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

To solve the equation \( \frac{5}{6} - y = \frac{1}{3} + \frac{1}{5} \), we first need to simplify the right-hand side. 1. Find a common denominator for \( \frac{1}{3} \) and \( \frac{1}{5} \). The least common multiple (LCM) of 3 and 5 is 15. 2. Rewrite the fractions with the common denominator: - \( \frac{1}{3} = \frac{5}{15} \) - \( \frac{1}{5} = \frac{3}{15} \) 3. Add these two fractions together: \[ \frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \] Now, rewrite the equation: \[ \frac{5}{6} - y = \frac{8}{15} \] 4. To isolate \( y \), first move \( y \) to one side of the equation and the other terms to the other side: \[ -y = \frac{8}{15} - \frac{5}{6} \] 5. Find a common denominator for \( \frac{8}{15} \) and \( \frac{5}{6} \). The LCM of 15 and 6 is 30. 6. Rewrite the fractions with the common denominator: - \( \frac{8}{15} = \frac{16}{30} \) - \( \frac{5}{6} = \frac{25}{30} \) Now substitute these values back into the equation: \[ -y = \frac{16}{30} - \frac{25}{30} \] \[ -y = \frac{16 - 25}{30} = \frac{-9}{30} \] 7. Simplify \( \frac{-9}{30} \): \[ -y = \frac{-3}{10} \] 8. Therefore, multiply both sides by -1 to solve for \( y \): \[ y = \frac{3}{10} \] Thus, the solution to the equation is: \[ \boxed{\frac{3}{10}} \]