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

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, its usually better to leave improper fractions improper.

2 1

2 1x

ym

y

x

=

1

2 1

2

x x

y y 7 6 3m

41 2

1

5

= = = =

Summary: Slope of a Line

positive slope negative slope

zero slope undefined slope

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

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 4m 2

2 0 2

= = =

t

3 1 4m 2

1 1 2

= = =

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 6m 3

2 0 2

+= = =

q

0 1 1 1m

3 0 3 3

= = =

p qm m 1= i p q

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 1m

8 2 6 2 ( )

= = =

6 10 16m 2

5 3 8

= = =

10 12 22m 2

10 1 11

= = =

28 4 24 12m

0 22 22 11

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 dont have to simplify it any further.

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

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=

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 43

= 3

y x 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 43

1

3( ) = +

= +

= +

1y 7 x 3

31

y 7 x 131

y x 83

b) perpendicular to

slope =

= 3

y x 12

2

3

( ) = +

=

= +

2y 7 x 3

32

y 7 x 232

y x 53