Cramer's rule gives the solutions to a linear system using determinants. Specifically,
x = det(A_1)/det(A)
y = det(A_2)/det(A)
A represent the coefficient matrix and A_1 and A_2 represent the A matrix with the ith column substituted with the vector b.
Then,
x = det(A_1)/det(A) = [tex]\left|\begin{array}{ccc}0&1\\22&4\end{array}\right|[/tex]/ [tex]\left|\begin{array}{ccc}2&1\\-3&4\end{array}\right|[/tex]= (0(4)-(1)(22)) / (2(4)-(1)(-3)) = -2
y = det(A_2)/det(A) =[tex]\left|\begin{array}{ccc}2&0\\-3&22\end{array}\right|[/tex]/ [tex]\left|\begin{array}{ccc}2&1\\-3&4\end{array}\right|[/tex]=(2(22)-(0)(-3)) / (2(4)-(1)(-3)) = 4
