Notice that (A-x)*y=B (I changed the names of the variables for simplicity) can be written component-wise as (Ai-xi)*yj = Bij, where i and j denote the indexes. This equation is then showing that, for i constant, Bij / yj does not depends on j. In other words, Bij = ki * yj, where ki is a constant that depends on i. Thus, B must be the result of an outer product between two vectors for the equation to be solvable. To find the vectors that generate B, you can use singular value decomposition, which gives you B = U * D * V^T, being D a diagonal matrix. If D has only one value different from zero, then B is effectively an outer product. You can then construct the vectors k and y as k = alpha * u and y = beta * v, where u and v are the columns of U and V associated with the nonzero singular value, and alpha and beta are constants which product must equal the nonzero singular value. x can then be found by computing x = A-k. Notice that there are infinite solutions, since you can choose alpha arbitrarily (beta being bound by the relation to the nonzero singular value).
Hope this helps.
