Philraphsonking Correlation Method for solving Simultaneous Equations Incorporating Problem-based Learning and Context-based Learning in Contrast with Cramer's Rule and inversion Method

Abstract

This paper introduces the PHILRAPHSONKING Correlation Method, a novel approach to solving systems of linear equations. Developed by Philip Raphael and refined by Prof. R. W. Gimba, this method diverges from traditional techniques such as matrix inversion and Cramer's Rule by incorporating a correlation-based framework. The approach categorizes systems into three distinct cases based on the nature of derived values—whole numbers, similar decimals, or dissimilar decimals—and provides structured charts detailing the problem, process, and product levels. Each case follows a stepwise process of comparison, division, and trial evaluation to determine "true values," offering a new interpretive layer: if correlation is present among the equations, the solutions are considered true; if absent, the system may still have mathematical solutions under other methods, but lacks relational coherence. The method is particularly accessible due to its simplicity, iterative nature, and visual clarity through tabular representation. Notably, it can still provide insights even when traditional methods fail—such as in cases where the determinant of the coefficient matrix is zero. This makes it a viable alternative for educational purposes and small scale business applications, especially in production planning and resource allocation. However, its reliance on correlation means it may not always yield solutions, and the outcomes vary based on the initial characteristics of the equations. Potential users— educators, business analysts, and students—are encouraged to adopt this method for its pedagogical clarity and real-world applicability. Further research is recommended to explore its utility in broader domains, such as engineering, logistics, and sustainability planning.