The trick to this question is to remember that to calculate an average, you add up all the values and then divide by how many values there are. This is difficult to see with values that are given as variables rather than actual numbers.

If you look at Statement (1), you are given THREE values: x, y, and y-2. To find the average of these three values, you would need to add them up and divide by 3.

$$\frac{x+y+y-2}{3}$$

You are told that the average is equal to x. So now you can set up an equation:
$$\frac{x+y+y-2}{3}=x$$

Now solve the equation for x:
Multiply both sides by 3: $$3\cdot\frac{x+y+y-2}{3}=x\cdot3$$
Simplify: $$x+2y-2=3x$$
Subtract x from both sides: $$x+2y-2-x=3x-x$$
Divide both sides by two: $$\frac{2y-2}{2}=\frac{2x}{2}$$

Finally, x is equal to y-1. This means that SINCE the difference between x and y is an integer (1), and x is an integer, then y must be an integer as well. (If x was not an integer, y would also have to be a non-integer).