3-6 Parallel Lines in the Coordinate Plane

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

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 4 feet 10 feet = = 0.4

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). y 2 – y 1 x 2 – x 1 m = (0, 6) 5 – 4 2 – 6 5 – 0 = 4 5 = – (5, 2) 4 5 – The slope of the line is .

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 j1 has a slope of m 1 = 4 2 = 2 2 Line j2 has a slope of 4 1 m 2 = 2 1 = 2 2 Because the lines have the same slope, j1 || j2.

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). 0 – 6 2 – 0 = m 1 6 2 – = = –3 Find the slope of k 2. Line k 2 passes through (–2, 6) and (0, 1). m 2 = 1 – 6 0 – (– 2) 5 0 + 2 – = 5 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). 0 – 5 – 4 – (– 6) m 3 = = 5 – 4 + 6 – 5 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.

y = m x + b 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 = m x + b The y-intercept is the y-coordinate of the point where the line crosses the y-axis.

Solve for b. Use (x, y) = (2, 3) and m = 5. 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 = m x + b Slope-intercept form 3 y = m x + b 5 (2) Substitute 2 for x, 3 for y, and 5 for m. 3 = 10 + b Simplify. –7 = b Subtract. Write an equation. Since m = 5 and b = –7, an equation of the line is y = 5x – 7.

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

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