Linear systems often involve coefficients that are uncertain or imprecise due to inherent variability and vagueness in the data. In scenarios where only approximate or vague knowledge of the system parameters is available, traditional fuzzy logic is commonly employed. However, conventional fuzzy logic may be inadequate when defining a membership degree with a single, precise value proves difficult. In such cases, Single-Valued Trapezoidal Neutrosophic Numbers (SVTrNNs) offer a more suitable framework, as they account for indeterminacy, alongside truth and falsity. The solution of Single-Valued Trapezoidal Neutrosophic Linear Equations (SVTrNLEs) was explored in this study using an embedding approach. The approach reformulates the SVTrNLEs into an equivalent crisp linear system, enabling the application of conventional solution methods. The solution was then obtained using either the matrix inversion method or the gradient descent optimization algorithm implemented in PyTorch. The robustness and adaptability of gradient-based optimization techniques were thoroughly assessed. The learning process minimizes the residual error iteratively, with convergence behaviour and numerical stability analyzed across various parameter configurations. The results demonstrate rapid convergence, proximity to exact solutions, and significant robustness to parameter variability, highlighting the efficacy of gradient descent for solving uncertain linear systems. These findings provide a foundation for the extension of gradient-based methods to more complex systems and broader applications. Furthermore, the existence and uniqueness of the neutrosophic solution to an $n\times n$ linear system were rigorously analyzed, with numerical examples provided to assess the reliability and efficiency of the proposed methods.
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