Question

A complex number (z) and its conjugate (bar{z}) satisfy the equation (z + 3bar{z} = 2(10 - i)), where (i = √'{-1}). Find (z).

a) (5 - 2i)
b) (2 + 4i)
c) (3 - 3i)
d) (1 + 5i)

Answer 1

To find the value of z, we can rearrange the equation and isolate z. By substituting the complex conjugate of z into the equation, we can solve for the real and imaginary parts of z. The correct value of z is 5 + i.

Step-by-step explanation:

To find the value of z, we can first rearrange the equation to isolate z.

We have z + 3(bar{z}) = 2(10 - i).

Next, we can substitute the complex conjugate of z, which is bar{z}, with its real and imaginary parts.

Let z = a + bi, where a and b are real numbers. Then bar{z} = a - bi.

Substituting these values into the equation, we get:

a + bi + 3(a - bi) = 20 - 2i

(a + 3a) + (bi - 3bi) = 20 - 2i

4a - 2bi = 20 - 2i

Now we can equate the real and imaginary parts of both sides of the equation:

4a = 20 → a = 5

-2b = -2 → b = 1

Therefore, z = 5 + i.

So, the correct option is a) (5 - 2i).