio between the two tensions.
However, I now have a different confusion. It no longer...
Yep, it's all in the PDF I linked to. The relation between the accelerations is on the last line of the first page; the acceleration of the second block is twice that of the first (and in the opposite direction).
That the one tension would be twice the other makes sense. But having the relative tensions doesn't produce an answer. To find an absolute acceleration we need an absolute tension, which we don't yet have.
I'm currently reading Kleppner and Kolenkow's "Introduction to Mechanics", and I'm working through the problem sets.
I'm only on chapter 2, but I happen to know Lagrangian mechanics. When I get stuck on a problem I occasionally take the Lagrangian approach to find the solution, and then I try...