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Hawkes Learning Systems:
College Algebra
3.3: Forms of Linear Equations
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Objectives
o Understand the meaning of and to be able to
calculate the slope of a line.
o Be able to write the equation of a line in
slope-intercept form.
o Be able to write the equation of a line in point-slope
form.
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The Slope of a Line
o There are several ways to characterize a given line in
the Cartesian plane.
o We have already used one way repeatedly: plotting
two distinct points in the Cartesian plane to
determine a unique line.
o Another approach is to identify just one point on the
line and to indicate how “steeply” the line is rising or
falling as we scan the plane from left to right. A
single number is sufficient to convey this notion of
“steepness”.
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The Slope of a Line
Rise and Run Between Two Points
y
 x2 , y2 

Rise  y2  y1
 x1 , y1 

  x2 , y1 
Run  x2  x1
x
y2  y1
As drawn above, the ratio x2  x1 is positive, and we say that the line has a positive
slope. If the rise and run have opposite signs, the slope of the line would be negative
and the line under consideration would be falling from the upper left to the lower
right.
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The Slope of a Line
Let L stand for a given line in the Cartesian plane,
and let  x1 , y1  and  x2 , y2  be the coordinates of
any two distinct points on L. The slope, m , of the
line, L is the ratio
y2  y1
m
x2  x1
which, can be described in words as “change in y
over change in x ” or “rise over run.”
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The Slope of a Line
Caution!
It doesn’t matter how you assign the labels  x1 , y1 
and  x2 , y2  to the two points you are using to calculate
slope, but it is important that you are consistent as you
apply the formula. That is, don’t change the order in
which you are subtracting as you determine the
y2  y1
numerator and denominator in the formula
.
x2  x1
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Example: Finding Slope Using Two Points
Determine the slopes of the line passing through the
following points.
y2  y1
m
x2  x1
8,1 and  2,3
3 1
m
2  8
2
m
10
1
m
5
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Example: Finding Slope Using Two Points
Determine the slopes of the line passing through the
following points.
y2  y1
m
x2  x1
 5,4 and 8,4
44
m
85
0
m
3
m0
Note: The two points lie on a horizontal line.
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Slopes of Horizontal Lines
Horizontal lines all have slopes of 0, and horizontal
lines are the only lines with slope equal to 0. The
equation of a horizontal line can be written in the
form y  c , where c is a constant.
y
yc
x
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Slopes of Vertical Lines
Vertical lines all have undefined slopes, and
vertical lines are the only lines for which the slope
is undefined. The equation of a vertical line can be
written in the form x  c where c is a constant.
y
xc
x
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Finding the Slope of a Line
o We already know how to identify any number of
ordered pairs that lie on a line, given the equation for
the line. Identifying just two such ordered pairs
allows us to calculate the slope of a line defined by
an equation.
o In the next example, we will first find two points on
the line. Then, we will use these points to determine
the slope.
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Example: Finding the Slope of a Line
Determine the slope of the line defined by the following
2 x  4 y  16
equation.
Solution: First, find two points on the line.
2 x  4  0   16
2  0   4 y  16
y4
x 8
x-intercept: 8,0 
y-intercept:  0,4 
Next, use these points to determine the slope.
40
m
08
4
1
m
m
8
2
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Example: Finding the Slope of a Line
Determine the slope of the line defined by the following
equation.
x5
First point:  5,2 
Second point:  5,8
y2  y1 8  2 6
m


Slope is undefined.
x2  x1 5  5 0
As soon as we realize that the line defined by the
equation is vertical, we can state that the slope is
undefined.
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Slope-Intercept Form of a Line
If the equation of a non-vertical line in x and y is
solved for y, the result is an equation of the form
y  mx  b.
The constant m is the slope of the line, and the line
crosses the y-axis at b ; that is, the y -intercept of
the line is  0,b . If the variable x does not appear in
the equation, the slope is 0 and the equation is
simply of the form y  b .
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Slope-Intercept Form of a Line
y
y2  y1
m
x2  x1
 x1 , y1 
y  mx  b
y  intercept, b
y2  y1
 x2 , y2 
x2  x1
x
The constant m is the slope of the line, and the line
crosses the y-axis at b ; that is, the y-intercept of the
line is  0,b  .
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Example: Graphing With Slope-Intercept Form
Use the slope-intercept form of the line to graph the
equation 4 x  3 y  6 .
4x  3y  6
3 y  4 x  6
4
y  x2
3
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Example: Graphing With Slope-Intercept Form
Find the equation of the line that passes through the
point  0,3 and has a slope of  3 . Then graph.
In Slope Intercept
Form:
y  mx  b
3
m
5
b3
3
y   x3
5
5
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Point-Slope Form of a Line
Given an ordered pair  x1 , y1  and a real number m,
an equation for the line passing through the
point  x1 , y1  with slope m is
y  y1  m  x  x1  .
Note that m, x1 , and y1 are all constants, and that x
and y are variables. Note also that since the line,
by definition, has slope m, vertical lines cannot be
described in this form.
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Example: Finding Slope-Intercept Form
Find the equation, in slope-intercept form, of the line
that passes through the point  4, 1 with slope 2.
x1 y1
m
y  y1  m  x  x1 
 4, 1 slope: 2
y   1  2  x  4 
y  1  2x  8
y  2x  9
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Example: Finding Slope-Intercept Form
Find the equation, in slope-intercept form, of the line
that passes through the two points  3,5 and  2,3.
53
m
3 2
m2
y  5  2  x  3
y – 5 = 2x – 6
y = 2x - 1