5/6-y=1/3+1/9

To solve the equation \( \frac{5}{6} - y = \frac{1}{3} + \frac{1}{9} \), we should first combine the fractions on the right side. 1. **Find a common denominator for the fractions \( \frac{1}{3} \) and \( \frac{1}{9} \)**. The least common multiple of 3 and 9 is 9. 2. **Convert \( \frac{1}{3} \) to have a denominator of 9**: \[ \frac{1}{3} = \frac{3}{9} \] 3. **Now, add \( \frac{3}{9} \) and \( \frac{1}{9} \)**: \[ \frac{3}{9} + \frac{1}{9} = \frac{4}{9} \] So, the equation now looks like this: \[ \frac{5}{6} - y = \frac{4}{9} \] 4. **To isolate \( y \), we first move \( y \) to one side and \( \frac{4}{9} \) to the other side**: \[ -y = \frac{4}{9} - \frac{5}{6} \] 5. **Next, we need to find a common denominator for \( \frac{4}{9} \) and \( \frac{5}{6} \)**. The least common multiple of 9 and 6 is 18. 6. **Convert \( \frac{4}{9} \) and \( \frac{5}{6} \) to have a denominator of 18**: \[ \frac{4}{9} = \frac{8}{18} \quad \text{(multiply top and bottom by 2)} \] \[ \frac{5}{6} = \frac{15}{18} \quad \text{(multiply top and bottom by 3)} \] 7. **Now, substitute those values back into the equation**: \[ -y = \frac{8}{18} - \frac{15}{18} \] 8. **Subtract the fractions**: \[ -y = \frac{8 - 15}{18} = \frac{-7}{18} \] 9. **Multiply by -1 to solve for \( y \)**: \[ y = \frac{7}{18} \] Thus, the solution to the equation is: \[ y = \frac{7}{18} \]