Determine the complex number z satisfying the euqation $3z + 4i \overline{z} = 1 - 8i + 2 + 5i$.
+129845 z = a + bi z (complement) = a - bi i^2 = -1
3 (a + bi) + 4i (a - bi) = 3 - 3i
3a + 3bi + 4ai - 4bi^2 = 3 - 3i
3a + (4a + 3b)i + 4b = 3 - 3i
(3a + 4b) + (4a + 3b) i = 3 - 3i implies that
3a + 4b = 3 ⇒ 12a + 16b = 12 (1)
4a + 3b = -3 ⇒ -12a -9b = 9 (2)
Add ( 1) and (2)
7b = 21
b = 3
3a + 4 (3) = 3
3a + 12 = 3
3a = 3 - 12
3a = -9
3a = -9 / 3
a = -3
z = -3 + 3 i
