246 Chapter 3 Systems of Linear Equations and Inequalities Section 3.2 Solving Systems of Linear Equations 1. The Substitution Method Graphing a system of equations is one method to find the solution of the system. However, sometimes it is difficult to determine the solution using this method because of limitations in the accuracy of the graph. This is particularly true when the coordinates of a solution are not integer values or when the solution is a point not sufficiently close to the origin. Identifying the coordinates of the point or , for example, might be difficult from a graph. In this section and Section 3.3, we will present two algebraic methods to solve a system of equations.The first is called the substitution method.This technique is particularly important because it can be used to solve more advanced problems including nonlinear systems of equations. The first step in the substitution process is to isolate one of the variables from one of the equations. Consider the system x y 16 x y 4 Solving the first equation for x yields Then, because x is equal to the expression may replace x in the second equation.This leaves the second equation in terms of y only. First equation: Solve for x. x y 16 x 16 y 116 y2 x 16 y. y 4 Second equation: Substitute Solve for y. 16 2y 4 2y 12 x 16 y y 6 To find x, substitute back into the expression x 16 y. Check the ordered pair (10, 6) in both original equations. x y 16 x y 4 1102 162 16 ✔ True 1102 162 4 ✔ True x 16 162 x 10 The solution set is 5110, 626. y 6 16 y, 16 y x 16 y. 1251, 83492 1 3 17, 23 9 2 v Concepts 1. The Substitution Method 2. Solving Inconsistent Systems and Systems of Dependent Equations by the Substitution Method
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