1. **Problem statement:** Simplify the algebraic expression $2x - 2y - 6x + 5y$. 2. **Formula and rules:** Combine like terms by adding or subtracting coefficients of the same variable. 3. **Step-by-step simplification:** $$2x - 2y - 6x + 5y = (2x - 6x) + (-2y + 5y)$$ $$= -4x + 3y$$ 4. **Explanation:** We grouped the $x$ terms and the $y$ terms separately, then combined their coefficients. 5. **Final answer:** $$\boxed{-4x + 3y}$$