Determining Whether Two Lines Are PerpendicularAre the lines with equations and perpendicular?
Find the slope of each line by solving each equation for y.
Divide by 2. Subtract 2x.Slope Divide by 3.SlopeSince the product of the slopes is the lines are perpendicular.
NOW TRYNOTE In Example 7,alternatively, we could have found the slope of each line by
using intercepts and the slope formula. For the graph of the equation
the x-intercept is and the y-intercept is
See Example 7.Find the intercepts of the graph of and use them to confirm the
slope Since the slopes are negative reciprocals, the lines are perpendicular.
Determining Whether Two Lines Are Parallel,
Perpendicular, or NeitherDetermine whether the lines with equations and are
parallel, perpendicular,or neither.
Find the slope of each line by solving each equation for y.
Subtract 2x. Subtract 2x.Divide by. Divide by 5.Slope SlopeThe slopes, and are not equal, and they are not negative reciprocals because their
product is - 254 ,not - 1. Thus, the two lines are neither parallel nor perpendicular.
(^25) - (^25) ,
y= -
2
5
x+
8
5
y= - 5
2
5
x-
8
5
- 5 y=- 2 x+ 8 5 y=- 2 x+ 8
2 x- 5 y= 8 2 x+ 5 y= 8
2 x- 5 y= 8 2 x+ 5 y= 8
EXAMPLE 8
-
23.
2 x+ 3 y=- 6
m=
0 - 1 - 32
2 - 0
=
3
2
1 2, 0 2 1 0, - 32.
2 y= 3 x- 6,
32 A-^
23 B =-1,
y= -
2
3
x- 2
y= 3 y=- 2 x- 6
3
2
x- 3
2 y= 3 x- 6 2 x+ 3 y=- 6
2 y= 3 x- 6 2 x+ 3 y=- 6
EXAMPLE 7
SECTION 3.2 The Slope of a Line 153
Slopes of Perpendicular LinesIf neither is vertical, perpendicular lines have slopes that are negative reciprocals—
that is, their product is Also, lines with slopes that are negative reciprocals
are perpendicular.
A line with 0 slope is perpendicular to a line with undefined slope.
- 1.
NOW TRYNOW TRY
EXERCISE 7
Are the lines with these
equations perpendicular?
2 x=y- 4x+ 2 y= 7NOW TRY ANSWERS
- yes
- neither
NOW TRY
EXERCISE 8
Determine whether the lines
with these equations are
parallel, perpendicular,or
neither.
2 x+y= 62 x-y= 4