I'm not sure what your question is. You have two independent equations in two unknowns (note: the equation P0= (4/5)P0 +(3/4)P1 is the same as P1 = (1/5)P0+(1/4)P1, so only one of those equations is important, and the other independ equation is P1+P0=1). The solution for P0 and P1 is as you stated: P0=15/1 and P1=4/19. So - what's your question?