Alignment in math equations

You can use

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{alignat*}{3}
  x_{1}'(t) &=  &x_{1}(t) &{}+{}& 2&x_{2}(t)\\
  x_{2}'(t) &= 3&x_{1}(t) &     &  &
\end{alignat*}

\end{document}

Notice that I have added three alignment points in case you need to vertically align the second factor on the right-hand side too.

I hope it's clear how to extend this to more than three alignment points. (Fore n alignment points, 2n-1 &s are needed.)

P.S. Remember the use of {} to get the correct spacing around the +.

Update

In case you don't need more than two alignment points, you can use the following:

\begin{alignat*}{2}
  x_{1}'(t) &={}&  &x_{1}(t) + 2x_{2}(t)\\
  x_{2}'(t) &={}& 3&x_{1}(t)
\end{alignat*}

Especially if your system of equations is fairly small -- as is the case with the example you've posted -- you could simply use \phantom directives: They insert whitespace equivalent to their arguments. (However, if the system of equations gets larger, it may be worth incurring the overhead associated with the other proposed methods.)

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
    \begin{split}
        x_1'(t) &= \phantom{3}x_1(t)+2 x_2(t) \\
        x_2'(t) &= 3 x_1(t)
    \end{split}
\end{equation}
\end{document}

Simple array:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
%    \begin{split}
\begin{array}{r@{\;}r@{}l}
        x_1'(t) =& x_1(t)&{}+2 x_2(t) \\
        x_2'(t) =& 3 x_1(t)&
%    \end{split}
\end{array}
\end{equation}
\end{document}

Probably vertical spacing needs some correction by adding a suitable length after \\. BTW: one can induce that an image in your example is an effect of just array, but without the correction of \arraycolsep. The spacing around + and = signs is too big.