1. **State the problem:** Solve the equation $$\frac{2}{u} + \frac{1}{6} = \frac{7}{6u}$$ for $u$. 2. **Identify the common denominator:** The denominators are $u$, $6$, and $6u$. The least common denominator (LCD) is $6u$. 3. **Multiply both sides of the equation by the LCD $6u$ to eliminate the denominators:** $$6u \times \left(\frac{2}{u} + \frac{1}{6}\right) = 6u \times \frac{7}{6u}$$ 4. **Distribute multiplication:** $$6u \times \frac{2}{u} + 6u \times \frac{1}{6} = 6u \times \frac{7}{6u}$$ 5. **Simplify each term:** $$6 \cancel{u} \times \frac{2}{\cancel{u}} + \cancel{6} u \times \frac{1}{\cancel{6}} = \cancel{6} \cancel{u} \times \frac{7}{\cancel{6} \cancel{u}}$$ which simplifies to $$6 \times 2 + u = 7$$ 6. **Simplify further:** $$12 + u = 7$$ 7. **Isolate $u$ by subtracting 12 from both sides:** $$u = 7 - 12$$ 8. **Calculate the result:** $$u = -5$$ **Final answer:** $$u = -5$$