Kifilideen’s Rule: A Groundbreaking Approach to Solving Product Index Nonlinear Simultaneous Equations with Two, Three, Four, and 𝒏 Variables

Many real-world and social problems give rise to product index nonlinear simultaneous equations, which traditional solution methods, whether direct or iterative, fail to solve effectively. These equations appear in diverse fields, necessitating a specialized approach to ensure accurate and efficient solutions. Existing methods for solving linear simultaneous equations are insufficient for addressing product index nonlinear simultaneous equations. Given their frequent occurrence in practical applications, there is a need to develop a systematic and efficient approach tailored to solving these complex equations. This study introduces Kifilideen’s Rule, a groundbreaking approach for solving product index nonlinear simultaneous equations involving two, three, four, and 𝑛v ariables. The objective is to establish a reliable, efficient, and mathematically rigorous method that overcomes the limitations of conventional techniques. The formulation of Kifilideen’s Rule is based on elimination methods, the laws of indices, and matrix theory, including cofactors and determinants. These mathematical principles are systematically applied to construct a general framework for solving product index nonlinear simultaneous equations. The proposed method successfully establishes a general solution for 𝑛 −variable product index nonlinear simultaneous equations, demonstrating its superiority over traditional techniques in handling multiplication-based systems. Kifilideen’s Rule offers an efficient, precise, and practical solution for complex nonlinear simultaneous equations. Its direct and exact approach makes it highly valuable for students, researchers, and professionals working in fields that require highly accurate mathematical modeling.