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Equations of Lines
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At the end of this lesson
you will be able to:
Write equations for non-vertical lines.
Write equations for horizontal lines.
Write equations for vertical lines.
Use various forms of linear equations.
Calculate the slope of a line passing
through two points.
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Before we begin.
Let’s review some vocabulary.
Slope (m): The measure of the steepness of a line; it is the ratio
of vertical change (DY) to horizontal change (DX).
Vertical change (DY)
Slope (m) =
Horizontal change (DX)
Y-intercept (b): The y-coordinate of the point where the
graph of a line crosses the y-axis.
X-intercept (a): The x-coordinate of the point where the graph
of a line crosses the x-axis.
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Equations of
Non-vertical Lines.
Let’s look at a line with a y-intercept of
b, a slope m and let (x,y) be any point
on the line.
Y-axis
(x,y)
(0,b)
X-axis
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Slope Intercept Form
The equation for the non-vertical line is:
y = mx + b ( Slope Intercept Form )
Where m is:
m=
DY
DX
=
Y-axis
(x,y)
(y – b)
(x – 0)
DY
(0,b)
DX
X-axis
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More Equations of
Non-vertical Lines.
Let’s look at a line passing through
Point 1 (x1,y1) and Point 2 (x2,y2).
Y-axis
(x2,y2)
(x1,y1)
X-axis
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Point Slope Form
The equation for the non-vertical line is:
y – y1 = m(x – x1) ( Point Slope Form )
Where m is:
m=
DY
DX
=
Y-axis
(x2,y2)
(y2 – y1)
DY
(x2 – x1)
(x1,y1)
DX
X-axis
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Equations of
Horizontal Lines.
Let’s look at a line
with a y-intercept of b,
a slope m = 0, and let
(x,b) be any point on
the Horizontal line.
(0,b)
Y-axis
(x,b)
X-axis
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Horizontal Line
The equation for the horizontal line is still
y = mx + b ( Slope Intercept Form ).
Where m is:
m=
DY
DX
=
Y-axis
(b – b)
(x – 0)
=0
(x,b)
(0,b)
DX
DY = 0
X-axis
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Horizontal Line
Because the value of m is 0,
y = mx + b becomes
Y-axis
y=b
(A Constant Function)
(0,b)
(x,b)
X-axis
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Equations of
Vertical Lines.
Let’s look at a line with
no y-intercept b, an xintercept a, an undefined
slope m, and let (a,y) be
any point on the vertical
line.
Y-axis
(a,y)
(a,0)
X-axis
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Vertical Line
The equation for the vertical line is
x = a ( a is the X-Intercept of the line).
Because m is:
Y-axis
(a,y)
m=
DY
DX
=
(y – 0)
= Undefined
(a – a)
(a,0)
X-axis
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Vertical Line
Because the value of m is undefined, caused by the
division by zero, there is no slope m.
x = a becomes the equation
Y-axis
x=a
(The equation of a vertical line)
(a,y)
(a,0)
X-axis
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Example 1: Slope Intercept Form
Find the equation for the line
with m = 2/3 and b = 3
Because b = 3
The line will pass through (0,3)
Y-axis
Because m = 2/3
DY = 2
DX = 3
The Equation for the line is:
y = 2/3 x + 3
DY = 2
(0,3)
DX = 3
X-axis
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Slope Intercept Form Practice
Write the equation for the lines using Slope Intercept
form.
1.) m = 3 & b = 3
2.) m = 1 & b = -4
3.) m = -4 & b = 7
4.) m = 2 & b = 0
5.) m = 1/4 & b = -2
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Example 2: Point Slope Form
Let’s find the equation for the line passing through
the points (3,-2) and (6,10)
First, Calculate m :
Y-axis
(6,10)
(10 – -2)
m=
=
=
DX
(6 – 3)
DY
12
= 4
3
DY
(3,-2)
X-axis
DX
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Example 2: Point Slope Form
To find the equation for the line passing through
the points (3,-2) and (6,10)
Next plug it into Point Slope From :
y – y1 = m(x – x1)
Select one point as P1 :
Let’s use (3,-2)
Y-axis
(6,10)
The Equation becomes:
y – -2 = 4(x – 3)
DY
(3,-2)
X-axis
DX
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Example 2: Point Slope Form
Simplify the equation / put it into Slope Intercept Form
Distribute on the right side and the equation becomes:
y + 2 = 4x – 12
Subtract 2 from both sides gives.
y + 2 = 4x – 12
-2 =
-2
Y-axis
(6,10)
y = 4x – 14
DY
(3,-2)
X-axis
DX
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Point Slope Form Practice
Find the equation for the lines passing through the
following points using Point Slope form.
1.) (3,2) & ( 8,-2)
2.) (-5,4) & ( 10,-12)
3.) (1,-5) & ( 7,7)
4.) (4,2) & ( -8,-4)
5.) (5,3) & ( 7,9)
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Example 3: Horizontal Line
Let’s find the equation for the line passing through
the points (0,2) and (5,2)
y = mx + b ( Slope Intercept Form ).
Where m is:
Y-axis
m=
DY
DX
=
(2 – 2)
(5 – 0)
=0
(5,2)
(0,2)
DX
DY = 0
X-axis
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Example 3: Horizontal Line
Because the value of m is 0,
y = 0x + 2 becomes
Y-axis
y=2
(A Constant Function)
(0,2)
(5,2)
X-axis
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Horizontal Line Practice
Find the equation for the lines passing through the
following points.
1.) (3,2) & ( 8,2)
2.) (-5,4) & ( 10,4)
3.) (1,-2) & ( 7,-2)
4.) (4,3) & ( -2,3)
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Example 4: Vertical Line
Let’s look at a line with no yintercept b, an x-intercept a,
passing through (3,0) and
(3,7).
Y-axis
(3,7)
(3,0)
X-axis
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Example 4: Vertical Line
The equation for the vertical line is:
x = 3 ( 3 is the X-Intercept of the line).
Because m is:
Y-axis
(3,7)
m=
DY
DX
=
(7 – 0)
(3 – 3)
=
7
0
= Undefined
(3,0)
X-axis
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Vertical Line Practice
Find the equation for the lines passing through the
following points.
1.) (3,5) & ( 3,-2)
2.) (-5,1) & ( -5,-1)
3.) (1,-6) & ( 1,8)
4.) (4,3) & ( 4,-4)
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Equation Internet Activity
Click on each of the links below and follow
the directions to complete problems.
Slope Intercept Form Information and Practice
SparkNotes: Slope Intercept Form
Point Slope Form Information and Practice
SparkNotes: Point Slope Form
Graphing Equations Conclusions
What are the similarities you see in the
equations for Parallel lines?
What are the similarities you see in the
equations for Perpendicular lines?
Record your observations on your sheet.
Equation Summary
Slope:
Vertical change (DY)
Slope (m) =
Horizontal change (DX)
Slope-Intercept Form:
y = mx + b
Point-Slope Form:
y – y1 = m(x – x1)