1. **State the problem:** Simplify the expression $$\frac{2x + 2y + 3(x - y) - 2(x + y + 3)}{3}$$. 2. **Expand the terms inside the numerator:** $$2x + 2y + 3x - 3y - 2x - 2y - 6$$ 3. **Combine like terms:** $$2x + 3x - 2x = 3x$$ $$2y - 3y - 2y = -3y$$ So the numerator becomes: $$3x - 3y - 6$$ 4. **Rewrite the expression:** $$\frac{3x - 3y - 6}{3}$$ 5. **Factor out 3 from the numerator:** $$\frac{3(x - y - 2)}{3}$$ 6. **Cancel the common factor 3:** $$\frac{\cancel{3}(x - y - 2)}{\cancel{3}} = x - y - 2$$ **Final answer:** $$x - y - 2$$