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What is a Line?
? A line is the set of points forming a straight
path on a plane
? The slant (slope) between any two points on
a line is always equal
? A line on the Cartesian plane can be
described by a linear equation
x-axis
y-axis

Definition - Linear Equation
? Any equation that can be put into the form
Ax + By ? C = 0, where A, B, and C are
Integers and A and B are not both 0, is called
a linear equation in two variables.
? The graph will be a straight line.
? The form Ax + By ? C = 0 is called standard
form (Integer coefficients all on one side = 0)

Definition - Linear Equation
? The equation of a line describes all of the
points on the line
? The equation is the rule for any ordered pair
on the line
1. 3x + 2y – 8 = 0
(4, -2) is on the line
(5, 1) is not on the line
2. x – 7y + 2 = 0
(4, -2) is not on the line
(5, 1) is on the line
Examples:
Test the point by plugging the x and y into the equation

Slope
Slope describes the
direction of a line.

Guard against 0 in
the denominator
Slope
If x1 ? x2, the slope of the line
through the distinct points P1(x1, y1)
and P2(x2, y2) is:
1
2
1
2
x
x
y
y
x
in
change
y
in
change
run
rise
slope
?
?
?
?
?
Why is
this
needed
?

x-axis
y-axis
Find the slope between (-3, 6) and (5, 2)
Rise
Run
-4
8
-1
2
= =
(-3, 6)
(5, 2)

Calculate the slope between (-3, 6) and (5, 2)
1
2
1
2
x
x
y
y
m
?
?
?
)
3
-
(
)
5
(
)
6
(
)
2
(
?
?
?
m
8
4
-
?
2
1
-
?
x1 y1 x2 y2
We use the letter m
to represent slope
m

Find the Slopes
(5, -2)
(11, 2)
(3, 9)
1
2
1
2
x
x
y
y
m
?
?
?
3
11
9
2
1
?
?
?
m
Yellow
5
11
)
2
-
(
2
2
?
?
?
m
Blue
3
5
9
2
-
3
?
?
?
m
Red
8
7
-
?
3
2
?
2
11
-
?

Find the slope between (5, 4) and (5, 2).
1
2
1
2
x
x
y
y
m
?
?
?
)
5
(
)
5
(
)
4
(
)
2
(
?
?
?
m
0
2
-
?
STOP
This slope is undefined.
x1 y1 x2 y2

x
y
Find the slope between (5, 4) and (5, 2).
Rise
Run
-2
0
Undefined
= =

Find the slope between (5, 4) and (-3, 4).
1
2
1
2
x
x
y
y
m
?
?
?
)
5
(
)
3
-
(
)
4
(
)
4
(
?
?
?
m
8
-
0
?
This slope is zero.
x1 y1 x2 y2
0
?

x
y
Rise
Run
0
-8
Zero
= =
Find the slope between (5, 4) and (-3, 4).

From these results we
can see...
?The slope of a vertical
line is undefined.
?The slope of a
horizontal line is 0.

Find the slope of the line
4x - y = 8
)
0
(
)
2
(
)
8
-
(
)
0
(
?
?
?
m
2
8
?
Let x = 0 to
find the
y-intercept.
8
-
8
-
8
)
0
(
4
?
?
?
?
y
y
y Let y = 0 to
find the
x-intercept.
2
8
4
8
)
0
(
4
?
?
?
?
x
x
x
(0, -8) (2, 0)
4
?
First, find two points on the line
x1 y1 x2 y2

Find the slope of the line
4x ? y = 8 Here is an easier way
Solve
for y.
8
4 ?
? y
x
8
4
-
- ?
? x
y
8
4 ?
? x
y
When the equation is solved for y the
coefficient of the x is the slope.
We call this the slope-intercept form
y = mx + b
m is the slope and b is the y-intercept

x
y
Graph the line that goes through (1, -3) with
(1,-3)
4
3
-
?
m

Sign of the Slope
Which have a
positive slope?
Green
Blue
Which have a
negative slope?
Red
Light Blue
White
Undefined
Zero
Slope

Slope of Parallel Lines
? Two lines with the
same slope are parallel.
? Two parallel lines have
the same slope.

Are the two lines parallel?
L1: through (-2, 1) and (4, 5) and
L2: through (3, 0) and (0, -2)
)
0
(
)
3
(
)
2
-
(
)
0
(
2
?
?
?
m
)
2
-
(
)
4
(
)
1
(
)
5
(
1
?
?
?
m
6
4
?
3
2
?
3
2
?
2
1
2
1
L
L
m
m
?
?
This symbol means Parallel

x
y
Perpendicular Slopes
4
3
-
1 ?
m
3
4
2 ?
m
4
3
What can we say
about the intersection
of the two white lines?

Slopes of Perpendicular Lines
? If neither line is vertical then the slopes of
perpendicular lines are negative reciprocals.
? Lines with slopes that are negative reciprocals
are perpendicular.
? If the product of the slopes of two lines is -1
then the lines are perpendicular.
? Horizontal lines are perpendicular to vertical
lines.

Write parallel, perpendicular or neither for the
pair of lines that passes through (5, -9) and (3, 7)
and the line through (0, 2) and (8, 3).
)
5
(
)
3
(
)
9
-
(
)
7
(
1
?
?
?
m
)
0
(
)
8
(
)
2
(
)
3
(
2
?
?
?
m
2
-
16
? 8
-
?
8
1
? ?
?
?
?
?
?
1
8
-
?
?
?
?
?
?
8
1
8
8
-
? 1
-
?
2
1
2
1 1
-
L
L
m
m
?
?
?
?
This symbol means Perpendicular

Objectives
? Write the equation of a line, given its
slope and a point on the line.
? Write the equation of a line, given two
points on the line.
? Write the equation of a line given its
slope and y-intercept.

Objectives
? Find the slope and the y-intercept of a
line, given its equation.
? Write the equation of a line parallel or
perpendicular to a given line through a
given point.

Slope-intercept Form
Objective
Write the equation of a line, given its slope
and a point on the line.
y = mx + b
m is the slope and b is the y-intercept

Write the equation of the line
with slope m = 5 and y-int -3
Take the slope intercept form y = mx + b
Replace in the m and the b y = mx + b
y = 5x + -3
y = 5x – 3
Simplify
That’s all there is to it… for this easy question

Find the equation of the line
through (-2, 7) with slope m = 3
Take the slope intercept form y = mx + b
Replace in the y, m and x y = mx + b
7 = mx + b
x y m
7 = 3x + b
7 = 3(-2) + b
7 = -6 + b
Solve for b
7 + 6 = b
13 = b
Replace m and b back into
slope intercept form y = 3x + 13

Write an equation of the line
through (-1, 2) and (5, 7).
First calculate the slope.
b
?
? )
1
-
(
2 6
5
1
2
1
2
x
x
y
y
m
?
?
?
)
1
-
(
5
2
7
?
?
?
6
5
?
Now plug into y, m and x into
slope-intercept form.
(use either x, y point)
Solve for b
Replace back into slope-intercept form
b
mx
y ?
?
b
?
? 6
5
-
2
b
?
? 6
5
2
b
?
6
17
6
17
6
5 ?
? x
y
Only replace
the m and b

Horizontal and
Vertical Lines
? If a is a constant,
the vertical line
though (a, b) has
equation x = a.
? If b is a constant,
the horizontal line
though ( a, b,) has
equation y = b.
(a, b)

Write the equation of the line
through (8, -2); m = 0
2
-
?
y
Slope = 0 means the line is horizontal
That’s all there is!

Find the slope and
y-intercept of
2x – 5y = 1
Solve for y and
then we will be
able to read it from
the answer.
1
5
2 ?
? y
x
y
x 5
1
2 ?
?
y
x ?
?
5
1
5
2
5
1
x
5
2
y ?
?
5
2
?
m
5
1
-
5 5 5
Slope: y-int:

Write an equation for the line
through (5, 7) parallel to 2x – 5y = 15.
5
2
?
?m
15
5
2 ?
? y
x
y
x 5
15
2 ?
?
5
5
5
15
5
2 y
x
?
?
y
x ?
? 3
5
2

We know the slope and
we know a point.
)
7
,
5
(
5
2
?
m
b
?
? )
5
(
7 5
2 b
mx
y ?
?
7 = 2 + b
7 – 2 = b
5 = b
5
5
2 ?
? x
y

3
5
2
?
? x
y
5
5
2
?
? x
y
15
5
2 ?
? y
x

The slope of the perpendicular.
? The slope of the perpendicular line is the
negative reciprocal of m
? Flip it over and change the sign.
3
2
Examples of slopes of perpendicular lines:
-2
5
1
2
7
-
2.4
Note: The product of perpendicular slopes is -1
2
3
1
5
= -5 -2
1 2
1
? 12
5
-7
2 7
2
?

What about the special cases?
? What is the slope of
the line perpendicular
to a horizontal line?
1
0
?
Well, the slope of a
horizontal line is 0…
So what’s the negative
reciprocal of 0?
0
0
1
Anything over
zero is undefined
The slope of a line
? to a horizontal
line is undefined.

Write an equation in for the line through (-8, 3)
perpendicular to 2x – 3y = 10.
We know the perpendicular
slope and we know a point.
3
2
?
slope
)
3
,
8
-
(
2
3
-
2 ?
m
Isolate y to find the slope: 2x – 3y = 10
2x = 10 + 3y
2x – 10 = 3y
3 3 3
b
?
? )
8
-
(
3 2
3
- b
mx
y ?
?
3 = 12 + b
3 – 12 = b
-9 = b
9
2
-3
: ?
? x
y
answer

Write an equation in standard form for the
line through (-8, 3) perpendicular to
2x - 3y = 10.
3
10
3
2
?
? x
y
9
2
3
-
?
? x
y

Summary
b
mx
y ?
?
? Slope-intercept form
? y is isolated
? Slope is m.
? y-intercept is (0, b)

Summary
? Vertical line
– Slope is undefined
– x-intercept is (a, 0)
– no y-intercept
? Horizontal line
– Slope is 0.
– y-intercept is (0, b)
– no x-intercept
a
x ?
b
y ?

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