Upon trying to solve this particular system , I've encountered a few problems. $$ y'=5y+4z $$ $$z'=-4y-3z$$
After solving for eigenvalues the quadratic yielded a double root at $\lambda=1$ . But I got stuck after getting at $a=-b$ . This prevented me from solving any further because I eventually arrived at $0=1$ .
My method looks exactly like presented on this site adapted to the case in which $\lambda$ is a double solution.
I'm very interested in a solution.