Prove that for any rationals x, y, with y not equal to 0, x+y.sqrt(2) is an irrational number.

For this, I wrote:

Suppose x+y.sqrt(2) is rational. Then, since x and y are rational, y.sqrt(2) is also rational, since the difference of two rationals is numbers is also rational.

Then from there I proved by contradiction that sqrt(2) is irrational, and said that this implied that x+y.sqrt(2) is also irrational.

Is this the correct method? Any help would be appreciated