Transcript pps
Slide 1
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 2
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 3
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 4
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 5
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 6
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 7
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 8
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 9
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 10
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 11
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 12
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 13
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 14
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 15
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 16
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 17
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 18
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 19
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 20
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 21
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 22
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 23
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 24
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 25
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 26
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 27
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 28
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 29
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 30
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 2
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 3
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 4
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 5
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 6
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 7
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 8
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 9
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 10
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 11
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 12
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 13
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 14
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 15
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 16
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 17
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 18
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 19
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 20
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 21
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 22
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 23
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 24
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 25
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 26
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 27
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 28
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 29
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
21
Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
22
Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
23
Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
24
Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
25
Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
26
Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
27
Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
28
Additional Example from Text
Page 27 # 62
29
Additional Example from Text
Page 27 # 64
30
Slide 30
Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
1
The Cartesian Coordinate System
The Cartesian coordinate system was named after
_____________________________________.
It consists of two real number lines which meet in a right angle
at a point called the ___________.
The two number lines divide the plane into four areas called
______________________.
The quadrants are numbered using Roman numerals, as shown
on the next slide. Each point in the plane corresponds to one and
only one ordered pair of numbers (x,y). Two ordered pairs are
shown.
2
The Cartesian Coordinate System
(continued)
II
I
Two points, (-1,-1)
and (3,1), are plotted.
(3,1)
x
III
(-1,-1)
The four quadrants
are as labeled.
IV
y
3
Linear Equations in Two Variables
A linear equation in two variables is an equation that can be
written in the standard form _________________________,
where A, B, and C are constants (A and B not both 0), and x
and y are variables.
A __________________ of an equation in two variables is an
ordered pair of real numbers that satisfy the equation. For
example, (4,3) is a ____________ of 3x - 2y = 6.
The graph of the solution set (all of the ordered pair solutions)
of a linear equation is a _____________.
4
Standard Form of a Linear Equation
Ax + By = C
If A is not equal to zero and B is not equal to zero, then
A
C
Ax + By = C can be written as
y
This is known as
SLOPE-INTERCEPT FORM.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
x
B
B
y mx b
y
C
B
x
C
A
5
Intercepts of a Line
The INTERCEPTS of the graph of an equation are the points at
which the graph intersects the horizontal (x) and/or vertical (y)
axes.
6
Finding the Intercepts Algebraically
1. The x-intercept of an equation can be found by algebraically by
7
Finding the Intercepts Algebraically
2. The y-intercept of an equation can be found by algebraically by
8
Using Intercepts to Graph a Line
Find the intercepts of the graph of the equation 2x - 6y = 12
algebraically, and use them to graph the line (on next slide).
9
Using Intercepts to Graph a Line
Graph 2x - 6y = 12.
x
y
0
x-intercept
0
y-intercept
3
check point
10
Using a Graphing Calculator
1) Graph 2x - 6y = 12 on a graphing calculator and find the
intercepts.
(See handout)
To Find Intercepts Using Graphing Calc
11
Other Examples
2) Graph 5x – 3y = 8 on a graphing calculator and find the
intercepts. Write as ordered pairs.
3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the
intercepts. Write as ordered pairs.
12
Special Case: Vertical Line
The graph of x = k is the graph of a VERTICAL LINE
k units from the y-axis.
Example:
Graph x = -7
y
x
13
Special Case: Horizontal Line
The graph of y = k is the graph of the HORIZONTAL LINE
k units from the x-axis.
Example:
Graph y = 3
y
x
14
Writing Equations of Horizontal and
Vertical Lines
Example
Graph the equations of the vertical and horizontal lines
through the point (-5.5, 3), and then write the equations of
each.
15
Horizontal and Vertical Lines
Example
Write each of the following equations in standard form and
describe their graphs.
1. 5 y 13 21
2.
6
1
2
3
x 0
4
16
Slope of a Line
Given any two points in a plane, we can calculate the slope of the
line through those points.
Slope of a line: m
y 2 y1
x 2 x1
rise
run
x1 , y 1
rise
x2 , y2
run
17
Slope of a Line
Find the slope of the lines containing the following pairs of points:
1.
8, 7
and
2, 1
2.
2.8,
3.1 a n d
1.8,
2.6
18
Slope of a Line
Find the slope of the lines containing the following pairs of points:
3.
5,
2 and
1, 2
4.
7,
4 and
7 ,1 0
19
Slope-Intercept Form
The equation
y = mx + b
is called the SLOPE-INTERCEPT FORM of an equation of
a line.
The letter m represents the _____________ and
b represents the __________________________.
20
Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line whose
equation is 5x - 2y = 10. Then graph this line.
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Point-Slope Form
The point-slope form of the equation of a line is
y y1 m ( x x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y 2 y1
x 2 x1
m
Cross-multiply and
substitute the more general
x for x2 and y for y2
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Example 1
Find the EQUATION of the line through the points (-5, 7) and (4, 16).
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Example 2
Find the EQUATION of the line through the points (2, 3) and (-3, 7).
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Example 3
Find the EQUATION of the line through the points (9, 0) and (9, -5).
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Example 4
Find the EQUATION of the line through the points (-8, 1) and (-3, 1).
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Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years.
1. Since it is a linear function, we are looking for an equation in the
form y = mx + b.
Instead of x and y, our example uses the variables t and V.
t is the ___________________________ variable.
V is the __________________________ variable.
So our answer will have the form of _______________________.
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Application (continued)
Office equipment was purchased for $20,000 and will have a scrap value of
$2,000 after 10 years. If its value is depreciated linearly, find the linear
equation that relates value (V) in dollars to time (t) in years.
2. Find the slope of the equation (rate of change):
m=
3. Do we know the y-intercept (i.e. Do we know the value, V, when
time, t, is 0?)
4. Substitute the values for m and b into the equation V = mt + b.
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Additional Example from Text
Page 27 # 62
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Additional Example from Text
Page 27 # 64
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