Analysis of Solving Linear Equations Using Gauss Elimination Method and Cramer's Rule Anggi Dwiyanto Department of Electrical Engineering Nusa Putra University Sukabumi, Indonesia anggi.dwiyanto_te20@nusaputra.ac.id Abstract— This article discusses the Gauss Elimination Method Analysis and Cramer's Rule in solving systems of linear equations. The purpose of this study is to compare the effectiveness of the Gauss Elimination Method and Cramer's Rule in solving a system of linear equations. The type of research carried out is library research by collecting some literature in the form of books and journals related to this research. The results obtained are the Gauss Elimination Method is more effective than Cramer's Rule. This comparison can be seen from the number of steps of completion, speed, and accuracy in solving a system of linear equations. Keywords— Gaussian Elimination, Cramer's Rule, System of Linear Equations I. INTRODUCTION Linear Algebra theory is a branch of mathematics. Linear algebra has applications in various fields of natural and social sciences and technology, especially information and communication technology, which is currently growing rapidly. One of the subjects studied in Linear Algebra is Linear Equations and Systems of Linear Equations. A Linear Equation is an equation when the curve is drawn in the form of a straight line, while the Linear Equation System is a system that consists of at least 2 linear equations [1]. Linear equations can consist of m equations and n variables or can consist of n equations and n variables. The solution of this form can be solved through matrices. The system of Linear Equations is a part of mathematics that studies how to solve engineering problems using linear algebra [2]. A matrix is defined as a rectangular array of numbers arranged in rows and columns and delimited by brackets. The numbers in the array are called entries or elements in the matrix [3]. Whereas in mathematics, matrices can be used to deal with linear models, such as finding solutions to systems of linear equations. The application of the Linear Algebra and Matrix system is used to predict rainfall [4], determine the closing price of Bitcoin [5], determine tube volume [5], collision energy protection [7], for price detection systems [8], create efficient wireless network designs [9], make employee performance decisions [10], recognize facial expressions [11]-[12], system automatic gas control [13], color database image segmentation [14], temperature disturbance handling systems [15], data mining classification methods [16], lighting control systems [17], virtual reality applications [18], and many more. Problems that often arise in finding solutions to systems of linear equations are usually related to the size of the matrix. The bigger the matrix, the more complicated the calculation, so the right method is needed. Solving a system of linear equations with m equations and n variables can use Gaussian and Gauss-Jordan Elimination, while for n equations and n variables, several methods can be used, including Gaussian Elimination, Gauss-Jordan Method, Inverse Matrix Method, Cramer's Rule, LU Decomposition and Crout Decomposition. In this study, the authors use several methods, including the Gauss Elimination Method and Cramer's Rule. Gaussian Elimination Method is an elimination process using row- echelon operations or converting a linear system into a triangular matrix, then solved by a backward substitution. Cramer's rule is a method of finding the value of a variable by using the determinant. Cramer's rule gives us an easy method for writing the solution to a system of n × n linear equations with determinants [19]. II. SOLVING THE SYSTEM OF LINEAR EQUATIONS USING THE GAUSS ELIMINATION METHOD AND CRAMER'S RULE A. Gaussian Elimination Method Determining the solution to the solution of a system of linear equations using the Gauss Elimination Method, using generally accepted steps, so that the solution can be done consistently. The system of linear equations is written in matrix form as follows: The above matrix is brought to the upper triangular matrix form so that it becomes. After the upper triangular matrix can be done back substitution so that the solution of the linear equation is obtained. An example of the Gaussian Elimination Method for solving a system of linear equations is shown below.