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\[
x_j  = \frac{{\det \left( {\begin{array}{*{20}c}
   {a_{1,1} } &  \cdots  & {a_{1,j - 1} } & {b_1 } & {a_{1,j + 1} } &  \cdots  & {a_{1,n} }  \\
    \vdots  & {} &  \vdots  &  \vdots  &  \vdots  & {} &  \vdots   \\
   {a_{n,1} } &  \cdots  & {a_{n,j - 1} } & {b_n } & {a_{n,j + 1} } &  \cdots  & {a_{n,n} }  \\
\end{array}} \right)}}{{\det \left( A \right)}}
\]