Question 1170874

Please help me solve this question

Given (4+2i)m - (1-i)n = -9+9i. Find the complex numbers m and n if m is the conjugate of n.

thank you.
<pre>(4 + 2i)m - (1  -  i)n = - 9 + 9i

Let m be a + bi
Then n is: a - bi, since n is the conjugate of m


We then get: (4 + 2i)(a + bi) - (1 - i)(a - bi) = - 9 + 9i
{{{matrix(7,3, 4a + 4bi + 2ai + 2bi^2  -  (a  -  bi - ai + bi^2), "=", - 9 + 9i,
4a + 4bi + 2ai + 2bi^2  -  a + bi + ai - bi^2, "=", - 9 + 9i,
4a  -  a + 2ai + ai + 4bi + bi + 2bi^2 - bi^2, "=", - 9 + 9i,
3a + 3ai + 5bi + bi^2, "=", - 9 + 9i,
3a + 3ai + 5bi + b(- 1), "=", - 9 + 9i,
3a  -  b + 3ai + 5bi, "=", - 9 + 9i,
(3a  -  b) + (3a + 5b)i, "=", - 9 + 9i)}}}

3a - b = - 9 ---- Equating 1st terms ------ eq (i)
3a + 5b = 9 ------- Equating 2nd terms ------ eq (ii)
     6b = 18 ------ Subtracting eq (i) from eq (ii)
     {{{matrix(1,5, b, "=", 18/6, "=", 3)}}}
3a - 3 = - 9 ------- Substituting 3 for b in eq (i)
    3a = - 6
    {{{matrix(1,5, a, "=", (- 6)/3, "=", - 2)}}}

{{{highlight_green(matrix(2,5, m, "=", "a + bi,", so, highlight(matrix(1,3, m, "=", - 2 + 3i)), n, "=", "a - bi,", so, highlight(matrix(1,3, n, "=", - 2 - 3i))))}}}</pre>