Linear algebra plays a pivotal role in solving systems of linear equations, which arise naturally in various branches of science, engineering, economics, and social sciences. This research paper presents a comprehensive study of how linear algebraic concepts such as matrices, determinants, vector spaces, rank, linear transformations, and eigenvalues are applied to analyse and solve systems of linear equations. Both theoretical foundations and practical solution methods are discussed in detail. The paper also highlights computational techniques and real-world applications, making it suitable for academic and research purposes.