Monday, December 14, 2015

Determine the complex number z if (3z-2z')/6=-5? z' is the conjugate of z.

We'll write the rectangular form of the complex number
z:


z = a + bi


Since z' is the
conjugate of z, then  z' = a - b.


To determine z, we'll
have to find out it's coefficients:


We'll multiply by 6
both sides:


3(a + bi) - 2(a - bi) = -
30


We'll remove the
brackets:


3a + 3bi - 2a + 2bi = -
30


We'll move all terms to the left
side:


3a + 3bi - 2a + 2bi + 30 =
0


We'll combine real parts and imaginary
parts:


a + 5bi + 30 = 0


The
real part of the complex number from the left side
is:


Re(z) = a + 30


The real
part of the complex number from the right side is:


Re(z) =
0


Comparing, we'll get:


a + 30
= 0


a = -30


The imaginary part
of the complex number from the left side is:


Im(z) =
5b


The imaginary part of the complex number from the right
side is:


Im(z) = 5b


Comparing,
we'll get:


5b = 0


b =
0


The requested complex number z is:
z = -30 (whose imaginary part is
0).

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