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Slope of Parallel Lines

Slope of Parallel Lines. Slope =. y. rise. run. y 2 – y 1. m =. x. x 2 – x 1. 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 ),.

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Slope of Parallel Lines

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  1. Slope of Parallel Lines Slope = y rise run y2 – y1 m = . x x2 – x1 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 (x1, y1) and (x2, y2), then the slope is given by (x2, y2) (x1, y1) y2 – y1 x2 – x1 Slope is usually represented by the variable m.

  2. Finding the Slope of Train Tracks Slope = rise 4 feet = run 10 feet 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 = 0.4

  3. Finding the Slope of a Line 4 5 4 5 = – y2–y1 2 – 6 m = = x2 – x1 5 – 0 – The slope of the line is . Find the slope of the line that passes through the points (0, 6)and (5, 2). SOLUTION Let (x1, y1) = (0, 6) and (x2, y2) = (5, 2). (0, 6) 5 –4 (5, 2)

  4. 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 k1 and k2 have the same slope.

  5. Deciding Whether Lines are Parallel 4 2 2 1 m1= m2= Because the lines have the same slope, j1|| j2. Find the slope of each line. Is j1 || j2? SOLUTION Line j1 has a slope of = 2 2 Line j2 has a slope of 1 4 = 2 2

  6. Identifying Parallel Lines 5 2 – – 5 0 + 2 6 2 – 0 – 6 m1 = = 2 – 0 1 – 6 m2= = = 0 – (– 2) Find the slope of each line. Which lines are parallel? SOLUTION Find the slope of k1. Line k1 passes through (0, 6) and (2, 0). = –3 Find the slope of k2. Line k2 passes through (–2, 6) and (0, 1).

  7. Identifying Parallel Lines – 5 –4 + 6 5 2 – = 0 – 5 m3 = –4 – (– 6) = Compare the slopes. Find the slope of each line. Which lines are parallel? SOLUTION Find the slope of k3. Line k3passes through (–6, 5) and (–4, 0). Because k2 andk3 have the same slope, they are parallel. Line k1 has a different slope, so it is not parallel to either of the other lines.

  8. 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.

  9. Writing an Equation of a Line Since m = 5 and b = –7, an equation of the line is y = 5x– 7. 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=mx+b 5 (2) Substitute 2 for x, 3 for y, and 5 for m. 3 = 10 + b Simplify. –7 = b Subtract. Write an equation.

  10. Writing an Equation of a Parallel Line 1 3 Line n1 has the equation . – x – 1 y = The slope of n1 is . 1 3 1 3 – – Because parallel lines have the same slope, the slope of n2 is also . Line n2 is parallel to n1and passes through the point (3, 2). Write an equation of n2. SOLUTION Find the slope.

  11. Writing an Equation of a Parallel Line 1 3 1 3 – – x + 3 x – 1 y = y = 1 3 – Solve for b. Use (x, y)= (3, 2) and m= . 1 3 – 1 3 2=–(3)+b Because m =and b = 3, an equation of n2 is . Line n1 has the equation . Line n2 is parallel to n1and passes through the point (3, 2). Write an equation of n2. SOLUTION y=mx+b 2 = –1 + b 3 = b Write an equation.

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