How to use \alignedat? (Aligning a system of linear equations)

Question

I've seen the post on aligning a system of equations with systeme and array; is there a way to do something similar with alignedat?

I am using KaTeX and it has array (but without arraycolsep support at present) and align and alignedat. I can get pretty close to what I want with alignedat, but I don't understand how it works.

Page 8 of the amsmath user's guide mentions alignat and there's a later reference to alignedat with no explanation of how it differs from alignat... can anyone help?

Here is a sample document I am using to help understand.

\documentclass{article}
\usepackage{systeme, array, amsmath}
\begin{document}
\systeme*{
  X_\alpha=\frac{2}{3}X_A - \frac{1}{3}X_B- \frac{1}{3}X_C,
  X_\beta=\frac{1}{\sqrt{3}}X_B - \frac{1}{\sqrt{3}}X_C
}
\par
This is the best I can do within LaTeX:
[
\setlength\arraycolsep{0pt}
\begin{array}{ r @{{}={}} r  >{{}}c<{{}} cr  >{{}}c<{{}}  cr }
  X_\alpha &\frac{2}{3}X_A &-& \frac{1}{3}&X_B       &-& \frac{1}{3}&X_C \[2pt]
  X_\beta  &               & &\frac{1}{\sqrt{3}}&X_B &-& \frac{1}{\sqrt{3}}&X_C
\end{array}
]
With no arraycolsep:
[
\begin{array}{ r @{{}={}} r  >{{}}c<{{}} cr  >{{}}c<{{}}  cr }
  X_\alpha &\frac{2}{3}X_A &-& \frac{1}{3}&X_B       &-& \frac{1}{3}&X_C \[2pt]
  X_\beta  &               & &\frac{1}{\sqrt{3}}&X_B &-& \frac{1}{\sqrt{3}}&X_C
\end{array}
]
Alignedat:
[
\begin{alignedat}{4}
   X_\alpha &=\frac{2}{3}X_A &-& \frac{1}{3}X_B &- \frac{1}{3}&X_C \
   X_\beta &= && \frac{1}{\sqrt{3}}X_B &- \frac{1}{\sqrt{3}}&X_C
\end{alignedat}
]
\end{document}