1. **State the problem:** Solve the equation $x^3 + y^3 = 1$ for $y$ in terms of $x$. 2. **Recall the formula:** The sum of cubes factorization is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. 3. **Apply the formula:** Here, $x^3 + y^3 = 1$ can be rewritten as $x^3 + y^3 - 1^3 = 0$. 4. **Rewrite the equation:** Using the sum of cubes factorization, $$x^3 + y^3 - 1^3 = (x + y - 1)(x^2 - xy + y^2 + x - y + 1) = 0$$ 5. **Solve for $y$ from the first factor:** $$x + y - 1 = 0 \implies y = 1 - x$$ 6. **Alternatively, solve for $y$ from the original equation:** $$y^3 = 1 - x^3$$ $$y = \sqrt[3]{1 - x^3}$$ 7. **Summary:** The solutions for $y$ in terms of $x$ are: $$y = 1 - x$$ or $$y = \sqrt[3]{1 - x^3}$$ These represent the implicit and explicit forms of the solution.