Calculate z if z+3z'=3i. z=a+b*i

Expert Answers
hala718 eNotes educator| Certified Educator

z + 3z' = 3i   

z = a+bi

First ley us determine z':

==> z' = a-bi

Now substitute:

z + 3z' = 3i

(a+bi) + 3(a-bi) = 3i

==> a + bi + 3a - 3bi = 3i

==> 4a - 2bi = 3i

==> 4a = 0 ==> a = 0

==> -2bi = 3i

==> -2b = 3 ==> b = -3/2

==> z= 0 + (-3/2)i

==> Z = -(3/2)i

neela | Student

z+3z' = 3i .

To find z.

Solution:

Let z = a+bi. then z' =  a-bi.

So the given equation becomes:

a+bi+3(a-bi) = 3i = 0+3i

4a  -2 bi = 0 +3i. Equate real parts on both sides . Equate imagonary parts on both sides.

4a = 0 and -2bi = 3i

a = 0 b = -3/2.

So z = a+bi = 0+(-3/2)i = -1.5i

z = -1.5i

giorgiana1976 | Student

Considering that z'=a-b*i is the conjugate of z, the given relation could be re-written:

z+3z'=3i

a+b*i +3(a-b*i)=3i

a+b*i + 3a - 3*b*i - 3i = 0

We'll combine real terms and imaginary terms:

4a + i*(b-3b-3) = 0

4a + i*(-2b-3) = 0

We'll consider the right side as a complex number:

0 = 0 + 0*i

4a + i*(-2b-3) = 0 + 0*i

Since the both sides are equivalent, we'll write:

4a = 0

a = 0

-2b-3 = 0

-2b = 3

b = - 3/2

The complex number z will be:

z = 0 + i*(-3/2)

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