+1243 Determine the complex number z satisfying the equation \(2z-3i\bar{z}=-7+3i\). Note that \(\bar{z} \) denotes the conjugate of z.
+128656 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
