College Algebra 9th txtbk

SECTION 2–3 Equations of a Line 141
Z Finding Slopes of Parallel or Perpendicular Lines
From geometry, we know that two vertical lines are parallel to each other and that a horizontal
line and a vertical line are perpendicular to each other. How can we tell when two
nonvertical lines are parallel or perpendicular to each other? Theorem 5, which we state
without proof, provides a convenient test.
Z THEOREM 5 Parallel and Perpendicular Lines
Given two nonvertical lines L 1 and L 2 with slopes m 1 and m 2 , respectively, then
L 1 L 2 if and only if m 1 m 2
L 1 L 2 if and only if m 1 m 2 1
The symbols and mean, respectively, “is parallel to” and “is perpendicular to.” In
the case of perpendicularity, the condition m 1 m 2 = 1 also can be written as
m 2 1 m 1
or
m 1 1 m 2
Therefore,
Two nonvertical lines are perpendicular if and only if their slopes are the
negative reciprocals of each other.
EXAMPLE 7 Parallel and Perpendicular Lines
Given the line L: 3x 2y = 5 and the point P (3, 5), find an equation of a line through
P that is
(A) Parallel to L
(B) Perpendicular to L
Write the final answers in the slope–intercept form y = mx b.
SOLUTIONS
First, find the slope of L by writing 3x 2y = 5 in the equivalent slope–intercept form
y = mx b:
3x 2y 5
2y 3x 5
3
y 3 2 x 5 2
So the slope of L is 2. The slope of a line parallel to L is the same, 2, and the slope of a
line perpendicular to L is 2 3. We now can find the equations of the two lines in parts
A and B using the point–slope form.
3
(A) Parallel (m 3 2):
(B) Perpendicular (m 2 3):
y y 1 m(x x 1 )
y 5 3 2 (x 3)
y 5 3 2 x 9 2
y 3 2 x 19 2
y y 1 m(x x 1 )
y 5 2 3 (x 3)
y 5 2 3x 2
y 2 3x 3
Substitute for x 1 , y 1 , and m.
Distribute.
Add 5 to both sides.