The randomized Kaczmarz method, the randomized Gauss-Seidel method, the randomized extended Kaczmarz method, and the randomized extended Gauss-Seidel method are four efficient randomized iteration methods for solving large-scale systems of linear equations. We point out that the randomized extended Gauss-Seidel method is mathematically equivalent to the randomized extended Kaczmarz method, and find the intrinsic connection between the randomized-Kaczmarz-type methods and the randomized-Gauss-Seidel-type methods. In addition, by classifying a linear system into four cases according to its consistency and the column-rank of its coefficient matrix, we give the preferred method among the four randomized iteration methods in each case. With these results, we can make full use of the appropriate randomized iteration method to solve the linear system and obtain new efficient randomized iteration methods.