How Many Integers to Solve an Equation?

[Sternhund]

Today I ran into this little abomination:

How many pairs of integers (x,y) will solve the equation 6x^2-3xy-13x+5y=-11?

I was able to find that the answer was two, the solutions being at x=1 and x=2 and their corresponding integer y-values, but I solved them through some off-the-wall calculator use that I don't care to explain. What I'm asking is the "proper" way of solving this.

Any takers?

[The3point14]

Here is AuSmith's explaination of this problem.

[Sternhund]

Aha! Thank you! His method was actually very similar to my own, but it's good to see it drawn out cleanly. :)