we have :

(z'+7i)/z=6

Let z= x+yi

==> z'= x-yi

Now substitute:

(z'+7i)/z= 6

(x-yi +7i)/x+yi =6

[x-yi +7i]/(x+yi)=6

Multiply by z= (x+yi)

==> (x-yi +7i]= 6(x+yi)

==> x-yi +7i= 6x +6yi

==> x-yi +7i -6x -6yi=0

==> -5x -7yi +7i=0

==> -5x +(-7y+7)i =0

==> x=0

==> -7y +7=0

==> y=1

Then z= x+yi = 0+(1)i=i

==> **z= i**

Let z = x+yi.

Then Z' = x-yi.

Therefore (z'+7i) = (x-yi +7i)/(x+yi) = 6

x-yi +7i = 6(x+yi).

x +(7-y)i = 6x+6yi.

Equating real and imaginary parts:

Real part x= 6x. implies x= 0.

Imaginary parts: (7-y)i = 6yi. Or

7-y = 6y. Or

7 = 7y. Or

1 = y.

Thus z = x+yi = 0+i = i.

Let's note z = a + b*i and z' = a - b*i

We'll multiply the expresion by (1/z), so that:

z'+7i =6z

a - b*i + 7i = 6(a + b*i )

a - 6a - b*i - 6b*i + 7i = 0

-5a + i(-7b + 7) = 0

We could write the right side as: 0 = 0+0*i

The expressions from both side are identically if and only if the real parts and imaginary parts are the same.

-5a = 0

We'll divide by -5:

**a = 0**

-7b + 7 = 0

We'll subtract 7 both sides:

-7b = -7

We'll divide by -7:

**b = 1**

So, z = 0 + 1*i

**z = i**