It is given that $f(0) = 0, g(0) = 0$, and for integer $n > 0$, $f(n)\pi + g(n) = (\frac{f(n - 1)}{2} + 5)\pi + 3g(n-1) - 1$. Would it be correct to separate the part without $\pi$ and the part with $\pi$ similar to how it is done on imaginary and real part with $xi + y = ai + b \iff x = a \land y = b$ ? Like this:
$f(n)\pi + g(n) = (\frac{f(n - 1)}{2} + 5)\pi + 3g(n) - 1 \iff f(n) = \frac{f(n - 1)}{2} + 5 \land g(n) = 3g(n - 1) - 1$
And what would be the general property? Saying $ax + b = px + q \iff a = p \land b = q$ would not be the correct because for example $3(-5) + 2 = 2(-5) - 3 \iff 3 = 2 \land 2 = -3$ is not true