Question

Solve for x and y 1/(x+yi) + 2/(x-yi) = 1+i

Answer 1

Explanation:

x+i*y≠0 ==> x≠0 and y≠0

x-i*y≠0 ==> x≠0 and y≠0

We must exclude (0,0) as solution.

`(1)/(x+i*y) +(2)/(x-i*y) =1+i\\\\(1*(x-i*y))/((x+i*y)(x-i*y)) +(2*(x+i*y))/((x-i*y)(x+i*y)) =1+i\\\\\\(x-i*y+2x+2i*y)/(x^2+y^2) =1+i\\\\3x+i*y=(1+i)(x^2+y^2)\\\\3x+i*y=(x^2+y^2)+i(x^2+y^2+i*(x^2+y^2)\\\\\left\{\begin{array}{ccc}3x&=&x^2+y^2\\y&=&x^2+y^2\\\end {array}\right.\\\\\\\left\{\begin{array}{ccc}y&=&3x\\3x&=&x^2+9x^2\\\end {array}\right.\\\\\\\left\{\begin{array}{ccc}y&=&3x\\x(10x-3)&=&0\\\end {array}\right.\\\\`

`\left\{\begin{array}{ccc}x&=&0\\y&=&0\\\end {array}\right.\ (to\ exclude) \ or \ \left\{\begin{array}{ccc}x&=&(3)/(10)\\\\y&=&(9)/(10)\\\end {array}\right.\\\\Sol=\{((3)/(10),(9)/(10))\}\\`