$( x \cdot y ) \mod 37 = 1$ I am doing a paper for my security class. I have this equation which I'm trying to understand
                                  $$( x \cdot  y ) \mod 37 = 1 $$
e.g.  if  $x = 8$ and $y = ?$ ; which has to be the inversion of x . then y in this case is 14. 
My question is how do I solve an equation like this? and is it possible that value of $y$ is a negative number? if so, then an example would be great.
Thank you in advance !

Instead of $37$ let's use a smaller number, $7$.  We have $xy \mod 7 = 1$ if the remainder of $xy$ upon division by $7$ is $1$.  Equivalently, if $xy-1$ is a multiple of $7$.
For example, $x=2, y=4$ satisfies $xy \mod 7 =1$.  This is also true if we replace either $x$ or $y$ by its sum with any number of $7$'s.  For example, $x=9=2+7, y=4$ still satisfies $xy \mod 7 = 1$.  We could just as well do $x=-5=2-7, y=4$ and still have $xy \mod 7 = 1$.  
If we know $x$ and want to find $y$, we may use the Euclidean algorithm to find a solution to Bezout's identity.  That is, we find integers $a,b$ such that $xa+7b=1$.  Now, you can see that $xa-1$ is a multiple of $7$, so the $a$ we found is actually the desired $y$.
