1. **State the problem:** Solve the equation $$(y - 7)^2 - 9 = (y - 10)(y - 4)$$ for $y$. 2. **Recall formulas and rules:** - Use the expansion formula for squares: $$(a - b)^2 = a^2 - 2ab + b^2$$ - Use distributive property to expand products. - Simplify and solve the resulting quadratic equation. 3. **Expand the left side:** $$(y - 7)^2 - 9 = (y^2 - 2 \cdot y \cdot 7 + 7^2) - 9 = y^2 - 14y + 49 - 9 = y^2 - 14y + 40$$ 4. **Expand the right side:** $$(y - 10)(y - 4) = y^2 - 4y - 10y + 40 = y^2 - 14y + 40$$ 5. **Set the equation:** $$y^2 - 14y + 40 = y^2 - 14y + 40$$ 6. **Subtract right side from left side:** $$y^2 - 14y + 40 - (y^2 - 14y + 40) = 0$$ 7. **Simplify:** $$\cancel{y^2} - 14y + 40 - \cancel{y^2} + 14y - 40 = 0$$ $$(-14y + 14y) + (40 - 40) = 0$$ $$0 = 0$$ 8. **Interpretation:** The equation simplifies to a true statement for all $y$, so the original equation holds for all real numbers $y$. **Final answer:** The equation is an identity; all real values of $y$ satisfy it.