Determine the complex number z satisfying the equation \(2z-3i\bar{z}=-7+3i\). Note that \(\bar{z} \) denotes the conjugate of z.

Lightning Apr 14, 2019

#1**+1 **

2(a + bi) - 3i ( a - bi) = -7 + 3i

2a + 2bi - 3ai - 3b = -7 +3i

Equate real and imaginary parts

(2a - 3b) = 7 ⇒ 6a - 9b = 21

(-3a + 2b) = 3 ⇒ -6a + 4b = 6

Add the last two equations and we have

-5b = 27

b = -27/5

And

2a - (3)(-27/5) = 7

2a + 81/5 = 35/5

2a = [ 35 -81] / 5

2a = -46 / 5

a = -46/10 = -23/5

So

z = -23/5 - (27/5) i

CPhill Apr 14, 2019