If you read just a little farther down on the Wiki entry, it says $$x_i = \frac{\det A_i}{\det A}.$$
If $\det A = 0$, we're dividing by zero.
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More generally, if a matrix has a determinant equal to zero, it is called "singular." This means it is non-invertible, so you cannot compute $x = A^{-1}b$, because the inverse $A^{-1}$ does not exist.
There are a number of ways to show this, and this fact is actually fundamentally important in linear algebra, but the best way to see is to note that the inverse of a matrix is formed by a permutation of the elements, and the whole matrix is the multiplied by one over the determinant.
For instance
$$\begin{pmatrix} a & b \\\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\\ -c & a\end{pmatrix}$$
where $\det = ad-bc$.