A Markov chain on state space {1,2,3,4,5} has transition matrix

$\displaystyle \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\

0 & 0 & 4/{5} & 1/{5} & 0 \\

0 & 1/{6} & 2/{3} & 0 & 1/{6} \\

0 & 0 & 0 & 1 & 0 \\

0 & 0 & 0 & 0 & 1 \\

\end{pmatrix}$

The process starts in state 1.

d) Calculate the expectation of the number of visits to state 2 before absorption.

Solution: $\displaystyle v_1=v_3$

$\displaystyle v_2=1+(4/5)v_3$

$\displaystyle v_3=(1/6)v_2+(2/3)v_3$

I am not sure how to get the above three equation? can you help please?

These are easily solved to give $\displaystyle E(V)=v_1=5/6$