Question

Simplify the complex fraction.
StartStartFraction StartFraction 6 Over y EndFraction plus y OverOver StartFraction 5 Over y EndFraction minus y EndEndFraction

Answer 1

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:

\[
\frac{6 + y^2}{5 - y^2}.
\]