Statement 1 tells us that x + y = xy. Let's try to manipulate (x-1)(y-1) to give us either x + y or xy so we can use this information. We'll start by FOILing to expand:
xy - x - y + 1
Then pulling out a -1 from the two middle terms gives:
xy - (x + y) + 1
Then plugging in xy for x + y gives:
xy - xy + 1
1
So (x-1)(y-1) = 1. Sufficient.
Statement 2 tells us that x = y. This means that (x -1)(y - 1) = (x - 1)^2 = (y - 1)^2.
Let's see what x and y must equal for (x-1)(y-1) to equal 1. Setting (x - 1)^2 equal to 1, we can solve for x:
(x -1 )^2 = 1
x^2 - 2x + 1 = 1
x^2 - 2x = 0
x (x - 2) = 0
x = 0 or x = 2
So (x - 1)(y - 1) = 1, x and y must both equal either 0 or 2.
So what do we know about x and y, other than that they are the same? We know that multiplied together, they are greater than zero. This tells us that x and y can't equal zero - 0 * 0 = 0, and xy > 0. However, any negative OR positive number multiplied by itself will give a product greater than zero. So based on this statement, x and y can be any number other than zero. This means x and y could equal 2, making (x - 1)(y - 1) = 1, or it could equal any other number other than 0, making (x - 1)(y - 1) not equal to 1. Insufficient.
