CHAPTER NINE - SYSTEMS OF EQUATIONS AND INEQUALITIES

Abstract

This chapter discusses some of the basic methods for solving systems of linear equations. It discusses both Gaussian elimination and Gauss–Jordan elimination, methods that can be extended to systems of linear equations of any size. Similar techniques can be applied to systems of equations that include at least one second-degree equation. The chapter presents a graphical technique for solving systems of linear inequalities. When one solves a system of linear equations by graphing, one must estimate the coordinates of the point of intersection. If one requires the answers to be accurate to, say, five decimal places, it is clear that graphing would not suffice. The method of substitution provides one with exact answers but suffers from the disadvantage that it is difficult to program for use in a digital computer. The method of elimination overcomes these difficulties. The strategy of the method is to obtain an equation in just one variable that is easily solved. The method of substitution and the method of elimination can both be applied to systems of linear equations in three unknowns and, more generally, to systems of linear equations in any number of unknowns. There is a method known as Gaussian elimination that is ideally suited for computers and can be applied to solving linear systems in three unknowns. The objective of Gaussian elimination is to transform a given linear system into triangular form.