* Find the slope of a line. * Use slopes to identify parallel and perpendicular lines. * Write the equation of a line through a given point - parallel to a given line - perpendicular to a given line

1. 1.3.2C Equations of Lines
The student is able to (I can):
• Find the slope of a line.
• Use slopes to identify parallel and perpendicular lines.
• Write the equation of a line through a given point
— parallel to a given line
— perpendicular to a given line

2. slope The ratio of riseriseriserise to runrunrunrun. If (xxxx1111, yyyy1111) and
(xxxx2222, yyyy2222) are any two points on a line, the
slope of the line is
So, for the previous example, substitute
(1111, 1111) for (xxxx1111, yyyy1111) and (5555, 7777) for (xxxx2222, yyyy2222):
Note: Always reduce fractions to their
simplest forms. Also, it’s usually better to
leave improper fractions improper.
2 1
2 1x
y
m
y
x
−
−
=
1
2 1
2
x x
y y 7 6 3
m
41 2
1
5
− −
= = = =
− −

3. Summary: Slope of a Line
positive slope negative slope
zero slope undefined slope

4. horizontal line
vertical line
reciprocal
The slope of a horizontal linehorizontal linehorizontal linehorizontal line is 0000.
The slope of a vertical linevertical linevertical linevertical line is undefinedundefinedundefinedundefined.
The reciprocalreciprocalreciprocalreciprocal of is . The product of
a number and its reciprocal is 1.
a
b
b
a

5. Parallel Lines
Theorem
In a coordinate plane, two nonvertical lines
are parallel if and only if they have the
same slope.
Any two vertical lines are parallel
x
y
ms = mt ⇒ s t
s t
s
1 3 4
m 2
2 0 2
− − −
= = =
− − −
t
3 1 4
m 2
1 1 2
− − −
= = =
− − −

6. Perpendicular
Lines Theorem
In a coordinate plane, two nonvertical lines
are perpendicular if and only if the product
of their slopes is —1 (negative reciprocals).
Vertical and horizontal lines are
perpendicular.
x
y
p
q
p
2 4 6
m 3
2 0 2
+
= = = −
− − −
q
0 1 1 1
m
3 0 3 3
− −
= = =
− − −
p qm m 1= − ⇒ ⊥i p q

7. Practice
Find the slopes between the following points:
1. (2, —1) and (8, —4) 2. (—3, 10) and (5, —6)
3. (1, 12) and (—10, —10) 4. (22, 4) and (0, 28)
( )− − − −
= = = −
−
4 1 3 1
m
8 2 6 2 ( )
− − −
= = = −
− −
6 10 16
m 2
5 3 8
− − −
= = =
− − −
10 12 22
m 2
10 1 11
−
= = = −
− −
28 4 24 12
m
0 22 22 11

8. Point-Slope
Form
Given the slope, mmmm, and a point on the line
(xxxx1111, yyyy1111), the equation of the line is
y — yyyy1111 = mmmm(x — xxxx1111)
Example: Write the equation of the line
whose slope is 2222, which goes through the
point (1111, 6666)
y — 6666 = 2222(x — 1111)
In point-slope form, you can leave it like this
— you don’t have to simplify it any
further.

9. Slope-
Intercept Form
Given the slope, mmmm, and bbbb, the y-intercept,
the equation of the line is
y = mmmmx + bbbb
Example: For mmmm = ————3333 and y-intercept 7777,
find the equation of the line.
y = ————3333x + 7777

10. Horizontal Line
Vertical Line
For a horizontal line (mmmm = 0000), the equation
of the line is
y = bbbb
For a vertical line (mmmm = undefinedundefinedundefinedundefined), the
equation of the line is
x = xxxx1111
Notice that this equation does not start
with “y=“

11. To write the equation of a line through a
given point that is parallel or perpendicular
to a given line, determine the slope from the
given line and then write the equation as
before.
Example: Write the equation of the line
that goes through (—3, 7) that is
a) parallel to
b) perpendicular to
= +
1
y x 4
3
= −
3
y x 1
2

12. To write the equation of a line through a
given point that is parallel or perpendicular
to a given line, determine the slope from the
given line and then write the equation as
before.
Example: Write the equation of the line
that goes through (—3, 7) that is
a) parallel to
slope =
= +
1
y x 4
3
1
3
( )− = +
− = +
= +
1
y 7 x 3
3
1
y 7 x 1
3
1
y x 8
3

13. b) perpendicular to
⊥ slope =
= −
3
y x 1
2
−
2
3
( )− =− +
− =− −
=− +
2
y 7 x 3
3
2
y 7 x 2
3
2
y x 5
3