Bipolar Junction Transistor problem. Where did I go wrong?
This is more of a comment than an answer, but I don't have enough reputation to comment just yet.

A partial answer to "where did I go wrong" is a simple arithmetic error early on in the equations:


  V_1/100000 = 0.01(3 - V_2)/9100
  V_1/100000 = 91(3 – V2)


In this step it appears you multiplied 0.01 by 9100 rather than divide. Once this changes, everything downstream needs to change too.