+1245 Determine the complex number satisfying the equation \(3z + 4i \overline{z} = 1 - 8\)i.
+129899 3z +4i (z conjugate) = 1 - 81
3 ( a + bi) - 4 i ( a - bi) = 1 - 8i
3a + 3bi - 4a i - 4b = 1 - 8i
(3a - 4b) + (3b - 4a) i = 1 - 8i
Equate coefficients on real , imaginary parts
3a - 4b = 1 ⇒ 12a - 16b = 4
-4a + 3b = - 8 ⇒ -12a + 9b = -24 add the last two equations
-7 b = - 20 ⇒ b = 20/7
And
3a - 4(20/7) = 1
3a - 80/7 = 7/7
3a = 87/7
a = 87/21 = 29/7
So
z = 29/7 + (20/7) i
