Slope of Parallel Lines The slope of a nonvertical line is the ratio of the vertical change (the rise) to the horizontal change (the run). If the line passes through the points ( x 1, y 1 ) and ( x 2, y 2 ), Slope = rise run y 2 – y 1 x 2 – x 1 m =. Slope is usually represented by the variable m. x 2 – x 1 y 2 – y 1 x y (x 1, y 1 ) (x 2, y 2 ) then the slope is given by

Finding the Slope of Train Tracks COG RAILWAY A cog railway goes up the side of Mount Washington, the tallest mountain in New England. At the steepest section, the train goes up about 4 feet for each 10 feet it goes forward. What is the slope of this section? SOLUTION Slope = rise run = feet 10 feet =

Finding the Slope of a Line Find the slope of the line that passes through the points (0, 6) and (5, 2). (5, 2) – 4– 4 5 SOLUTION Let (x 1, y 1 ) = (0, 6) and (x 2, y 2 ) = (5, 2). y 2 – y 1y 2 – y 1 x 2 – x 1 m = 2 – 6 5 – 0 = 4545 = – 4545 – The slope of the line is. (0, 6)

The slopes of two lines can be used to tell whether the lines are parallel. POSTULATE Postulate 17 Slopes of Parallel Lines In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. Lines k 1 and k 2 have the same slope. Slope of Parallel Lines

Deciding Whether Lines are Parallel Find the slope of each line. Is j 1 || j 2 ? SOLUTION Line j 1 has a slope of m 1 =m 1 = 4242 = 2 Line j 2 has a slope of m 2 =m 2 = 2121 = 2 Because the lines have the same slope, j 1 || j 2.

Identifying Parallel Lines Find the slope of each line. Which lines are parallel? SOLUTION Find the slope of k 1. Line k 1 passes through (0, 6) and (2, 0). Find the slope of k 2. Line k 2 passes through (–2, 6) and (0, 1). 0 – 6 2 – 0 = m 1m 1 m 2 =m 2 = 1 – 6 0 – ( – 2) 5252 – = – = – = = –3

Identifying Parallel Lines Find the slope of each line. Which lines are parallel? SOLUTION Find the slope of k 3. Line k 3 passes through (–6, 5) and (– 4, 0). = 5 – – 0 – 5 – 4 – ( – 6) m 3 = 5252 – = Compare the slopes. Because k 2 and k 3 have the same slope, they are parallel. Line k 1 has a different slope, so it is not parallel to either of the other lines.

Writing Equations of Parallel Lines You can use the slope m of a nonvertical line to write an equation of the line in slope-intercept form. y = m x + by = m x + b The y -intercept is the y -coordinate of the point where the line crosses the y -axis. slope y- intercept

Writing an Equation of a Line Write an equation of the line through the point (2, 3) that has a slope of 5. SOLUTION y = m x + b 35 3 = 10 + b –7 = b Write an equation. Since m = 5 and b = –7, an equation of the line is y = 5x – 7. Slope-intercept form Substitute 2 for x, 3 for y, and 5 for m. Simplify. Subtract. Solve for b. Use (x, y) = (2, 3) and m = 5. (2)

Line n 1 has the equation – x – 1 y = Line n 2 is parallel to n 1 and passes through the point (3, 2). Write an equation of n 2. Find the slope – The slope of n 1 is. Because parallel lines have the same slope, the slope of n 2 is also – Writing an Equation of a Parallel Line SOLUTION

Line n 1 has the equation – x – 1 y = Line n 2 is parallel to n 1 and passes through the point (3, 2). Write an equation of n 2. y = m x + b 2 = –1 + b 3 = b Write an equation. 2 = – (3) + b 1313 Because m = and b = 3, an equation of n 2 is – x + 3 y = 1313 – 1313 – Solve for b. Use (x, y) = (3, 2) and m =. Writing an Equation of a Parallel Line SOLUTION