Computing all the values that solve an equation

Question

Given

$\frac{1}{2}=\frac{x+y-2}{xy}$

how can I compute all the $(x,y)$ values that satisfy such equation? Is there a general rule?

From the plot https://www.desmos.com/calculator/ym9ddpbika?lang=it it is not very clear which are all the solutions.( One is $x=3,y=2$ and viceversa)

Answer 1

$xy=2(x+y-2)$.

$y(x-2)=2(x-2)$.

So all pairs $(x,y)$ with $x=2$ are solutions. All pairs $(x,y)$ with $y=2$ are also solutions. There are no other solutions.

Answer 2

Your equation is equivalent to $$(x-2)(y-2)=0,\quad xy\ne0.$$ The set of solutions $(x,y)$ is therefore $$(\{2\}\times\Bbb R^*)\cup(\Bbb R^*\times\{2\}).$$ This is consistent with your plot, where the exclusion of the two pairs $(2,0)$ and $(0,2)$ is of course not viewable.