How does this fraction $\frac{1}{y} = \frac{1}{a}$ become $y = a$? I was learning SET. To solve a problem for Y at some point I stopped right here  $\frac{1}{y} = \frac{1}{a}$, and I couldn't go further. Then I opened the solution book and there I found that everything I did was right up-to this $\frac{1}{y} = \frac{1}{a}$ line. And I also found that, the solution book solved the problem just adding this line $y = a$ after my line ($\frac{1}{y} = \frac{1}{a}$), but there is no any explicit explanation. Can anybody expert in mathematics please tell me how this fraction ($\frac{1}{y} = \frac{1}{a}$) has been converted to $y = a$? Thanks in advance guys! 

There’s plenty of ways. You can take reciprocals on both sides, since $r=s$ implies $\frac{1}{r} = \frac{1}{s}$ when they are not zero. So you get
$$\frac{1}{\quad\frac{1}{y}\quad} = \frac{1}{\quad\frac{1}{a}\quad}$$
and this is the same as $y=a$.
Or you can multiply through by $ay$ and then cancel. You get
$$\begin{align*}
\frac{1}{y} &= \frac{1}{a}\\
\frac{ay}{y} &= \frac{ay}{a}\\
\frac{a}{1} &= \frac{y}{1}\\
a &= y
\end{align*}$$
