Properties of Cramer's Rule in Matrix Linear Systems Given a system of the form 
$$\begin{cases}
ax+by=s \\ cx+dy=t \\
\end{cases}$$
I acknowledge, assuming $a$, $b$, $c$, and, $d$ are real numbers and $ad-bc\ne0$, that we can say 
$$x=\frac{\begin{vmatrix} s & b \\ t & d\end{vmatrix}}{\begin{vmatrix} a & b \\ c & d\end{vmatrix}}, y=\frac{\begin{vmatrix} a & s \\ c & t\end{vmatrix}}{\begin{vmatrix} a & b \\ c & d\end{vmatrix}}
; \text{in other words,} \ x=\frac{Dx}{D} \text{and} \ y=\frac{Dy}{D}$$
My question is: Why is the the $y$ column and the constant column swapped in $Dx$ compared to $Dy$?Whenever I solve systems using Cramer's rule I always have to remind myself to make this swap when finding the minors of an $m \times n$ discriminant, which is frustrating.
 A: In matrix form,
$$\begin{bmatrix}a & b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}s\\t\end{bmatrix}$$
When $ax+by = s$ and $cx+dy = t$, the values of $x$ and $y$ can be found by cross-multiplication or by substitution method as follows.

$$x = \frac{ds-bt}{ad-bc} \ , \ y = \frac{at-cs}{ad-bc}$$

Now, the determinant of the matrices,
$\det\begin{bmatrix}a & b\\c&d \end{bmatrix}= ad-bc$
$\det\begin{bmatrix}s & b\\t&d \end{bmatrix}= ds-bt$
and $\det\begin{bmatrix}a & s\\c&t\end{bmatrix}= at-cs$
So,

$$x = \frac{\begin{vmatrix}s & b\\t&d \end{vmatrix}}{\begin{vmatrix}a & b\\c&d \end{vmatrix}}\ , \ y = \frac{\begin{vmatrix}a & s\\c&t\end{vmatrix}}{\begin{vmatrix}a & b\\c&d \end{vmatrix}}$$

Due to this, the columns are swapped.
To remember, there's a simple rule.

$\bullet$ If you need to find $x$ replace the column consisting of the co-efficients of $x$ by the column in the RHS.
$\bullet$ If you need to find $y$ replace the column consisting of the co-efficients of $y$ by the column in the RHS.

