The align environment is used for two or more equations when vertical alignment is desired; usually binary relations such as equal signs are aligned (see Table 2.1).
To have several equation columns side-by-side, use extra ampersands to separate the columns:
\begin{align}
x&=y & X&=Y & a&=b+c\\
x'&=y' & X'&=Y' & a'&=b\\
x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
\end{align}
Line-by-line annotations on an equation can be done by judicious
application of text inside an align environment:
\begin{align}
x& = y_1-y_2+y_3-y_5+y_8-\dots
&& \text{by \eqref{eq:C}}\\
& = y'\circ y^* && \text{by \eqref{eq:D}}\\
& = y(0) y' && \text {by Axiom 1.}
\end{align}
A variant environment alignat allows the space between equation
columns to be explicitly specified. Here the number of equation
columns must also be specified (where the number of `columns' is
calculated as 
with 
number of & markers on any line).
\begin{alignat}{2}
x& = y_1-y_2+y_3-y_5+y_8-\dots
&\quad& \text{by \eqref{eq:C}}\\
& = y'\circ y^* && \text{by \eqref{eq:D}}\\
& = y(0) y' && \text {by Axiom 1.}
\end{alignat}