How to solve the system of linear equations to recover original file using "Erasure Coding"?
The original data are $X_1$ and $X_2$.  The shares are

\begin{align*}
  A_1 &amp;= X_1, \\
  A_2 &amp;= X_2, \\
  A_3 &amp;= X_1 + X_2 \\
  A_4 &amp;= X_1 + 2 X_2.
\end{align*}

You store the shares $A_1$, $A_2$, $A_3$, and $A_4$ on disk, or over the internet and through the hills on someone else's disk, or buried in a tree somewhere for safekeeping.  If a forest fire takes out the tree and someone else's house is washed away in a flood (or, if a malicious storage server tries to modify one of the shares and you detect it because you're using cryptography and throw that modified share away), but you still have two of the four shares, say $A_2$ and $A_4$, you can recover

\begin{align*}
  X_2 &amp;= A_2, \\
  X_1 &amp;= A_4 - 2 A_2,
\end{align*}

and get your original file $X_1 \mathbin\| X_2$ back.