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2.1 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
.
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
Write the point-slope form of the equation
of the line passing through (1,3) with a slope
of 4. Then slove the equation for .y
1 1
We use the point-slope equation of a line
with
Solut
4, 1, and 3.
ion
m x y
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.
Example 4: Writing the Point-Slope Equation of a Line
Write the point-slope form of the equation
of the line passing through the points (3,5)
and (8,15). Then slove the equation for .y
First find the slope to use the
point-slop
Solu
e f
tion
orm.
Given (3,5) and (8,15).
15 5
8 3m
10
5 2
1 1 1 1
We can take either point on the line to
be ( , ). Let's use ( , ) (3,5).x y x y 15 2( 3)y x 15 2 6y x
2 9y 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 slove 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 slove the equation for .y
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
2 RiseSlope 2
1 Runm
First use the -intercept 2, to
plot the point (0,2). Starting
at (0,2), move 2 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.
The graph of 3 2.y x
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
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.
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