Explanation of this step in a modular arithmetic problem 
The multiplicative inverse of $5$ is $7$, when using mod $34$.
  $$\begin{align*}
5\cdot x&=3\\[0.1in]
7\cdot 5\cdot x &=7\cdot 3\\[0.1in]
1\cdot x &=7\cdot 3\\[0.1in]
x&=21
\end{align*}$$

I don't understand this part:
$$\begin{align*}
7\cdot 5\cdot x &=7\cdot 3\\[0.1in]
1\cdot x &=7\cdot 3
\end{align*}$$
How is 7*5*x the same as 1*x?

Recall that $a\equiv b\bmod n$ precisely when $n$ divides the difference $a-b$. Therefore, we have
$$35\equiv 1\bmod 34.$$
It is also true that if $a\equiv b \bmod n$, then $ac\equiv bc\bmod n$ for any $c$. Therefore, whatever $x$ is,
$$35x\equiv x\bmod 34,$$
so that we can go from
$$35x\equiv 21\bmod 34$$
to
$$x\equiv 21\bmod 34.$$
