AoPS - Problem

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Problem

Source: 2004 Estonia National Olympiad Final Round grade 9 p1

Tags: equation, algebra

parmenides51

25.03.2020 03:49

Find all pairs of real numbers $(x, y)$ that satisfy the equation $\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}$

Prod55

05.08.2020 18:39

We must have $xy\neq 0$ . Multiply by $xy$ : $x^2+6x+13=4y-y^2\Leftrightarrow x^2+6x+9+y^2-4y+4=0\Leftrightarrow (x+3)^2+(y-2)^2=0\Leftrightarrow \boxed{(x,y)=(-3,2)}$