# Thread: All reals numbers

1. ## All reals numbers

Find all reals numbers x , y : x + y = x y

2. my attempt
x+y=xy for real number
(0,0),(2,2),(-1,1/2),(1/2,-1)

Edit\\
$x+y=xy$
$\Rightarrow \frac{1}{x}+\frac{1}{y}=1 \ which\ is\ a\ geometrical\ hyperbola$

$\Rightarrow \frac{1}{y}=1 - \frac{1}{x} \quad or \quad \frac{1}{x}=1 - \frac{1}{y}$
$\quad \Rightarrow \quad y=\frac{x}{x-1}\quad or \quad x=\frac{y}{y-1}$
therefore $x \neq 1\ and\ y \neq 1$ so,there are many solutions.

3. Originally Posted by ramiee2010
my attempt
x+y=xy for real number
(0,0),(2,2),(-1,1/2),(1/2,-1)

Edit\\
$x+y=xy$
$\Rightarrow \frac{1}{x}+\frac{1}{y}=1 \ which\ is\ a\ geometrical\ hyperbola$

$\Rightarrow \frac{1}{y}=1 - \frac{1}{x} \quad or \quad \frac{1}{x}=1 - \frac{1}{y}$
$\quad \Rightarrow \quad y=\frac{x}{x-1}\quad or \quad x=\frac{y}{y-1}$
therefore $x \neq 1\ and\ y \neq 1$ so,there are many solutions.
Hello THANK YOU
I'can write the set of solutions is :

$S = \left\{ {} \right.\left( {x,\frac{x}{{x - 1}}} \right),\left( {\frac{x}{{x - 1}},x} \right)....\left. {x \in \Re ..\left( {x \ne 1} \right)} \right\}$