Find real `x` and `y` , if

Question

Find real `x` and `y` , if `(x -iy)(3+5i)` is the conjugate of `-6-24i`

Solution:

Let

`z=-6-24i`

`\bar z = \overline(-2-24i}`

And

`\bar z = (x-iy)(3+5i)`

`-6+24i = ( x-iy)(3+5i)`

or

`(x-iy)(3+5i) = -6+24i`

`(x-iy) = \frac{-6+24i}{3+5i} \times \frac{3-5i}{3-5i}`

`\left(x - iy\right) = \frac{\left( - 6 + 24i \right)\left(3 - 5i\right)}{\left(3 + 5i\right)\left(3 - 5i\right)}`

`\left(x - iy\right) = \frac{- 6\left(3 - 5i\right) + 24i\left(3 - 5i\right)}{\left( 3 \right)^2 - \left( 5i \right)^2}`

`\left(x - iy\right) = \frac{ - 18 + 30i + 72i - 120i^2}{9 - 25i^2}`

`\left(x - iy\right) = \frac{ - 18 + 102i + 120}{9 + 25}`

`\left(x - iy\right) = \frac{102 + 102i}{34}`

`\left(x - iy\right) = \frac{102}{34} + \frac{102i}{34}`

`\left(x - iy \right) = 3 + 3i`

`x = 3 and - y =  - 3`

`x = 3 and y = 3`