Although Cramer's Rule provides an elegant way to solve a consistent system of equations, there is arguably no good reason to use it in practice; it's just too computationally expensive compared to Gaussian elimination/back substitution.
The only case in which use of Cramer's Rule might be more justified is if you have a large system of equations, but are interested only in the value of one of the unknowns. In this case, you're computing only two determinants. On the other hand, one of the more computationally efficient ways of computing a determinant involves using Gaussian elimination to cut down on the number of multiplies (but the details of this belong in another node). So you're really not buying yourself anything.
In my opinion, this is true even for hand computations, but your preferences may vary.