Solve the equation in z : 2z+6i = z/2i+5i-7.

We'll have to determine z = a +
bi.

In other words, we'll have to determine the real part
and the imaginary part of the complex number.

We'll move
the terms in z to the left side:

2z - z/2i = 5i - 6i -
7

We'll combine like terms:

2z
- z/2i = -i - 7

We'll multiply by
2i:

4iz - z = -2i^2 - 14i

z(4i
- 1) = 2 - 14i

We'll divide by 4i -
1:

z = (2 - 14i)/(4i -
1)

We'll multiply by the conjugate of
denominator:

z = (2 - 14i)(-1 - 4i)/(4i - 1)(-1 -
4i)

z = (-2 - 8i + 14i -
56)/(1+16)

z = (-58 + 6i)/17

z
= -58/17 + (6/17)*i

The real part is: Re(z) =
-58/17

The imaginary part is: Im(z)
=6/17

z = -58/17 +
(6/17)*i