Chapter 6
Concept
Lines
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y = 2x - 3 -2x + y = -3
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y = 2x - 3 -2x + y = -3
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
2x - 3
- x + 4 3 2 2x + 3y = 12
-2x + y = -3
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
2x - 3
- x + 4 3 2 2x + 3y = 12
-2x + y = -3
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
y =
2x - 3
- x + 4
x 1
3 2
2 - x + y = 0
2x + 3y = 12
-2x + y = -3
1 2
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
y =
2x - 3
- x + 4
x 1
3 2
2 - x + y = 0
2x + 3y = 12
-2x + y = -3
1 2
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2 - x + y = 0
2x + 3y = 12
-2x + y = -3
y = 4
1 2
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2 - x + y = 0
2x + 3y = 12
-2x + y = -3
y = 4
1 2
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2 - x + y = 0
2x + 3y = 12
-2x + y = -3
y = 4
N/A x = -4
1 2
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2 - x + y = 0
2x + 3y = 12
-2x + y = -3
y = 4
x = -4
1 2
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2 - x + y = 0
2x + 3y = 12
-2x + y = -3
y = 4
x = -4
Ax + By = C y = mx + b
1 2
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2 - x + y = 0
2x + 3y = 12
2x - y = 3
y = 4
x = -4
Ax + By = C y = mx + b
1 2
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2 x - 2y = 0
2x + 3y = 12
2x - y = 3
y = 4
x = -4
Ax + By = C y = mx + b
Slope-Intercept form Standard Form
Ax + By = C y = mx + b
Slope
Gradient
how “steep” the line is
Slope-Intercept form Standard Form
Ax + By = C y = mx + b
Slope
Gradient
how “steep” the line is
m -A B
Slope of this
line is 1 = 1
1
The slope is becoming
less steep - slope is
getting smaller than 1.
The slope is becoming
less steep - slope is
getting smaller than 1.
The slope is becoming
less steep - slope is
getting smaller than 1.
The slope is becoming
less steep - slope is
getting smaller than 1.
The slope is becoming
so small - what is the
slope of this line?
The slope is becoming
so small - what is the
slope of this line?
Slope = 0
Slope of this
line is 1 = 1
1
The slope is becoming
more steep - slope is
getting bigger than 1.
The slope is becoming
more steep - slope is
getting bigger than 1.
The slope is becoming
more steep - slope is
getting bigger than 1.
The slope is becoming
more steep - slope is
getting bigger than 1.
The slope is becoming
more steep - slope is
getting bigger than 1.
The slope is
becoming so big-
what is the slope
of this line?
The slope is
becoming so big-
what is the slope
of this line?
Slope = doesn’t exist
The slope is
becoming so big-
what is the slope
of this line?
Slope = undefined
Slope-Intercept form Standard Form
Ax + By = C y = mx + b
Slope
Gradient
how “steep” the line is
m -A B
Slope-Intercept form Standard Form
y = 2x - 3
y = mx + b Ax + By = C
slope = m slope = - A B
2x - y = 3
y =
y =
2x - 3
- x + 4 3 2 2x + 3y = 12
Slope-Intercept form Standard Form
slope = m slope = -
Ax + By = C y = mx + b
A B
2x - y = 3
y =
y =
y =
2x - 3
- x + 4
x 1
3 2
2
2x + 3y = 12
Slope-Intercept form Standard Form
slope = m slope = -
y = mx + b Ax + By = C
A B
2x - y = 3
x - 2y = 0
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2
2x + 3y = 12
Slope-Intercept form Standard Form
slope = m slope = -
y = mx + b Ax + By = C
A B
2x - y = 3
x - 2y = 0
y = 4
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2
2x + 3y = 12
x = -4
Slope-Intercept form Standard Form
slope = m slope = - A B
Ax + By = C y = mx + b
Undefined slope y = mx + b
2x - y = 3
x - 2y = 0
y = 4
(x,y)
point A (3,6) point B (1,2)
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2 6-2 4
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2
(x,y)
point A (3,6) point B (1,2)
6-2 4
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2
(x,y)
point A (3,6) point B (1,2)
6-2 4
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2
(x,y)
point A (3,6) point B (1,2)
6-2 4
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2
(x,y)
point A (3,6) point B (1,2)
6-2 4
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2
(x,y)
point A (3,6) point B (1,2)
6-2 4
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2 6-2 4
(x,y)
point A (3,6) point B (1,2)
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2
(x,y)
point A (3,6) point B (1,2)
6-2 4
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = 2 3-1 2
(x,y)
point A (3,6) point B (1,2)
6-2 4
slope = = = 2 2-6 -4 1-3 -2
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = - -4-4 -8 2 1--3 4 1
(x,y)
point A (-4,1) point B (4,-3)
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = - -4-4 -8 2
(x,y)
point A (-4,1) point B (4,-3)
1--3 4 1
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = - -4-4 -8 2
(x,y)
point A (-4,1) point B (4,-3)
1--3 4 1
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = - -4-4 -8 2
(x,y)
point A (-4,1) point B (4,-3)
1--3 4 1
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = - -4-4 -8 2
(x,y)
point A (-4,1) point B (4,-3)
1--3 4 1
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = - -4-4 -8 2
(x,y)
point A (-4,1) point B (4,-3)
1--3 4 1
A
B
slope = change in y-values change in x-values
slope = difference in y-values difference in x-values
slope = x y
slope = = = - -4-4 -8 2
(x,y)
point A (-4,1) point B (4,-3)
1--3 4 1
-3-1 -4 1 slope = = = - 4--4 8 2
A
B
slope = = = = x -6-4 -10 5 y 0-4 -4 2 .
(x,y)
point A (-6,0) point B (4,4)
A
B
slope = = = = x -6-4 -10 5 y 0-4 -4 2 .
(x,y)
point A (-6,0) point B (4,4)
A
B
slope = = = = x -6-4 -10 5 y 0-4 -4 2 .
(x,y)
point A (-6,0) point B (4,4)
A
B
slope = = = = x -6-4 -10 5 y 0-4 -4 2 .
(x,y)
point A (-6,0) point B (4,4)
A
B
slope = = = = x -6-4 -10 5 y 0-4 -4 2 .
(x,y)
point A (-6,0) point B (4,4)
A
B
slope = = = = x -6-4 -10 5 y 0-4 -4 2 .
(x,y)
point A (-6,0) point B (4,4)
slope = = = = y 4-0 4 2 . x 4--6 10 5
A
B
slope = = = = 7 x -3--1 -2 y -9-5 -14
(x,y)
point A (-3,-9) point B (-1,5)
A
B
slope = = = = 7 x -3--1 -2 y -9-5 -14
(x,y)
point A (-3,-9) point B (-1,5)
A
B
slope = = = = 7 x -3--1 -2 y -9-5 -14
(x,y)
point A (-3,-9) point B (-1,5)
A
B
slope = = = = 7 x -3--1 -2 y -9-5 -14
(x,y)
point A (-3,-9) point B (-1,5)
A
B
slope = = = = 7 x -3--1 -2 y -9-5 -14
(x,y)
point A (-3,-9) point B (-1,5)
slope = = = = 7 x -1--3 2 y 5--9 14
B A
slope = = = = 0 x 3--3 6 y 3-3 0
(x,y)
point A (3,3) point B (-5,3)
B A
slope = = = = 0 x 3--5 6 y 3-3 0
(x,y)
point A (3,3) point B (-5,3)
B A
slope = = = = 0 x 3--5 8 y 3-3 0
(x,y)
point A (3,3) point B (-5,3)
B A
slope = = = = 0 x 3--5 8 y 3-3 0
(x,y)
slope = = = = 0 y 3-3 0 x -5-3 -8
point A (3,3) point B (-5,3)
B
A
slope = = = = undefined x 3-3 0 y 3--3 6
(x,y)
point A (3,3) point B (3,-3)
Can’t divide by 0
Zero cannot be on bottom of fraction
B
A
slope = = = = undefined x 3-3 0 y 3--3 6
(x,y)
point A (3,3) point B (3,-3)
Can’t divide by 0
Zero cannot be on bottom of fraction
B
A
slope = = = = undefined x 3-3 0 y 3--3 6
(x,y)
point A (3,3) point B (3,-3)
Can’t divide by 0
Zero cannot be on bottom of fraction
B
A
x 3-3 0 y 3--3 6
(x,y)
point A (3,3) point B (3,-3)
Can’t divide by 0
Zero cannot be on bottom of fraction
slope = = = = undefined
B
A
slope = = = = undefined x 3-3 0 y 3--3 6
(x,y)
point A (3,3) point B (3,-3)
Can’t divide by 0
Zero cannot be on bottom of fraction
B
A
slope = = = = undefined x 3-3 0 y 3--3 6
(x,y)
point A (3,3) point B (3,-3)
Can’t divide by 0
Zero cannot be on bottom of fraction
B A
slope = = = = 0 x 3--5 8 y 3-3 0
(x,y)
point A (3,3) point B (-5,3)
Yes,
Zero can be on top of fraction
B
A
slope = = = = undefined x 3-3 0 y 3--3 6
(x,y)
point A (3,3) point B (3,-3)
NO
Zero cannot be on bottom of fraction
Write Equations of Lines 2 different ways:
Slope-Intercept form Standard Form
y =
y =
y =
y =
2x - 3
- x + 4
x
4
1
3 2
2
2x + 3y = 12
x = -4
2x - y = 3
x - 2y = 0
y = 4
y = 2x + 2
y = 2x + 2 y = 2x - 3
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 7
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 7
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 7
y = 2x - 9
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 7
y = 2x - 9
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 7
y = 2x - 9
y = 2x + 0
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 7
y = 2x - 9
y = 2x + 0
y = 2x + 2 y = 2x - 3
y = 2x - 6
y = 2x + 7
y = 2x - 9
y = 2x + 0
Parallel lines have the same slope.
y = 2x + 2
y = 2x + 2
y = 2x + 2
y =- x + 4 1 2
y = 2x + 2
y =- x + 4 1
Slopes of perpendicular lines are
1) Reciprocals 2) Opposite signs
2
y = x + 4 3 2
Slopes of perpendicular lines are
1) Reciprocals 2) Opposite signs
y = x + 4 3 2
Slopes of perpendicular lines are
1) Reciprocals 2) Opposite signs
y = x + 4
y =- x + 5 2 3
3 2
Slopes of perpendicular lines are
1) Reciprocals 2) Opposite signs
y = x + 4
y =- x + 5
y = x - 2
2 3
5 2
3 2
Slopes of perpendicular lines are
1) Reciprocals 2) Opposite signs
y = x + 4
y =- x + 5
y = x - 2
2 3
5 2
3 2
Slopes of perpendicular lines are
1) Reciprocals 2) Opposite signs
y = x + 4
y =- x + 5
y =- x - 4
y = x - 2
2 3
5 2
3 2
2 5
Slopes of perpendicular lines are
1) Reciprocals 2) Opposite signs