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Welcome to Section 2.1 - Lines in the Plane & Slope
This section is a review of linear equations that you learned about
in previous courses.
Definition of a Linear Equation
A linear equation is any equation that may be written in the form
y = mx + b where
m=the slope of the line and (x=0, y=b) is the y-intercept of the graph.
Definition of Slope
The slope of a given line is the ratio of the distance units up divided by the
distance units right when moving from point to point on the line.
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Example: If the slope of a line is m=2, and a point on the line is (3,-4),
find another point on the line.
Since m=2, we may express this as m=2/1 which means that you move UP 2, and
to the RIGHT 1 inorder to find another point.
If you move UP 2 and RIGHT 1,
the point obtained is (4,-2).
You may also move DOWN 2
and LEFT 1 to obtain a point of
(2,-6) since m=2/1 is equal to
m=(-2)/(-1). If there is a
negative component of the slope
we interpret it as down or left.
Example: a slope of m=-3
is equal to m=(-3)/1 which
means Down 3, Right 1 or
m=3/(-1) which means Up 3,
Left 1.
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Slope Formula
Slope indicates the change in y divided by the change in x. The formula
for the slope of a line through 2 points (x1, y1) & (x2, y2) is
Example: Find the slope of the line that passes through (-3, 2) and (-4,
1).
The slope is
Point-Slope Formula for the Equation of a Line
Using the formula above, find an equation for the line that passes
through the points (2,1) and (x,y) where the line has slope m=3.
Note that (x,y) are the coordinates of any point on this line. Simply
substitute the points (2,1) and (x,y) & m=3 into the formula for m.
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If you substitute into the slope formula where (x1,y1) = (2,1),
(x2,y2) = (x,y), and m=3, you obtain
Point-Slope Formula
In general, the equation of a line that has a specified slope of m and
passes through a specified point (x1,y1) will have equation
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Example: Find the equation of the line with slope m=-2 that
passes through (-3,4). The equation is
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Slope-Intercept Form
The slope-intercept form of an equation is y=mx + b where m is the
slope and the y-intercept is (0,b).
Example: Rewrite the equation 3x + 2y -7 = 0 in slope-intercept form and
then determine the slope, the y-intercept, and then plot the
graph.
The slope is m=-3/2 and the
y-intercept is x=0, y=7/2.
Plot this intercept and use
the slope to find another
point.
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Parallel & Perpendicular Lines
Parallel lines have the same slopes.
Example: y = 3x + 4 & y= 3x - 6 are parallel because they both have
a slope of m=3. Note: They both "slant" the same way.
Perpendicular lines have slopes that are negative reciprocals of each other.
Example: y = 3x + 2 & y = -x/3 - 1 are perpendicular because the
slope of the first line is m=3 and the slope of the second
is m=-1/3 and -1/3 is the negative reciprocal of 3.
*The product of slopes of perpendicular lines will always be -1 also*
Putting All the Concepts Together
Find the equation of a line that passes through (3,2)
that is
a) Parallel to the line 4x - 2y = 6
b) Perpendicular to the line 4x - 2y = 6
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The slope-intercept form of 4x - 2y = 6
is obtained.
-2y = -4x + 6
y = 2x - 3
So the slope of the given line is m=2.
This means that the slope of the line through (3,2) that is parallel is ALSO
m=2.
We may substitute this info into the Point-Slope Formula to get
The slope of the perpendicular line is the negative reciprocal of m=2 which is
m=-1/2 and it also contains the point (3,2). Using the Point-Slope Formula,
the equation is
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Different Forms of Equations
The last 2 equations may be written in slope-intercept form.
y-2 = 2(x - 3) may be written as y = 2x - 4 by solving for y.
y-2 = (-1/2) (x - 3) may be written y = (-1/2)x + (7/2) by solving for y.
General Form
The general form of a linear equation is Ax + By + C = 0 where A, B, & C
are positive or negative integers (whole numbers). In other words, the
general form has ALL terms on one side of the equation with all fractions
cleared out.
Example: Write the 2 previous equations in "general" form.
y = 2x - 4 is written as -2x + y + 4 = 0 by subtracting 2x and adding 4 to
both sides. We may also multiply both sides by -1 to get 2x - y - 4 = 0.
Try to find the "general" form of the second equation.
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y = (-1/2)x + 7/2 may be written as
(1/2)x + y - 7/2 = 0 by adding (1/2)x and subtracting (7/2) from both sides.
Now, multiply both sides by 2 to get x + 2y - 7 = 0.
*Your text uses this "general" form in many problems*
Horizontal & Vertical Lines
Horizontal lines always have a slope of m=0 and their equation may always
be written in the form y=k where "k" is the y-value of all points on the line.
Example: The line y=3 is a horizontal line where ALL points on the line have
a y-value of y=3. The slope of this line is m = 0.
Vertical linese always have an UNDEFINED slope and their equation may
always be written in the form x = k where "k" is the x-value of all points on
the line.
Example: The line x=2 is a vertical line where ALL points on the line have
an x-value of x=3. The slope of this line is undefined.
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