To simplify the complex fraction
\[ \frac{\frac{6}{y} + y}{\frac{5}{y} - y}, \]
we first simplify the numerator and the denominator separately.
Step 1: Simplify the numerator
The numerator is
\[ \frac{6}{y} + y. \]
To combine these terms, let's make \(y\) into a fraction:
\[ y = \frac{y^2}{y}. \]
So, we have:
\[ \frac{6}{y} + \frac{y^2}{y} = \frac{6 + y^2}{y}. \]
Step 2: Simplify the denominator
The denominator is
\[ \frac{5}{y} - y. \]
Again, we can express \(y\) as a fraction:
\[ y = \frac{y^2}{y}. \]
Now we can combine:
\[ \frac{5}{y} - \frac{y^2}{y} = \frac{5 - y^2}{y}. \]
Step 3: Rewrite the complex fraction
Now we substitute the simplified numerator and denominator back into the complex fraction:
\[ \frac{\frac{6 + y^2}{y}}{\frac{5 - y^2}{y}}. \]
Step 4: Simplify the complex fraction
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{6 + y^2}{y} \cdot \frac{y}{5 - y^2} = \frac{6 + y^2}{5 - y^2}. \]
Final Result:
The simplified form of the original complex fraction is:
\[ \frac{6 + y^2}{5 - y^2}. \]