Find all solution in <mi mathvariant="double-struck">R for the following system of equations:

Question

Find all solution in

R

for the following system of equations:

{\begin{cases} x + \frac{3 x - y}{x^{2} + y^{2}} = 3 \\ y - \frac{x + 3 y}{x^{2} + y^{2}} = 0 \end{cases}

Answer 1

Complexify. Let

z = x + i y

. Then for

x^{2} + y^{2} \neq 0

:

\begin{aligned} (x + \frac{3 x - y}{x^{2} + y^{2}}) + i (y - \frac{x + 3 y}{x^{2} + y^{2}}) & = (3 + i 0) \\ ⟺ & (x + i y) + \frac{(3 - i) (x - i y)}{x^{2} + y^{2}} & = 3 \\ ⟺ & z + \frac{3 - i}{z} & = 3 \\ ⟺ & z^{2} - 3 z + (3 - i) & = 0 \\ ⟺ & {(z - \frac{3}{2})}^{2} & = i - \frac{3}{4} \\ ⟺ & z & = \frac{3 \pm \sqrt{4 i - 3}}{2} . \end{aligned}

Now,

(1 + 2 i)^{2} = - 3 + 4 i

, so the solutions are

z = \frac{3 \pm (1 + 2 i)}{2},

either

z = 2 + i

or

z = 1 - i

, which, in real form, yields

(x, y) = (2, 1)

or

(x, y) = (1, - 1)

.

Answer 2

Great solution and answer. I didn't know that complexifying will provide a real solution. Thanks again.

Answers (2)