We can solve linear system of equations.
Equation $(a)$ can be converted to $$(0.2-1)\pi_1 + 0.1\pi_2 + 0.55\pi_3=0\tag{a'}$$
Similarly for $b$ and $c$.
Also, with the constraint $\pi_1+\pi_2+\pi_3=1$
We have $3$ variables and $4$ constraints.
$$\pi=P^T\pi$$ $$e^T\pi=1$$
$$\begin{bmatrix} P^T-I \\ e^T\end{bmatrix}\pi =\begin{bmatrix} 0_3 \\ 1\end{bmatrix}$$
You can perform Gaussian Elimination to get the solution.
Here is the Octave solution:
octave:1> A = [-0.8, 0.1, 0.55, 0; 0.3, -1, 0, 0; 0.5, 0.9, -0.55, 0; 1, 1, 1, 1]
A =
-0.80000 0.10000 0.55000 0.00000
0.30000 -1.00000 0.00000 0.00000
0.50000 0.90000 -0.55000 0.00000
1.00000 1.00000 1.00000 1.00000
octave:2> rref(A)
ans =
1.00000 0.00000 0.00000 0.37037
0.00000 1.00000 0.00000 0.11111
0.00000 0.00000 1.00000 0.51852
0.00000 0.00000 0.00000 0.00000