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