To solve the equation \( \frac{11}{y} + 9 = \frac{3}{y} \), let's first eliminate the fractions by multiplying both sides by \( y \) (assuming \( y \neq 0 \)):
\[ y \left( \frac{11}{y} + 9 \right) = y \left( \frac{3}{y} \right) \]
This simplifies to:
\[ 11 + 9y = 3 \]
Next, we will isolate \( y \) by rearranging the equation:
\[ 9y = 3 - 11 \]
\[ 9y = -8 \]
Now, divide both sides by 9:
\[ y = -\frac{8}{9} \]
Now, we have a potential solution. Let’s check this by substituting \( y = -\frac{8}{9} \) back into the original equation:
- Calculate \( \frac{11}{y} \):
\[ \frac{11}{-\frac{8}{9}} = 11 \cdot -\frac{9}{8} = -\frac{99}{8} \]
- Calculate \( \frac{3}{y} \):
\[ \frac{3}{-\frac{8}{9}} = 3 \cdot -\frac{9}{8} = -\frac{27}{8} \]
- Substitute the values into the original equation and simplify:
\[ -\frac{99}{8} + 9 \stackrel{?}{=} -\frac{27}{8} \]
To combine \( -\frac{99}{8} + 9 \), convert 9 to a fraction with a denominator of 8:
\[ 9 = \frac{72}{8} \]
Now add:
\[ -\frac{99}{8} + \frac{72}{8} = \frac{-99 + 72}{8} = \frac{-27}{8} \]
Thus:
\[ -\frac{27}{8} = -\frac{27}{8} \]
The left side equals the right side, confirming that our solution is correct.
Therefore, the solution to the equation is:
\[ \boxed{-\frac{8}{9}} \]