Solving complex equation for z?

Question

How do you solve equations involving $z = a + bi$ and imaginary units?

The one I am looking at right now:

$$\frac{z-2}{z+1} = 3i$$

If you could help me with this one, I think I can do the rest by myself.

Answer 1

Set $z=a+ib$ where $a,b$ are real

So, we have $a+ib-2=3i(a+1+i)$

$$a-2+ib=-3+i\cdot3(a+1)$$

Now equate the real & the imaginary parts

Answer 2

Rewrite as $z-2=3iz+3i$ and isolate $z$ as usual. We get $$z=\frac{2+3i}{1-3i}.$$ You may be expected to change the form of the answer by multiplying top and bottom by $1+3i$.