In this paper, which deals essentially with various summability concepts and summability techniques and shows how these concepts and techniques lead to a number of approximation results, we have used the new concept of weighted A-summability proposed by Mohiuddine (2016) and introduced the notions of statistically weighted B-summability and weighted B-statistical convergence with respect to the weighted regular method. We then prove a Korovkin type approximation theorem for functions of two variables and also present an example via generalized Meyer-Konig and Zeller type operator to show that our proposed method is stronger than its classical and statistical versions. Furthermore, the rate of convergence of approximating positive linear operators are estimated by means of the modulus of continuity and some Voronovskaja type results are investigated. Computational and geometrical approaches to illustrate some of our results are also presented. (C) 2017 Elsevier Inc. All rights reserved.