Question

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

Accepted Answer

We have to find the complex number z given that (3z - 2z')/6 = -5
Let z = a + ib, z' = a - ib
(3z - 2z')/6 = -5
=>(3(a + ib) - 2(a - ib)) = -30
=> 3a + 3ib - 2a + 2ib = -30
=> a + 5ib = -30
We see that for the imaginary part 5ib there is no equivalent on the right hand side, so b = 0
a = -30
The number z = -30

Answer

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

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