Slope of Parallel Lines The slope of a

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), then the slope is given by Slope = y rise run (x 2, y 2) (x 1, y 1) y 2 – y 1 x 2 – x 1 Slope is usually represented by the variable m. x m= y 2 – y 1 x 2 – x 1 .

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 = 4 feet rise = = 0. 4 run 10 feet

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

Slope of Parallel Lines 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.

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 = 4 = 2 2 2 Line j 2 has a slope of m 2 = 2 1 Because the lines have the same slope, j 1 || j 2. 4 2 1

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). m 1 = 0 – 6 2– 0 = – 6 = – 3 2 Find the slope of k 2. Line k 2 passes through (– 2, 6) and (0, 1). m 2 = 1– 6 = – 5 =– 5 2 0+2 0 – (– 2)

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). m 3 = – 5 0– 5 = = – 5 – 4 – (– 6) – 4 + 6 2 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. slope y-intercept y = mx + b The y-intercept is the y-coordinate of the point where the line crosses the y-axis.

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

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

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