Step 1 :The given expression is \(-\frac{1}{3} x y + \frac{4}{9} x y - \frac{1}{9} x y\).
Step 2 :The expression contains three terms, all of which are products of x and y with different coefficients.
Step 3 :To simplify the expression, we need to add the coefficients of the like terms. In this case, the like terms are the terms that contain the product of x and y.
Step 4 :The sum of the coefficients of the like terms is 0.
Step 5 :Therefore, the simplified form of the expression \(-\frac{1}{3} x y + \frac{4}{9} x y - \frac{1}{9} x y\) is 0.
Step 6 :\(\boxed{0}\) is the final answer.