Question

Given that
`z=x+iy`, find the value of x and of y such that
`z+4iz^*=-3+18i` where z* is the complex conjugate of z.

Answer 1

`z= 5-2i`

Explanation:

Start replacing!

`(x+iy) + 4i (x-iy) = -3+18i \\ x+iy +4ix -4i^2 y = -3 + 18i \\\\x + i (4x+y) +4y = -3 + 18i\\x+4y + (4x+y) i = -3 + 18i`

Now two complex numbers are equal if both the real and imaginary parts are equal, which gives you the system of equation
`\left \{x+4y=-3} \atop {4x+y=18} \right.`

Pick any method to solve it and you'll get
`x=5, y= -2`