1. **State the problem:** Solve the equation $$-\frac{(2-3y)^2}{54} = 0.666\ldots y + \frac{(y+0.5)(0.3333\ldots - y)}{6}$$ for $y$. 2. **Rewrite repeating decimals as fractions:** - $0.666\ldots = \frac{2}{3}$ - $0.3333\ldots = \frac{1}{3}$ So the equation becomes: $$-\frac{(2-3y)^2}{54} = \frac{2}{3} y + \frac{(y+0.5)\left(\frac{1}{3} - y\right)}{6}$$ 3. **Multiply both sides by 54 to clear denominators:** $$\cancel{54} \times \left(-\frac{(2-3y)^2}{\cancel{54}}\right) = 54 \times \left(\frac{2}{3} y + \frac{(y+0.5)\left(\frac{1}{3} - y\right)}{6}\right)$$ which simplifies to: $$-(2-3y)^2 = 54 \times \frac{2}{3} y + 54 \times \frac{(y+0.5)\left(\frac{1}{3} - y\right)}{6}$$ 4. **Simplify the right side:** $$54 \times \frac{2}{3} y = 36 y$$ $$54 \times \frac{(y+0.5)\left(\frac{1}{3} - y\right)}{6} = 9 (y+0.5)\left(\frac{1}{3} - y\right)$$ So the equation is: $$-(2-3y)^2 = 36 y + 9 (y+0.5)\left(\frac{1}{3} - y\right)$$ 5. **Expand the left side:** $$(2-3y)^2 = (2)^2 - 2 \times 2 \times 3y + (3y)^2 = 4 - 12 y + 9 y^2$$ So: $$-(4 - 12 y + 9 y^2) = -4 + 12 y - 9 y^2$$ 6. **Expand the right side second term:** $$(y+0.5)\left(\frac{1}{3} - y\right) = y \times \frac{1}{3} - y^2 + 0.5 \times \frac{1}{3} - 0.5 y = \frac{y}{3} - y^2 + \frac{1}{6} - 0.5 y$$ Simplify the $y$ terms: $$\frac{y}{3} - 0.5 y = \frac{y}{3} - \frac{1}{2} y = \frac{2y - 3y}{6} = -\frac{y}{6}$$ So the expression is: $$-\frac{y}{6} - y^2 + \frac{1}{6}$$ Multiply by 9: $$9 \times \left(-\frac{y}{6} - y^2 + \frac{1}{6}\right) = -\frac{9y}{6} - 9 y^2 + \frac{9}{6} = -\frac{3y}{2} - 9 y^2 + \frac{3}{2}$$ 7. **Rewrite the full equation:** $$-4 + 12 y - 9 y^2 = 36 y - \frac{3 y}{2} - 9 y^2 + \frac{3}{2}$$ 8. **Bring all terms to one side:** $$-4 + 12 y - 9 y^2 - 36 y + \frac{3 y}{2} + 9 y^2 - \frac{3}{2} = 0$$ Simplify terms: - $-9 y^2 + 9 y^2 = 0$ - $12 y - 36 y + \frac{3 y}{2} = (12 - 36 + 1.5) y = (-24 + 1.5) y = -22.5 y$ - $-4 - \frac{3}{2} = -4 - 1.5 = -5.5$ So: $$-22.5 y - 5.5 = 0$$ 9. **Solve for $y$:** $$-22.5 y = 5.5$$ $$y = \frac{5.5}{-22.5} = -\frac{11/2}{45/2} = -\frac{11}{45}$$ **Final answer:** $$y = -\frac{11}{45}$$