I know that in a complex number such as

"Re" is supposed to be the "real part", which is so that and that "Im" is supposed to be the "imaginary part", which is so that in the above example.

However, recently I stumpled upon a problem when trying to solve linear systems:

I'm not sure that I understand how exactly to execute this formula with the "Re" part.

Here is an example:

Can somebody explain to me what the "Re" did here?

And what about "Im"? If it had said "Im" instead, what would be different in the answer?