Linear system 2 unknowns There are $x$ white and $y$ black pearls and their ratio is $z$. If I add six black and six white pearles, the ratio doubles.
I did the following:
$ \frac{x+6}{y+6} = \frac{2x}{y}$ and then I get 
$xy -6(2x-y)=0$
I can find solutions by guessing. Is there any other way? 
ADDED:
Now I have to solve the problem for
$ \frac{c(x+y)+6}{x+y+12} = 2c $ and once again I am lost in factoring out variables. 

$\frac{x}{y} = z$ and $\frac{x+6}{y+6} = 2z$
$\implies x = zy$ and $\frac{zy+6}{y+6} = 2z$
$\implies 6 = z(y+12) \implies z = \frac{6}{y+12}$
$\implies x = zy = \frac{6y}{y+12}$
Since $x$ and $y$ are the number of pearls, they must be integers. That is
$(y+12) | (6y)$.
There are only 3 possible $y$ values by giving $x=2, x=3, x=4$. Other $x$ values gives non-integer or non-positive $y$'s.
Thus, all possible $(x,y)$ pairs are $(2,6), (3,12), (4,24)$
