SECTION 1.5 Lines 41 Showing That Two Lines Are Parallel Show that the lines given by the following equations are parallel. + = + = L x y L x y : 2 3 6 : 4 6 0 1 2 Solution EXAMPLE 9 To determine whether these lines have equal slopes and different y-intercepts, write each equation in slope-intercept form. Figure 66 Parallel lines 8 L2 L1 28 25 5 y 5 5 L1 L2 25 x + = = − + = − + L x y y x y x : 2 3 6 3 2 6 2 3 2 1 = − = y Slope 2 3 ; -intercept 2 + = = − = − L x y y x y x : 4 6 0 6 4 2 3 2 = − = y Slope 2 3 ; -intercept 0 Because these lines have the same slope, − 2 3 , but different y-intercepts, the lines are parallel. See Figure 66. Finding a Line That Is Parallel to a Given Line Find an equation for the line that contains the point ( ) − 2, 3 and is parallel to the line + = x y 2 6. Solution EXAMPLE 10 Since the two lines are to be parallel, the slope of the line containing the point ( ) − 2, 3 equals the slope of the line + = x y 2 6. Begin by writing the equation of the line + = x y 2 6 in slope-intercept form. + = = − + x y y x 2 6 2 6 The slope is −2. Since the line containing the point ( ) − 2, 3 also has slope −2, use the point-slope form to obtain its equation. y y m x x y x y x y x x y 3 2 2 3 2 4 2 1 2 1 1 1 ( ) ( ) ( ) − = − − − = − − + = − + = − + + = This line is parallel to the line + = x y 2 6 and contains the point ( ) − 2, 3 . See Figure 67. Point-slope form =− = =− m x y 2, 2, 3 1 1 Simplify. Slope-intercept form General form Now Work PROBLEM 67 Figure 67 y 6 25 6 26 (2, 23) 2x 1 y 5 1 2x 1 y 5 6 x
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