We need to solve the equation $\frac{1}{3}y - 9 = \frac{1}{9}y$ for $y$.

1. Problem Description

We need to solve the equation $\frac{1}{3}y - 9 = \frac{1}{9}y$ for $y$.

2. Solution Steps

First, let's subtract $\frac{1}{9}y$ from both sides of the equation:

$\frac{1}{3}y - \frac{1}{9}y - 9 = \frac{1}{9}y - \frac{1}{9}y$

$\frac{1}{3}y - \frac{1}{9}y - 9 = 0$

We need to find a common denominator for $\frac{1}{3}$ and $\frac{1}{9}$, which is $9$. So we can rewrite $\frac{1}{3}y$ as $\frac{3}{9}y$. Then we have

$\frac{3}{9}y - \frac{1}{9}y - 9 = 0$

Combine the $y$ terms:

$\frac{2}{9}y - 9 = 0$

Now, add $9$ to both sides of the equation:

$\frac{2}{9}y - 9 + 9 = 0 + 9$

$\frac{2}{9}y = 9$

To isolate $y$, we multiply both sides of the equation by $\frac{9}{2}$:

$\frac{9}{2} \cdot \frac{2}{9}y = 9 \cdot \frac{9}{2}$

$y = \frac{81}{2}$

3. Final Answer

$y = \frac{81}{2}$