Cramer's Rule is used to solve $Ax=B$, where $A$ is a $n\times n$ matrix and $B$ is a $n\times 1$ vector. And Cramer's Rule states that $x_k=\frac{|A(k;B)|}{|A|}$, where $A(k;B)$ is substituting the $k$th column of $A$ by $B$ vectors and $x_k$ is the $k$the coordinate in the solution vector.
I am wondering is there any similar versions for substituting the rows of A?
Consider the problem $$x^TC=D$$
Taking tranpose, we obtain $$C^Tx=D^T$$
Hence to solve for $x$, Cramer rules says that we can substitute columns of $D^T$ into columns of $C^T$.
This is equivalent to replacing the rows of $D$ into $rows$ of $C$.
Hence, the row version is meant to solve problem of the form of $$x^TC=D$$