1. 3-6 Parallel Lines in the Coordinate Plane

6. 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 =

7. 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

8. 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 .

9. 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.

10. 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.

11. 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 – =

12. 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.

13. 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.

14. 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.

15. 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 –

16. 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 =