356 CHAPTER 6 Algebra, Graphs, and Functions In Example 2, we graphed the same system of equations that we solved algebraically in Example 3. Notice we get the same answer graphically as we do when solving algebraically. y x 5 4 3 2 1 1 2 3 4 5 22 22 23 24 25 25 23 24 x 1 y 5 3 2x 1 2y 5 24 (0, 3) (1, 2) (3, 0) (0,22) (22, 0) (1, 23) 21 21 Figure 6.26 The system of equations from Example 4 is graphed in Fig. 6.26. Notice that the lines do not intersect, and therefore, there is no solution to the system. This agrees with the answer we obtained algebraically in Example 4. When solving a system of equations, if you obtain a false statement, such as 4 0 = or 2 0, − = the system is inconsistent and has no solution. Timely Tip When solving a system of equations, if you obtain a statement that is always • false, such as 6 4, = − the system is inconsistent and has no solution. The equations in this system represent parallel lines. • true, such as =6 6, the system is dependent and has an infinite number of solutions. The equations in this system both represent the same line. Example 4 No Solution by the Substitution Method Solve the following system of equations by substitution. x y x y 3 2 2 4 + = + = − Solution The numerical coefficients of the x- and y-terms in the equation x y 3 + = are both 1. Thus, we can solve this equation for either x or y. Let us solve for y in the first equation. x y x x y x y x 3 3 3 + = − + = − = − Subtract x from both sides of the equation. Now substitute x 3 − for y in the second equation. x y x x x x 2 2 4 2 2(3 ) 4 2 6 2 4 6 4 + = − + − = − + − = − = − Distributive property False Since 6 cannot be equal to 4, − there is no solution to the system of equations. Thus, the system of equations is inconsistent. 7 Now try Exercise 27 Example 5 An Infinite Number of Solutions by the Substitution Method Solve the following system of equations by substitution. y x y x 2 3 2 6 4 = − + = − Solution The first equation = − + y x 2 3 is already solved for y, so we will substitute x2 3 − + for y in the second equation. = − − + = − − + = − − + = − + = y x x x x x x x x x 2 6 4 2( 2 3) 6 4 4 6 6 4 4 4 6 6 4 4 6 6 Add x4 to both sides of the equation. True Since 6 6, = the system has an infinite number of solutions. Thus, the system of equations is dependent. 7 Now try Exercise 33 The system of equations from Example 5 is graphed in Fig. 6.27. Notice that the graph of both equations is the same line. Every point on the line represents a solution y x 5 3 2 1 1 3 4 5 21 21 22 22 23 24 25 25 23 24 y 5 22x 1 3 2y 5 6 2 4x (0, 3) (1, 1) (2, 21) (21, 5) (3, 23) (4, 25) Figure 6.27
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