Aligned equations

Here is how to TeX up an a``aligned equation'':

$$   % this starts the displayed math mode
\begin{array}{rcl}   % r for aligned right, c for center, and l (L) for aligned left
  b & =  &  b(aa^{-1}) \\       % & separates the columns; \\ starts a new line
  & =  & (ba)a^{-1}  \\
  & =  &   (ca)a^{-1} \qquad\hbox{by associativity}  \\
                                % you won't do this much, but here's how to add space and words
& =  &   c(aa^{-1}) \\
& =  &   c. % note the punctuation; we don't need a new line so no \\
\end{array}
$$