1. Problem statement: Simplify the expression $5y^2 - 2y^3 - 7y^2 + 7y^3$. 2. Formula and rules: To combine like terms, add the coefficients of terms that have the same power of $y$. Like terms are terms with the same variable raised to the same exponent. 3. Group like terms by power: Group the $y^2$ terms together and the $y^3$ terms together. $$5y^2 - 2y^3 - 7y^2 + 7y^3 = (5y^2 - 7y^2) + (-2y^3 + 7y^3)$$ 4. Combine the $y^2$ terms. $$5y^2 - 7y^2 = (5 - 7)y^2 = -2y^2$$ 5. Combine the $y^3$ terms. $$-2y^3 + 7y^3 = (-2 + 7)y^3 = 5y^3$$ 6. Reassemble the simplified expression by adding the results. $$5y^3 - 2y^2$$ 7. Optional check by factoring out $y^2$ to verify. $$5y^2 - 2y^3 - 7y^2 + 7y^3 = y^2(5 - 2y - 7 + 7y) = y^2(-2 + 5y) = y^2(5y - 2)$$ 8. Final answer: The simplified expression is $5y^3 - 2y^2$.