Development of Kifilideen's Rule to Solve Multiplication of Bi-Indexes, Tri- Indexes and 𝑛-Indexes Simultaneous Equations of Two Variables, Three Variables and 𝑛 Variables Respectively

Research Article http://dx.doi.org/10.4314/mejs.v17i2.6 Open Access Momona Ethiopian Journal of Science (MEJS), V17(2):302-323,2025 ©CNCS, Mekelle University,ISSN:2220-184X Submitted: 3rd September 2024 Accepted: 1st October 2024 Published: 15th December 2025 © CNCS Mekelle University. This article is licensed under a Creative Commons Attribution 4.0 International License. This license enables re-users to distribute, remix, adapt, and build upon the material in any medium or format, so long as attribution is given to the creator. The license allows for commercial use. To view the details of this license, visit http:// creativecommons.org/ licenses/ by/4. 0/. CC: Creative Commons; BY: credit must be given to the creator. Development of Kifilideen’s Rule to Solve Multiplication of Bi-Indexes, TriIndexes and 𝑛-Indexes Simultaneous Equations of Two Variables, Three Variables and 𝑛 Variables Respectively Kifilideen L. Osanyinpeju* Department of Agricultural and Biosystems Engineering, Bells University, Ota, Ogun State, Nigeria (*amkifilideenosanyinpej[email protected]m, kosanyinpej[email protected].ng; https://orcid.org/0000-0002-89957559). ABSTRACT Kifilideen’s Geometric Matrix Progression Sequence of infinite and finite terms generates multiplication of triindexes or bi – indexes simultaneous equations of its components’ migration level value, migration step value, and first term. There is a need to develop a simple, short, effective, and standardized rule or model to tackle such a problem. This study develops Kifilideen’s Rule to solve the multiplication of biindexes, tri-indexes, and -indices simultaneous equations of two variables, three variables, and variables respectively. Elimination method, laws of indices, and cofactor and determinant of matrix were deployed in establishing, formulating, and modeling the Kifilideen’s Rule to solve multiplication of bi-indexes and tri-indexes simultaneous equations of two variables and three variables, respectively. The Kifilideen’s Rule was implemented in the form of a model in solving problems involving multiplication of bi-indexes and triindexes simultaneous equations of two variables and three variables, respectively. The general solution to solve the multiplication of index simultaneous equations of variables using Kifilideen’s rule was inaugurated. The Kifilideen’s Rule is in the form of a model to solve multiplication of bi-indexes and triindexes simultaneous equations of two variables and three variables, respectively, which was found to be effective, straightforward, reliable, interesting, easy, and intuitive to understand for students and beginners. Keywords: Kifilideen’s Rule, Simultaneous Equations, Multiplication of Bi-Indexes, Multiplication of TriIndexes, Kifilideen’s Geometric Matrix Progression Sequence. 1. INTRODUCTION The concept of simultaneous equations, also known as systems of linear equations, has a rich history with contributions from ancient civilizations and mathematicians which involve finding solutions for multiple equations with multiple variables, dates back thousands of years and spans various cultures (Grcar, 2011a; Khushbu and Poonia, 2021). The earliest known evidence of solving simultaneous equations dates back to ancient Babylonian mathematicians where clay tablets showed solutions to systems of linear equations. They used algebraic methods to solve systems of linear equations, often in the context of practical problems, such as determining the dimensions of fields or distribution of goods (Zara, 2008). The Babylonians utilized geometric Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 303 ISSN: 2220-184X methods and tables to solve linear and quadratic equations, including systems involving two equations (Burton, 2011). In ancient Egypt (1650 BCE), the Rhind Papyrus contained problems that involved solving systems of linear equations (Fribera, 2008). The ancient Greeks, particularly Diophantus (fl. 3rd century CE), made significant contributions to the field of algebra, including solving simultaneous equations (Rizos and Gkrekas, 2022). Diophantus of Alexandria contributed to the development of algebraic methods. Though Diophantus primarily dealt with single-variable equations, his work laid the groundwork for future developments in algebra (Sfard, 1995). During the Middle Ages, Arabic mathematicians such as Al-Khwarizmi (780-850 CE) and Al-Kindi (801-873 CE) made significant contributions to algebra, including solving simultaneous equations (Rahaman, 2022). Islamic mathematicians such as Al-Khwarizmi and Omar Khayyam expanded on Greek and Indian mathematics, further developing methods for solving equations (Baki, 1992). While their work primarily focused on quadratic equations, their contributions to algebra helped set the stage for later developments in simultaneous equations (Kabar, 2023). Al-Khwarizmi's book "AlKitab al-mukhtasar fi hisab al-jabrwa'l-muqabala" (The Compendious Book on Calculation by Completion and Balancing) introduced algebraic methods for solving linear and quadratic equations, including simultaneous equations. In the 16th century, Italian mathematician Girolamo Cardano (1501-1576) developed methods for solving systems of linear equations (Heeffer and Rothman, 2014). René Descartes (1596-1650) introduced the concept of coordinates and graphing, which laid the foundation for modern methods of solving simultaneous equations (Neovius, 2013). René Descartes and Pierre de Fermat were pivotal figures in the development of analytical geometry, which links algebra and geometry (Oaks, 2021). Descartes’ work on Cartesian coordinates provided a geometric interpretation of simultaneous equations, allowing for the visualization of solutions as points of intersection between curves or lines (Neovius, 2013). In the 18th century, Swiss mathematician Leonhard Euler (1707-1783) developed the method of substitution for solving systems of linear equations (Carmine, 2021). In the 19th century, German mathematician Carl Friedrich Gauss (1777-1855) developed the method of elimination for solving systems of linear equations, which is still widely used today (Grcar, 2011a). The 20th century saw the development of matrix theory and the introduction of the concept of linear independence, which further advanced the solution of simultaneous equations (Grcar, 2011b). Today, simultaneous equations are an essential tool in various fields, including physics, engineering, economics, and computer science, and are solved using a variety of methods, Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 304 ISSN: 2220-184X including Cramer’s rule, Gaussian elimination, Jacobian method, Gauss-Seidel method, LU decomposition, and matrix inversion (Samuel, 2011; Woollard , 2015; Osanyinpeju, 2024a). Key figures in the history of simultaneous equations include Diophantus (fl. 3rd century CE), AlKhwarizmi (780-850 CE), Girolamo Cardano (1501-1576), René Descartes (1596-1650), Leonhard Euler (1707-1783) and Carl Friedrich Gauss (1777-1855). These mathematicians, along with many others, have contributed to the development of methods for solving simultaneous equations, which has had a profound impact on various fields of science and engineering. Kifilideen’s Geometric Matrix Progression Sequence of infinite and finite terms generates multiplication of tri – indexes or bi – indexes simultaneous equations of its components migration level value, 𝑘, migration step value, 𝑖, and first term, 𝑓. The Kifilideen’s General Term Formula of the Kifilideen’s Geometric Matrix Progression Sequence of infinite and finite terms is of the form: 𝑇𝑛=𝑘𝑎𝑖𝑛−𝑚𝑓, (1) Where, 𝑇𝑛 is the 𝑛𝑡ℎ term,𝑘 is the migration level value, 𝑖 is the migration step value and 𝑓 is the first term of the Kifilideen’s Geometric Matrix Progression Sequence of infinite and finite terms (Osanyinpeju, 2021; Osanyinpeju, 2024b). The Kifilideen’s Geometric Matrix Progression Sequence of infinite and finite terms is the geometric versions of the Kifilideen’s Arithmetic Matrix Progression Sequence of infinite and finite terms (Osanyinpeju, 2022; Osanyinpeju, 2023). The Kifilideen’s Arithmetic Matrix Progression Sequence of infinite and finite terms are originated and generalized from Kifilideen’s Trinomial Theorem of infinite and finite terms respectively (Osanyinpeju, 2020a; Osanyinpeju, 2020b). There is need to develop a simple, short, effective and standardized rule, method or model to tackle multiplication of tri – indexes or bi – indexes simultaneous equations generated from Kifilideen’s Geometric Matrix Progression Sequence of infinite terms. Such developed model can be applied to solve mathematical or World problems which have the same pattern or form as the Kifilideen’s Geometric Matrix Progression Sequence of infinite terms. With reference to method to solve multiplication of bi-indexes, tri-indexes and 𝑛-indexes simultaneous equations of two variables, three variables and 𝑛 variables respectively, there is no available study. This study develops Kifilideen’s Rule to solve multiplication of bi-indexes, tri-indexes and 𝑛-indexes simultaneous equations of two variables, three variables and 𝑛 variables respectively. Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 305 ISSN: 2220-184X 2. MATERIALS AND METHODS Elimination method, laws of indices, and cofactor and determinant of matrix were deployed in establishing, formulating and modelling the Kifilideen’s Rule to solve multiplication of bi-indexes and tri-indexes simultaneous equations of two variables and three variables respectively. 2.1. Establishment of Kifilideen’s Rule to Solve Multiplication of Bi-Indexes Simultaneous Equations of Two Variables For a given multiplication of bi-indexes simultaneous equations of two variables, say 𝑎 𝑎𝑛𝑑 𝑏 such that: 𝑎𝑥𝑏𝑦=𝑚, (2) 𝑎𝑤𝑏𝑣=𝑛, (3) The generation of the Kifilideen’s Rule to solve multiplication of bi-indexes simultaneous equations of two variables is illustrated as follow: From (2) and (3), I have: 𝑎𝑥 𝑦𝑏=𝑚1 𝑦, (4) 𝑎𝑤 𝑣𝑏=𝑛1 𝑣, (5) To find 𝑎, eliminate 𝑏 by dividing (4) with (5), 𝑎𝑥 𝑦−𝑤 𝑣=𝑚1 𝑦𝑛−1 𝑣, 𝑎𝑥𝑣−𝑤𝑦 𝑦𝑣 =𝑚1 𝑦𝑛−1 𝑣, 𝑎=𝑚1 𝑦(𝑦𝑣 𝑥𝑣−𝑤𝑦)𝑛−1 𝑣(𝑦𝑣 𝑥𝑣−𝑤𝑦), 𝑎=𝑚(𝑣 𝑥𝑣−𝑤𝑦)𝑛(−𝑦 𝑥𝑣−𝑤𝑦), (6) From (2) and (3), I have: 𝑎𝑏𝑦 𝑥=𝑚1 𝑥, (7) 𝑎𝑏𝑣 𝑤=𝑛1 𝑤, (8) To find 𝑏, eliminate 𝑎 by dividing (7) with (8), 𝑏𝑦 𝑥−𝑣 𝑤=𝑚1 𝑥𝑛−1 𝑤, 𝑏𝑦𝑤−𝑣𝑥 𝑥𝑤 =𝑚1 𝑥𝑛−1 𝑤, 𝑏=𝑚1 𝑥(𝑥𝑤 𝑦𝑤−𝑣𝑥)𝑛−1 𝑤(𝑥𝑤 𝑦𝑤−𝑣𝑥), 𝑏=𝑚(𝑤 𝑦𝑤−𝑣𝑥)𝑛(−𝑥 𝑦𝑤−𝑣𝑥), 𝑏=𝑚(−𝑤 𝑣𝑥−𝑦𝑤)𝑛(𝑥 𝑣𝑥−𝑦𝑤) , (9) Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 306 ISSN: 2220-184X From (2) and (3), the determinant of matrix of the input index system is represented as ∆𝑘𝑖𝑓 and is given as: ∆𝑘𝑖𝑓=|𝑥 𝑦 𝑤 𝑣|=𝑥𝑣−𝑤𝑦, (10) Also, let the matrix of the input index system represent 𝑘𝑖𝑓, so I have: 𝑘𝑖𝑓=(𝑥 𝑦 𝑤 𝑣)=(𝑑11 𝑑12 𝑑21 𝑑22), The components of the cofactor of 𝑘𝑖𝑓 are given as: 𝑘𝑖𝑓11 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑11 =𝑣, (11) 𝑘𝑖𝑓12 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑12 =−𝑤, (12) 𝑘𝑖𝑓21 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑21 =−𝑦, (13) 𝑘𝑖𝑓22 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑22 =𝑥, (14) Inputting (10), (11), (12), (13), and (14) in (6), I have: 𝑎=𝑚(𝑘𝑖𝑓11 ∆𝑘𝑖𝑓)𝑛(𝑘𝑖𝑓21 ∆𝑘𝑖𝑓), Inputting (10), (11), (12), (13), and (14) in (9), I have: 𝑏=𝑚(𝑘𝑖𝑓12 ∆𝑘𝑖𝑓)𝑛(𝑘𝑖𝑓22 ∆𝑘𝑖𝑓), Generally, the Kifilideen’s Rule to solve multiplication of bi-indexes simultaneous equations of two variables, say 𝑎 𝑎𝑛𝑑 𝑏 such that: 𝑎𝑥𝑏𝑦=𝑚 , 𝑎𝑤𝑏𝑣=𝑛 is given as: 𝑎=𝑚(𝑘𝑖𝑓11 ∆𝑘𝑖𝑓)𝑛(𝑘𝑖𝑓21 ∆𝑘𝑖𝑓) , (15) 𝑏=𝑚(𝑘𝑖𝑓12 ∆𝑘𝑖𝑓)𝑛(𝑘𝑖𝑓22 ∆𝑘𝑖𝑓) , (16) 2.2. Inauguration of Kifilideen’s Rule to Solve Multiplication of Tri-Indexes Simultaneous Equations of Three Variables For a given multiplication of tri-indexes simultaneous equations of two variables, say 𝑎,𝑏 𝑎𝑛𝑑 𝑐 such that: 𝑎𝑒𝑏𝑓𝑐𝑔=𝑚 , (17) 𝑎𝑟𝑏𝑠𝑐𝑡=𝑛 , (18) 𝑎𝑥𝑏𝑦𝑐𝑧=𝑝 , (19) Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 307 ISSN: 2220-184X The generation of the Kifilideen’s Rule to solve multiplication of tri-indexes simultaneous equations of three variables is illustrated below. From (17), I have: 𝑎𝑒 𝑔𝑏𝑓 𝑔𝑐=𝑚1 𝑔 , (20) From (18), I have: 𝑎𝑟 𝑡𝑏𝑠 𝑡𝑐=𝑛1 𝑡 , (21) And from (19), I have: 𝑎𝑥 𝑧𝑏𝑦 𝑧𝑐=𝑝1 𝑧 , (22) To eliminate 𝑐, divide (20) with (21) and (21) with (22), so we have: Dividing (20) with (21), so I have 𝑎𝑒 𝑔−𝑟 𝑡𝑏𝑓 𝑔−𝑠 𝑡=𝑚1 𝑔𝑛−1 𝑡 , 𝑎𝑒𝑡−𝑔𝑟 𝑔𝑡 𝑏𝑓𝑡−𝑔𝑠 𝑔𝑡 =𝑚1 𝑔𝑛−1 𝑡 , (23) 𝑎𝑒𝑡−𝑔𝑟 𝑓𝑡−𝑔𝑠𝑏=𝑚 𝑡 𝑓𝑡−𝑔𝑠𝑛−𝑔 𝑓𝑡−𝑔𝑠 , (24) Dividing (21) with (22), so I have 𝑎𝑟 𝑡−𝑥 𝑧𝑏𝑠 𝑡−𝑦 𝑧=𝑛1 𝑡𝑝−1 𝑧 , 𝑎𝑟𝑧−𝑡𝑥 𝑡𝑧 𝑏𝑠𝑧−𝑦𝑡 𝑡𝑧 =𝑛1 𝑡𝑝−1 𝑧 , (25) 𝑎𝑟𝑧−𝑡𝑥 𝑠𝑧−𝑦𝑡𝑏=𝑛 𝑧 𝑠𝑧−𝑦𝑡𝑝−𝑡 𝑠𝑧−𝑦𝑡 , (26) To find 𝑎, eliminate 𝑏 by dividing (24) with (26), so I have: 𝑎𝑒𝑡−𝑔𝑟 𝑓𝑡−𝑔𝑠−𝑟𝑧−𝑡𝑥 𝑠𝑧−𝑦𝑡 =𝑚 𝑡 𝑓𝑡−𝑔𝑠𝑛−𝑔 𝑓𝑡−𝑔𝑠−𝑧 𝑠𝑧−𝑦𝑡𝑝𝑡 𝑠𝑧−𝑦𝑡 , 𝑎(𝑒𝑡−𝑔𝑟)(𝑠𝑧−𝑦𝑡)−(𝑟𝑧−𝑡𝑥)(𝑓𝑡−𝑔𝑠) (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)=𝑚 𝑡 𝑓𝑡−𝑔𝑠𝑛−𝑔(𝑠𝑧−𝑦𝑡)−𝑧(𝑓𝑡−𝑔𝑠) (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)𝑝𝑡 𝑠𝑧−𝑦𝑡 , 𝑎𝑒𝑡𝑠𝑧−𝑒𝑡2𝑦−𝑔𝑟𝑠𝑧+𝑔𝑟𝑡𝑦−(𝑟𝑧𝑡𝑓−𝑟𝑧𝑔𝑠−𝑥𝑡2𝑓+𝑡𝑥𝑔𝑠) (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)=𝑚 𝑡 𝑓𝑡−𝑔𝑠𝑛−𝑔𝑠𝑧+𝑔𝑡𝑦−𝑧𝑓𝑡+𝑧𝑔𝑠 (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)𝑝𝑡 𝑠𝑧−𝑦𝑡 , 𝑎𝑒𝑡𝑠𝑧−𝑒𝑡2𝑦−𝑔𝑟𝑠𝑧+𝑔𝑟𝑡𝑦−𝑟𝑧𝑡𝑓+𝑟𝑧𝑔𝑠+𝑥𝑡2𝑓−𝑡𝑥𝑔𝑠 (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)=𝑚 𝑡 𝑓𝑡−𝑔𝑠𝑛𝑔𝑡𝑦−𝑧𝑓𝑡 (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)𝑝𝑡 𝑠𝑧−𝑦𝑡 , 𝑎𝑒𝑡𝑠𝑧−𝑒𝑡2𝑦+𝑔𝑟𝑡𝑦−𝑟𝑧𝑡𝑓+𝑥𝑡2𝑓−𝑡𝑥𝑔𝑠 (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)=𝑚 𝑡 𝑓𝑡−𝑔𝑠𝑛𝑡(𝑔𝑦−𝑧𝑓) (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)𝑝𝑡 𝑠𝑧−𝑦𝑡 , 𝑎𝑡(𝑒𝑠𝑧−𝑒𝑡𝑦+𝑔𝑟𝑦−𝑟𝑧𝑓+𝑥𝑡𝑓−𝑥𝑔𝑠) (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)=𝑚 𝑡 𝑓𝑡−𝑔𝑠𝑛𝑡(𝑔𝑦−𝑧𝑓) (𝑓𝑡−𝑔𝑠)(𝑠𝑧−𝑦𝑡)𝑝𝑡 𝑠𝑧−𝑦𝑡 , 𝑎𝑒𝑠𝑧−𝑒𝑡𝑦+𝑔𝑟𝑦−𝑟𝑧𝑓+𝑥𝑡𝑓−𝑥𝑔𝑠 =𝑚𝑠𝑧−𝑦𝑡𝑛𝑔𝑦−𝑧𝑓𝑝𝑓𝑡−𝑔𝑠 , 𝑎𝑒𝑠𝑧−𝑒𝑡𝑦−𝑓𝑠𝑟+𝑥𝑡𝑓+𝑔𝑟𝑦−𝑥𝑔𝑠 =𝑚𝑠𝑧−𝑦𝑡𝑛𝑔𝑦−𝑧𝑓𝑝𝑓𝑡−𝑔𝑠 , (27) Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 308 ISSN: 2220-184X From (17), (18), and (19), the determinant of matrix of the input index system is represented as ∆𝑘𝑖𝑓 and is given as: ∆𝑘𝑖𝑓=|𝑒 𝑓 𝑔 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧| (28a) ∆𝑘𝑖𝑓=𝑒|𝑠 𝑡 𝑦 𝑧|−𝑓|𝑟 𝑡 𝑥 𝑧|+𝑔|𝑟 𝑠 𝑥 𝑦| ∆𝑘𝑖𝑓=𝑒(𝑠𝑧−𝑦𝑡)−𝑓(𝑟𝑧−𝑥𝑡)+𝑔(𝑟𝑦−𝑠𝑥) , ∆𝑘𝑖𝑓=𝑒𝑠𝑧−𝑒𝑦𝑡−𝑓𝑧𝑟+𝑓𝑥𝑡+𝑔𝑟𝑦−𝑔𝑥𝑠 , (28b) Also, let the matrix of the input index system represent 𝑘𝑖𝑓, so I have: 𝑘𝑖𝑓=(𝑒 𝑓 𝑔 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧)=(𝑑11 𝑑12 𝑑13 𝑑21 𝑑22 𝑑23 𝑑31 𝑑32 𝑑33) , The components of the cofactor of 𝑘𝑖𝑓 are given as: 𝑘𝑖𝑓11 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑11=|𝑠 𝑡 𝑦 𝑧|=𝑠𝑧−𝑡𝑦 , (29) 𝑘𝑖𝑓12 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑12=−|𝑟 𝑡 𝑥 𝑧|=−(𝑟𝑧−𝑥𝑡)=𝑥𝑡−𝑟𝑧 , (30) 𝑘𝑖𝑓13 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑13=|𝑟 𝑠 𝑥 𝑦|=𝑟𝑦−𝑠𝑥, (31) 𝑘𝑖𝑓21 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑21 =−|𝑓 𝑔 𝑦 𝑧|=−(𝑓𝑧−𝑔𝑦)=𝑔𝑦−𝑓𝑧 , (32) 𝑘𝑖𝑓22 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑22 =|𝑒 𝑔 𝑥 𝑧|=𝑒𝑧−𝑔𝑥, (33) 𝑘𝑖𝑓23 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑23 =−|𝑒 𝑓 𝑥 𝑦|=−(𝑒𝑦−𝑓𝑥)=𝑓𝑥−𝑒𝑦, (34) 𝑘𝑖𝑓31 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑31 =|𝑓 𝑔 𝑠 𝑡|=𝑓𝑡−𝑔𝑠, (35) 𝑘𝑖𝑓32 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑32 =−|𝑒 𝑔 𝑟 𝑡|=−(𝑒𝑡−𝑟𝑔)=𝑟𝑔−𝑒𝑡, (36) 𝑘𝑖𝑓33 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑜𝑓𝑑33 =|𝑒 𝑓 𝑟 𝑠|=𝑒𝑠−𝑓𝑟, (37) Inputting (28), (29), (30), (31), (32), (33), (34), (35), (36), and (37) in (27), I have: 𝑎∆𝑘𝑖𝑓 =𝑚𝑘𝑖𝑓11𝑛𝑘𝑖𝑓21𝑝𝑘𝑖𝑓31, 𝑎=𝑚(𝑘𝑖𝑓11 ∆𝑘𝑖𝑓)𝑛(𝑘𝑖𝑓21 ∆𝑘𝑖𝑓)𝑝(𝑘𝑖𝑓31 ∆𝑘𝑖𝑓), (36) From (23), I have: Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 309 ISSN: 2220-184X 𝑎𝑒𝑡−𝑔𝑟 𝑔𝑡 𝑏𝑓𝑡−𝑔𝑠 𝑔𝑡 =𝑚1 𝑔𝑛−1 𝑡 , (23) 𝑎𝑏𝑓𝑡−𝑔𝑠 𝑒𝑡−𝑔𝑟 =𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛−𝑔 𝑒𝑡−𝑔𝑟, (37) From (25), I have: 𝑎𝑟𝑧−𝑡𝑥 𝑡𝑧 𝑏𝑠𝑧−𝑦𝑡 𝑡𝑧 =𝑛1 𝑡𝑝−1 𝑧 , (25) 𝑎𝑏𝑠𝑧−𝑦𝑡 𝑟𝑧−𝑡𝑥 =𝑛 𝑧 𝑟𝑧−𝑡𝑥𝑝−𝑡 𝑟𝑧−𝑡𝑥 , (38) To find𝑏, I eliminate 𝑎 by dividing (37) with (38), so I have: 𝑏𝑓𝑡−𝑔𝑠 𝑒𝑡−𝑔𝑟−𝑠𝑧−𝑦𝑡 𝑟𝑧−𝑡𝑥 =𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛−𝑔 𝑒𝑡−𝑔𝑟−𝑧 𝑟𝑧−𝑡𝑥𝑝𝑡 𝑟𝑧−𝑡𝑥 , 𝑏(𝑓𝑡−𝑔𝑠)(𝑟𝑧−𝑡𝑥)−(𝑒𝑡−𝑔𝑟)(𝑠𝑧−𝑦𝑡) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)=𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛−𝑔(𝑟𝑧−𝑡𝑥)−𝑧(𝑒𝑡−𝑔𝑟) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)𝑝𝑡 𝑟𝑧−𝑡𝑥 , 𝑏𝑓𝑡𝑟𝑧−𝑓𝑡2𝑥−𝑔𝑠𝑟𝑧+𝑔𝑠𝑡𝑥−(𝑒𝑡𝑠𝑧−𝑒𝑡𝑦𝑡−𝑔𝑟𝑠𝑧+𝑔𝑟𝑦𝑡) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)=𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛−𝑔𝑟𝑧+𝑔𝑡𝑥−𝑧𝑒𝑡+𝑧𝑔𝑟 (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)𝑝𝑡 𝑟𝑧−𝑡𝑥 , 𝑏𝑓𝑡𝑟𝑧−𝑓𝑡2𝑥+𝑔𝑠𝑡𝑥−𝑒𝑡𝑠𝑧+𝑒𝑡2𝑦−𝑔𝑟𝑦𝑡 (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)=𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛𝑔𝑡𝑥−𝑧𝑒𝑡 (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)𝑝𝑡 𝑟𝑧−𝑡𝑥 , 𝑏−𝑒𝑡𝑠𝑧+𝑒𝑡2𝑦−𝑔𝑟𝑡𝑦+𝑟𝑧𝑡𝑓−𝑥𝑡2𝑓+𝑡𝑥𝑔𝑠 (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)=𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛𝑡(𝑔𝑥−𝑧𝑒) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)𝑝𝑡 𝑟𝑧−𝑡𝑥 , 𝑏−𝑡(𝑒𝑠𝑧−𝑒𝑡𝑦+𝑔𝑟𝑦−𝑟𝑧𝑓+𝑥𝑡𝑓−𝑥𝑔𝑠) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)=𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛𝑡(𝑔𝑥−𝑧𝑒) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)𝑝𝑡 𝑟𝑧−𝑡𝑥 , 𝑏−𝑡(𝑒𝑠𝑧−𝑒𝑡𝑦+𝑔𝑟𝑦−𝑟𝑧𝑓+𝑥𝑡𝑓−𝑥𝑔𝑠) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)=𝑚 𝑡 𝑒𝑡−𝑔𝑟𝑛𝑡(𝑔𝑥−𝑧𝑒) (𝑒𝑡−𝑔𝑟)(𝑟𝑧−𝑡𝑥)𝑝𝑡 𝑟𝑧−𝑡𝑥 , 𝑏𝑒𝑠𝑧−𝑒𝑡𝑦+𝑔𝑟𝑦−𝑟𝑧𝑓+𝑥𝑡𝑓−𝑥𝑔𝑠 =𝑚−(𝑟𝑧−𝑡𝑥)𝑛−(𝑔𝑥−𝑧𝑒)𝑝−(𝑟𝑡−𝑔𝑟) , 𝑏𝑒𝑠𝑧−𝑒𝑡𝑦−𝑓𝑠𝑟+𝑥𝑡𝑓+𝑔𝑟𝑦−𝑥𝑔𝑠 =𝑚−(𝑟𝑧−𝑡𝑥)𝑛𝑒𝑧−𝑔𝑥𝑝−(𝑟𝑡−𝑔𝑟) , (39) Inputting (28), (29), (30), (31), (32), (33), (34), (35), (36), and (37) in (39), I have: 𝑏∆𝑘𝑖𝑓 =𝑚𝑘𝑖𝑓12𝑛𝑘𝑖𝑓22𝑝𝑘𝑖𝑓32, 𝑏=𝑚(𝑘𝑖𝑓12 ∆𝑘𝑖𝑓)𝑛(𝑘𝑖𝑓22 ∆𝑘𝑖𝑓)𝑝(𝑘𝑖𝑓32 ∆𝑘𝑖𝑓), (40) From (17), I have: 𝑎𝑒 𝑓𝑏𝑐𝑔 𝑓=𝑚1 𝑓 , (41) From (18), I have: 𝑎𝑟 𝑠𝑏𝑐𝑡 𝑠=𝑛1 𝑠 , (42) And from (19), I have: 𝑎𝑥 𝑦𝑏𝑐𝑧 𝑦=𝑝1 𝑦, (43) To eliminate 𝑏, divide (41) with (42) and (42) with (43), so we have: Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 310 ISSN: 2220-184X Dividing (41) with (42), so I have 𝑎𝑒 𝑓−𝑟 𝑠𝑐𝑔 𝑓−𝑡 𝑠=𝑚1 𝑓𝑛−1 𝑠 , 𝑎𝑒𝑠−𝑓𝑟 𝑓𝑠 𝑐𝑔𝑠−𝑓𝑡 𝑓𝑠 =𝑚1 𝑓𝑛−1 𝑠 , (44) 𝑎𝑐𝑔𝑠−𝑓𝑡 𝑒𝑠−𝑓𝑟 =𝑚 𝑠 𝑒𝑠−𝑓𝑟𝑛−𝑓 𝑒𝑠−𝑓𝑟 , (45) Dividing (42) with (43), so I have 𝑎𝑟 𝑠−𝑥 𝑦𝑐𝑡 𝑠−𝑧 𝑦=𝑛1 𝑠𝑝−1 𝑦 , 𝑎𝑟𝑦−𝑠𝑥 𝑠𝑦 𝑐𝑡𝑦−𝑠𝑧 𝑠𝑦 =𝑛1 𝑠𝑝−1 𝑦 , (46) 𝑎𝑐𝑡𝑦−𝑠𝑧 𝑟𝑦−𝑠𝑥 =𝑛 𝑦 𝑟𝑦−𝑠𝑥𝑝−𝑠 𝑟𝑦−𝑠𝑥 , (47) To find 𝑐, eliminate 𝑎 by dividing (45) with (47), so I have: 𝑐𝑔𝑠−𝑓𝑡 𝑒𝑠−𝑓𝑟−𝑡𝑦−𝑠𝑧 𝑟𝑦−𝑠𝑥 =𝑚 𝑠 𝑒𝑠−𝑓𝑟𝑛−𝑓 𝑒𝑠−𝑓𝑟−𝑦 𝑟𝑦−𝑠𝑥𝑝𝑠 𝑟𝑦−𝑠𝑥 , 𝑐(𝑔𝑠−𝑓𝑡)(𝑟𝑦−𝑠𝑥)−(𝑡𝑦−𝑠𝑧)(𝑒𝑠−𝑓𝑟) (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)=𝑚 𝑠 𝑒𝑠−𝑓𝑟𝑛−𝑓(𝑟𝑦−𝑠𝑥)−𝑦(𝑒𝑠−𝑓𝑟) (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)𝑝𝑠 𝑟𝑦−𝑠𝑥 , 𝑐𝑔𝑠𝑟𝑦−𝑔𝑠2𝑥−𝑓𝑡𝑟𝑦+𝑓𝑡𝑠𝑥−(𝑡𝑦𝑒𝑠−𝑡𝑦𝑓𝑟−𝑒𝑠2𝑧+𝑠𝑧𝑓𝑟) (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)=𝑚 𝑠 𝑒𝑠−𝑓𝑟𝑛−𝑓𝑟𝑦+𝑓𝑠𝑥−𝑦𝑒𝑠+𝑦𝑓𝑟 (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)𝑝𝑠 𝑟𝑦−𝑠𝑥 , 𝑐𝑔𝑠𝑟𝑦−𝑔𝑠2𝑥−𝑓𝑡𝑟𝑦+𝑓𝑡𝑠𝑥−𝑡𝑦𝑒𝑠+𝑡𝑦𝑓𝑟+𝑒𝑠2𝑧−𝑠𝑧𝑓𝑟 (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)=𝑚 𝑠 𝑒𝑠−𝑓𝑟𝑛𝑓𝑠𝑥−𝑦𝑒𝑠 (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)𝑝𝑠 𝑟𝑦−𝑠𝑥 , 𝑐𝑒𝑠2𝑧−𝑡𝑦𝑒𝑠+𝑔𝑠𝑟𝑦−𝑠𝑧𝑓𝑟+𝑓𝑡𝑠𝑥−𝑔𝑠2𝑥 (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)=𝑚 𝑠 𝑒𝑠−𝑓𝑟𝑛𝑠(𝑓𝑥−𝑦𝑒) (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)𝑝𝑠 𝑟𝑦−𝑠𝑥 , 𝑐𝑠(𝑒𝑠𝑧−𝑒𝑡𝑦+𝑔𝑟𝑦−𝑟𝑧𝑓+𝑥𝑡𝑓−𝑥𝑔𝑠) (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)=𝑚 𝑠 𝑒𝑠−𝑓𝑟𝑛𝑠(𝑓𝑥−𝑦𝑒) (𝑒𝑠−𝑓𝑟)(𝑟𝑦−𝑠𝑥)𝑝𝑠 𝑟𝑦−𝑠𝑥 , 𝑐𝑒𝑠𝑧−𝑒𝑡𝑦+𝑔𝑟𝑦−𝑟𝑧𝑓+𝑥𝑡𝑓−𝑥𝑔𝑠 =𝑚𝑟𝑦−𝑠𝑥𝑛𝑓𝑥−𝑦𝑒𝑝𝑒𝑠−𝑓𝑟 , 𝑐𝑒𝑠𝑧−𝑒𝑡𝑦−𝑓𝑠𝑟+𝑥𝑡𝑓+𝑔𝑟𝑦−𝑥𝑔𝑠 =𝑚𝑟𝑦−𝑠𝑥𝑛𝑓𝑥−𝑦𝑒𝑝𝑒𝑠−𝑓𝑟 , (48) Inputting (28), (29), (30), (31), (32), (33), (34), (35), (36), and (37) in (48), I have: 𝑐∆𝑘𝑖𝑓 =𝑚𝑘𝑖𝑓13𝑛𝑘𝑖𝑓23𝑝𝑘𝑖𝑓33 , 𝑐=𝑚(𝑘𝑖𝑓13 ∆𝑘𝑖𝑓)𝑛(𝑘𝑖𝑓23 ∆𝑘𝑖𝑓)𝑝(𝑘𝑖𝑓33 ∆𝑘𝑖𝑓) , (49) Generally, Kifilideen’s Rule or model to solve multiplication of tri-indexes simultaneous equations of three variables say 𝑎,𝑏𝑎𝑛𝑑 𝑐 of the form: 𝑎𝑒𝑏𝑓𝑐𝑔=𝑚 , 𝑎𝑟𝑏𝑠𝑐𝑡=𝑛 , 𝑎𝑥𝑏𝑦𝑐𝑧=𝑝 , is given as: Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 317 ISSN: 2220-184X The ∆𝑘𝑖𝑓= the determinant of matrix of the input index system ∆𝑘𝑖𝑓=|1 5 3 1 2 6 1 4 2| ∆𝑘𝑖𝑓=1|2 6 4 2|−5|1 6 1 2|+3|1 2 1 4| ∆𝑘𝑖𝑓=1(4−24)−5(2−6)+3(4−2), ∆𝑘𝑖𝑓=1(−20)−5(−4)+3(2), ∆𝑘𝑖𝑓=−20+20+6, ∆𝑘𝑖𝑓=6, 𝑘𝑖𝑓=the matrix of the input index system 𝑘𝑖𝑓=(153 126 142)=(𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33) The components of the cofactor of 𝑘𝑖𝑓 are given as: 𝑘𝑖𝑓11 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎11=|2 6 4 2|=4−24=−20, 𝑘𝑖𝑓12 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎12=−|1 6 1 2|=−(2−6)=4, 𝑘𝑖𝑓13 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎13 =|1 2 1 4|=4−2=2, 𝑘𝑖𝑓21 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎21 =−|5 3 4 2|=−(10−12)=2, 𝑘𝑖𝑓22 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎22 =|1 3 1 2|=2−3=−1, 𝑘𝑖𝑓23 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎23 =−|1 5 1 4|=−(4−5)=1, 𝑘𝑖𝑓31 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎31 =|5 3 2 6|=30−6=24, 𝑘𝑖𝑓32 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎32 =−|1 3 1 6|=−(6−3)=−3, 𝑘𝑖𝑓33 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎33 =|1 5 1 2|=2−5=−3 , Using Kifilideen’s Rule, I have: 𝑃𝑜=(432,000)(𝑘𝑖𝑓11 ∆𝑘𝑖𝑓)(1,458,000)(𝑘𝑖𝑓21 ∆𝑘𝑖𝑓)(72,000)(𝑘𝑖𝑓31 ∆𝑘𝑖𝑓), 𝑃𝑜=(432,000)(−20 6)(1,458,000)(2 6)(72,000)(24 6), 𝑃𝑜=((432,000)−20(1,458,000)2(72,000)24)1 6, Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 318 ISSN: 2220-184X 𝑃𝑜=(1.5625×1016)1 6, 𝑃𝑜=initial population of each of the bacteria=500, 𝑥=(432,000)(𝑘𝑖𝑓12 ∆𝑘𝑖𝑓)(1,458,000)(𝑘𝑖𝑓22 ∆𝑘𝑖𝑓)(72,000)(𝑘𝑖𝑓32 ∆𝑘𝑖𝑓), 𝑥=(432,000)(4 6)(1,458,000)(−1 6)(72,000)(−3 6), 𝑥=((432,000)4(1,458,000)−1(72,000)−3)1 6, 𝑥=(64)1 6, 𝑥=2 The factors, 𝑥 of the population growth =2 𝑦=(432,000)(𝑘𝑖𝑓13 ∆𝑘𝑖𝑓)(1,458,000)(𝑘𝑖𝑓23 ∆𝑘𝑖𝑓)(72,000)(𝑘𝑖𝑓33 ∆𝑘𝑖𝑓), 𝑦=(432,000)(2 6)(1,458,000)(1 6)(72,000)(−3 6), 𝑦=((432,000)2(1,458,000)1(72,000)−3)1 6, 𝑦=(729)1 6, 𝑦=3 The factors, 𝑦 of the population growth =3 (2) The initial salary of a staff in an organization is ₦𝑎. If the salary of the staff is increasing geometrically as the staff migrates from his initial organization to higher organizations due to the complexity of the nature of work carry out by the staff in the higher organizations. After the increase in salary of the staff by a factor of u for 4 times as the staff migrates to higher organizations, the staff is receiving ₦480,000. If after the increase in salary of the staff by a factor of u for 7 times as the staff migrates to higher organizations, the staff is receiving ₦3,840,000. Determine the (i) the initial salary of the staff, ₦𝑎 in the first organization, (ii) the factor, 𝑢, at which the salary of the staff is increasing. Solution The mathematical equations of the salary growth of staff are given as: 𝑎𝑢4=₦480,000, 𝑎𝑢7=₦3,840,000, The ∆𝑘𝑖𝑓= the determinant of matrix of the input index system ∆𝑘𝑖𝑓=|1 4 1 7|=7−4 , Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 319 ISSN: 2220-184X ∆𝑘𝑖𝑓=3 , 𝑘𝑖𝑓=the matrix of the input index system 𝑘𝑖𝑓=(1 4 1 7)=(𝑎11 𝑎12 𝑎21 𝑎22), The components of the cofactor of 𝑘𝑖𝑓 are given as: 𝑘𝑖𝑓11 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎11 =7, 𝑘𝑖𝑓12=𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎12 =−1, 𝑘𝑖𝑓21 =𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎21 =−4, 𝑘𝑖𝑓22=𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎22 =1, Using Kifilideen’s Rule, I have: ₦𝑎=(480,000)(𝑘𝑖𝑓11 ∆𝑘𝑖𝑓)(3,840,000)(𝑘𝑖𝑓21 ∆𝑘𝑖𝑓), ₦𝑎=(480,000)(7 3)(3,840,000)(−4 3) ₦𝑎=((480,000)7(3,840,000)−4)1 3 , ₦𝑎=(2.7×1013)1 3, ₦𝑎=₦30,000, The initial salary of the staff, ₦𝑎 in the first organization=₦30,000 𝑢=(480,000)(𝑘𝑖𝑓12 ∆𝑘𝑖𝑓)(3,840,000)(𝑘𝑖𝑓22 ∆𝑘𝑖𝑓), 𝑢=(480,000)(−1 3)(3,840,000)(1 3) 𝑢=((480,000)−1(3,840,000)1)1 3 , 𝑢=(8)1 3, 𝑢=2, The factor, 𝑢, at which the salary of the staff is increasing=2 Let us give the following theorem without proof Theorem. Let the following equation be the system. ∏𝑥1𝑖 𝑡1𝑖 𝑛 𝑖=1 =𝑏1, ∏𝑥1𝑖 𝑡2𝑖 𝑛 𝑖=1 =𝑏2 ,𝑤ℎ𝑒𝑟𝑒 𝑛∈𝑍 + . Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 320 ISSN: 2220-184X . . ∏𝑥1𝑖 𝑡𝑛𝑖 𝑛 𝑖=1 =𝑏𝑛 . If the solution of the system of equations has only one solution, then 𝑥11 =∏𝑏𝑖(𝑘𝑖𝑓𝑖1 ∆𝑘𝑖𝑓) 𝑛 𝑖=1 , 𝑥12 =∏𝑏𝑖(𝑘𝑖𝑓𝑖2 ∆𝑘𝑖𝑓) 𝑛 𝑖=1 , . . . 𝑥1𝑛 =∏𝑏𝑖(𝑘𝑖𝑓𝑖𝑛 ∆𝑘𝑖𝑓) . 𝑛 𝑖=1 4. CONCLUSION This study develops Kifilideen’s Rule to solve multiplication of bi-indexes, tri-indexes and 𝑛indexes simultaneous equations of two variables, three variables and 𝑛 variables respectively. Elimination method, laws of indices, and cofactor and determinant of matrix were deployed in establishing, formulating and modeling the Kifilideen’s Rule to solve multiplication of bi-indexes and tri-indexes simultaneous equations of two variables and three variables respectively. The Kifilideen’s Rule was implemented in form of a model in solving problems involving multiplication of bi-indexes and tri-indexes simultaneous equations of two variables and three variables respectively. The general solution to solve multiplication of 𝑛-indexes simultaneous equations of 𝑛 variables using Kifilideen’s rule was inaugurated. The Kifilideen’s Rule is in form of a model to solve multiplication of bi-indexes and tri-indexes simultaneous equations of two variables and three variables respectively which was found to be effective, straightforward, reliable, interesting, easy and initiative to understand for students and beginners. The general solution for 𝑛 to this rule is transferred to the discussion. 5. CONFLICT OF INTEREST There is no conflict of interest. Kifilideen L. Osanyinpeju (MEJS) Volume 17(2):302-323, 2025 © CNCS, Mekelle University 321 ISSN: 2220-184X 6. ACKNOWLEDGEMENTS The author wishes to express heartfelt gratitude to Allah for His mercy, guidance, and the wisdom granted to complete this work. 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