Finding two numbers given their sum and their product 
Which two numbers when added together yield $16$, and when multiplied together yield $55$.

I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I couldn't. 
In $x+y=16$, I know $x=16/y$ but when I plug it back in I get something like $16/y + y = 16$, then I multiply the left side by $16$ to get $2y=256$ and then ultimately $y=128$. Am I doing something wrong?  

Given  $$  x + y =  16   \tag{1} $$
and $$  x \cdot y =  55.   \tag{2} $$
We can use the identity:  $$ ( x-y)^2 = ( x+y)^2 - 4 \cdot x \cdot y \tag{3}  $$
to get
$$ x - y =   \sqrt{256 - 4 \cdot 55}   = \sqrt{36}  = \pm \, 6.   \tag{4} $$
Finding half sum and half difference of equations (1)  and (4) gives us
$$  (x,y) = (11,5) , \, (5,11)  \tag{5}  $$
