First, we need to multiply the first row by 2 and add it to the second row to eliminate the coefficient of 2 in the second row, as follows:
$$\left[\begin{array}{rr|r}
1 & -2 & -3 \\
2 & 3 & -1
\end{array}\right] \quad \xrightarrow{R_{2}-2R_{1}} \quad
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