(I know that there are numerous questions on this, but my problem is in actually solving the equations, which isn't the problem in other questions.)
I'm trying to figure out the steady state probabilities for a Markov Chain, but I'm having problems with actually solving the equations that arise. So,
$$ V = V\times \left[ { \begin{array}{ccc} 0 & 0.4 & 0 & 0.6\\ 0 & 0.5 & 0.5 & 0 \\ 0.4 & 0 & 0.2 & 0.4\\ 0 & 0 & 1 & 0 \end{array} }\right]$$ Where $V = \left[ \begin{array}{cccc}a & b & c & d \end{array} \right]$
So typing out the equations that I get from multiplying V with the transition matrix.
\begin{align} &0a &+ &0b &+ &0.4c &+ &0d&= a\\ &0.4a &+ &0.5b &+ &0c &+ &0d &= b\\ &0a &+ &0.5b &+ &0.2c &+ &d &=c\\ &0.6a &+ &0b &+ &0.4c &+ &0d &=d \end{align}
How do I solve these equations?
I know the answer for a, b, c, d but I get lost in trying to get to the answer from the equations.