If $A$ is square and $\det A\ne0$, the solution of $Ax=b$ has $x_j=\det(A_j(b))/\det A$, where $A_j(b)$ replaces column $j$ by $b$. For $2x+y=3$, $x+y=2$, the determinant is $1$ and the formula gives $x=y=1$. Cramer's rule is a theoretical identity, usually less efficient numerically than elimination.
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