Example and Introduction
Let's say you have the following:
xy - x - y = 119
You can factor the left side, making it
x(y-1) - y = 119
Now, it seems like you're stuck, but by adding 1 on each side, you can factor it.
x(y-1) - y + 1 = 120
x(y-1) - (y-1) = 120
(x-1)(y-1) = 120
Now, if the problem is asking for integers, you can just look through the possible factor pairs of 120, and figure out what x and y should be.
General Case and Rule
xy + ax + by = c
x(y + a) + by + ab = c + ab
x(y + a) + b(y + a) = c + ab
(x + b)(y + a) = c + ab
So, when you do those problems, you have to be careful with the signs.
This trick is very useful, and it is commonly asked in AMC competitions.
Find the length and width of a rectangle whose area is equal to its perimeter
We start with our equations:
P = 2a + 2b
A = ab
2a + 2b = ab
ab - 2a - 2b = 0
Now, we use Simon's Favorite Factoring Trick, and we get
(a-2)(b-2) = 4
So the pairs could be 1 * 4, 2 * 2, and 4 * 1 So, we get