SECTION 11.1 Systems of Linear Equations: Substitution and Elimination 757 Some examples of linear equations are + = − + = + − + = x y x y z x y z w 23 2523108825 0 If each equation in a system of equations is linear, we have a system of linear equations. The systems in Examples 2(a), (c), (d), and (e) are linear, but the system in Example 2(b) is nonlinear. In this chapter we solve linear systems in Sections 11.1 to 11.3. Nonlinear systems are discussed in Section 11.6. We begin by discussing a system of two linear equations containing two variables. Solving such a system can be viewed as a geometry problem. The graph of each equation in such a system is a line. So a system of two linear equations containing two variables represents a pair of lines. The lines may intersect, be parallel, or be coincident (that is, identical). • If the lines intersect, the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. See Figure 1(a). • If the lines are parallel, the system of equations has no solution, because the lines never intersect. The system is inconsistent. See Figure 1(b). • If the lines are coincident (the lines lie on top of each other), the system of equations has infinitely many solutions, represented by all of the points on the line. The system is consistent, and the equations are dependent. See Figure 1(c). Solving a System of Linear Equations Using a Graphing Utility Solve: + =− − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 1 4 6 42 (1) (2) Solution EXAMPLE 3 If necessary for your graphing utility, solve each equation for y. This is equivalent to writing each equation in slope-intercept form. Equation (1) in slope-intercept form is = − − Y x2 1. 1 Equation (2) in slope-intercept form is = + Y x 2 3 7. 2 Figure 2(a) shows the graphs using a TI-84 Plus CE.The lines intersect, so the system is consistent and the equations are independent. Using INTERSECT gives the solution ( ) −3, 5 . Figure 2(b) shows the result using Desmos. Figure 1 x y Parallel lines; system has no solution (b) x y Intersecting lines; system has one solution (a) x y Coincident lines; system has infinitely many solutions consisting of all points on the line. (c) Solution Consistent and Independent Inconsistent Consistent and Dependent (b) Figure 2 8 22 211 5 Y1 5 22x 2 1 Y2 5 x 1 7 2 – 3 (a)
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