Consider the system \(Ax = b\) where
\(A = \begin{bmatrix} 13 & 5 \\ 8 & 3 \end{bmatrix}\)
and \(b = \begin{bmatrix} 19 \\ 12\end{bmatrix}\).
What must be the value of \(x_2\)?
The answer is \(-4\).
By Cramer's rule,
\[x_2 = \frac{\left|\begin{array}{cc} 13 & 19\\ 8 & 12\end{array}\right|}
{\det(A)} =
\frac{13 \cdot 12 - 19\cdot 8}{13\cdot 3 - 5\cdot 8} =
\frac{156-152}{39-40} = \frac{4}{-1} = -4.\]
(Using Cramer's rule here seemed simpler than
solving by Gauss-Jordan elimination, for example.)