<p>Elementary operations with fuzzy numbers are essential for implementing predictive fuzzy modeling. Although operations with triangular fuzzy numbers are well-established, they are not adequately studied for predictive modeling with Gaussian fuzzy numbers, which are more flexible than their triangular counterparts. This study focuses on the elementary operations with Gaussian fuzzy numbers by incorporating principles from distribution theory and their application for linear regression modeling. Previously proposed approaches convert Gaussian fuzzy numbers into interval format to perform interval arithmetic or utilize membership functions and grades. These approaches lead to the loss of location and variance information, which are highly valuable in building predictive models. We aim to preserve the location and variance information of Gaussian fuzzy numbers while conducting elementary operations. This approach allows us to use the proposed elementary operations in a wide range of predictive models in practice. We illustrate their practical application with examples and provide proof of their essential properties. We showcase the use of operations for fuzzy linear regression modeling with two datasets and compare the estimation accuracies of Gaussian and triangular fuzzy numbers. Overall, we observe better estimation accuracy with Gaussian fuzzy numbers.</p>