Lines and Slopes

1 1 2 2

2 1

2 1

2 1

The of the line through the

distinct points ( , ) and ( , )

Change in Riseis

Change in Ru

slop

n

where 0.

e

x y x y

y yy

x x x

x x

Example 1: Find the SlopeFind the slope of the line passing through

the pair of points (2,1) and (3,4).

1 1 2 2Let ( , ) (2,1) and

Solu

( , ) (3,4).

tion

x y x y

2 1

2 1

Slopey y

mx x

4 1

3 2

3

1 3.

Possibilities for a Line’s Slope

Positive Slope

0m

Line rises from left to right.

Negative Slope

0m

Line falls from left to right.

Possibilities for a Line’s Slope

Zero Slope

0m

Line is horizontal.

Undefined Slope

is

undefined.

m

Line is vertical.

Example 2: Find the Slope

Find the slope of the line passing through

the pair of points ( 1,3) and (2,4) or state

that the slope is undefined. Then indicate

whether the line through the points rises,

falls, is horizontal, or is ve

rtical.

Solution

1 1 2 2Let ( , ) ( 1,3) and ( , ) (2,4).x y x y

2 1

2 1

Slopey y

mx x

4 3

2 ( 1)

1

3

The slope is and

the line from l

positi

eft to

ve,

r riises ght.

Practice Exercise

Find the slope of the line passing through

the points (4, 1) and (3, 1) or state

that the slope is undefined. Then indicate

whether the line through the points rises,

falls, is horizontal, or is vertical

.

Answer

The slope is zero.

Thus, the line is a horizontal line.

m

Point-slope Form of the Equation of a Line

1 1

The point-slope equation of a nonvertical

line of slope that passes through the

point ( , ) is

m

x y

1 1( .)y y m x x

Example 3: Writing the Point-Slope Equation of a Line

1 1

We use the point-slope equation of a line

with

Solut

4, 1, and 3.

ion

m x y

Write the point-slope form of the equation

of the line passing through (1,3) with a slope

of 4. Then solve the equation for .y

1 14, 1, and 3m x y

1 1( )y y m x x 3 4( 1)y x 3 4 4y x

4 1y x

Practice Exercise

Write the point-slope of the equation

of the line passing throuhg the points

(3,5) and (8,15). Then solve the

equation for y.

Answer

Point-slope form of the equation:

5 2( 3).

Then solve for gives:

2 1

y x

y

y x

Practice Exercises

1. Write the point-slope form of the equation

of the line passing through (4,-1) with a slope

of 8. Then solve the equation for .y

2. Write the point-slope form of the equation

of the line passing through the points ( 2,0)

and (0,2). Then solve the equation for .y

Answers to Practice Exercises

1. 8 33

2. 2

y x

y x

The Slope-Intercept Form of the Equation of a Line

The slope-intercept

equation of a

nonvertical line

with slope and

-intercept is

m

y b

y mx b

(0, )b

y

x

Y-intercept is b

Slope is m

A line with slope

and -intercept .

m

y b

Graphing y=mx+b Using the Slope and y-Intercept. Plot the y-intercept on the y-

axis. This is the point (0,b). Obtain a second point using

the slope, m. Write m as a fraction, and use rise over run starting at the y-intercept to plot this point

Graphing y=mx+b Using the Slope and y-Intercept. Use a straightedge to draw a

line through the two points. Draw arrowheads at the ends of line to show that the line to show that the line continues indefinitely in both directions.

Example 5: Graphing by Using the Slope and y-Intercept

Give the slope and the -intercept of the

line 3 2. Then graph the line.

y

y x

Solution 3 2y x

The slope

is 3

The -intercept

is 2.

y

The graph of 3 2.y x

First use the -intercept 2, to

plot the point (0,2). Starting

at (0,2), move 3 units up and

1 unit to the right. This gives

us the second point of the line.

Use a straightedge to draw a

line through the tw

y

o points.

Practice ExercisesGive the slope and -intercept

of each line whose equation is

given. Then graph the line.

y

1. 3 2

32. 3

4

y x

y x

Answers to Practice Exercises

1. 3, 2m b 32. , 3

4m b

Equation of a Horizontal Line

A horizontal line

is given by an

equation of the

form

where is the

-intercept.

b

b

y

y

Y-interceptis 40m

The graph of 4y

Equation of a Vertical LineA vertical line is

given by an

equation of the

form

where is the

-intercept.

a

x

x a

X-intercept is -5

Slope is

undefined

The graph of -5x

Example 6: Graphing a Horizontal Line

Graph 5 in the

rectangular coordinate system.

y

SolutionAll points on the graph

of 5 have a value of

that is always 5. Thus

it is a horizontal line

with -intercept 5.

y

y

y

Y-intercept is 5.

Example 7: Graphing a Vertical Line

Graph 5 in the

rectangular coordinate system.

x

No matter what the

-coordinate is, the

corresponding

-coordinate for every

point on the line is 5.

y

x

Solution

X-intercept is –5.

Practice Exercises

Graph each equation in the rectangular

coordinate system.

1. 4

2. 0

y

x

Answers to Practice Exercises

2..1

General Form of the Equation of a Line

0

Every line has an equation that can

be written in the general form

where, , , and are three

real numbers, and and

are not both zero.

A B C

A B

Ax By C

Equations of Lines

1 11. Point-slope form:

2. Slope-intercept form:

3. Horizontal line:

4. Vertical line:

5. Gene

(

ral form:

)

0

y y m x x

y mx b

y b

x a

Ax By C

Example 8: Finding the Slope and the y-InterceptFind the slope and the -intercept of the

line whose equation is 4 6 12 0.

y

x y

SolutionFirst rewrite the equation in slope-intercept

form . We need to solve for .y mx b y

4 6 12 0x y

6 4 12y x 4 12

6 6y x

22

3y x

23

The coefficient of ,

, is the slope and

the constant term, 2,

is the -intercept.

x

y

23 , 2.m b

Practice Exercises

a. Rewrite the given equation in

slope-intercept form.

b. Give the slope and y-intercept.

c. Graph the equation.

1. 6 5 20 0

2. 4 28 0

x y

y

Answers to Practice Exercises

651. 4

6slope

5-intercept 4.

y x

m

y b

Answers to Practice Exercises

2. 7

slope 0

-intercept 7.

y

m

y b