The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.

1. The Rectangular Coordinate System and Lines
Frank Ma © 2011

2. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The Rectangular Coordinate System

3. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The Rectangular Coordinate System

4. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The Rectangular Coordinate System

5. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The horizontal axis is called
the x-axis.
The Rectangular Coordinate System

6. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis.
The Rectangular Coordinate System

7. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
The Rectangular Coordinate System

8. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
The Rectangular Coordinate System

9. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
x = amount to move
right (+) or left (–).
The Rectangular Coordinate System

10. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grids where each point in the plane is addressed
by an ordered pair of numbers (x, y).
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
The Rectangular Coordinate System

11. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
The Rectangular Coordinate System

12. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3)
The Rectangular Coordinate System

13. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right,
The Rectangular Coordinate System

14. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
(4, –3)
P
The Rectangular Coordinate System

15. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The Rectangular Coordinate System

16. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
The Rectangular Coordinate System

17. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B
The Rectangular Coordinate System

18. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B
C
The Rectangular Coordinate System

19. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) that corresponds to a point is called
the coordinate of the point, x is the x-coordinate and y is
the y-coordinate.
A
B
C
The Rectangular Coordinate System

20. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) that corresponds to a point is called
the coordinate of the point, x is the x-coordinate and y is
the y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R.
P
Q
R
The Rectangular Coordinate System

21. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) that corresponds to a point is called
the coordinate of the point, x is the x-coordinate and y is
the y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R.
P(4, 5),
P
Q
R
The Rectangular Coordinate System

22. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) that corresponds to a point is called
the coordinate of the point, x is the x-coordinate and y is
the y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R.
P(4, 5), Q(3, -5),
P
Q
R
The Rectangular Coordinate System

23. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) that corresponds to a point is called
the coordinate of the point, x is the x-coordinate and y is
the y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R.
P(4, 5), Q(3, -5), R(-6, 0)
P
Q
R
The Rectangular Coordinate System

24. The coordinate of the
origin is (0, 0).
(0,0)
The Rectangular Coordinate System

25. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(0,0)
The Rectangular Coordinate System

26. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)
(0,0)
The Rectangular Coordinate System

27. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
(0,0)
The Rectangular Coordinate System

28. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).(0,0)
The Rectangular Coordinate System

29. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).
(0, 6)
(0,0)
The Rectangular Coordinate System

30. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).
(0, -4)
(0, 6)
(0,0)
The Rectangular Coordinate System

31. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
III
III IV
The axes divide the plane
into four parts and they are
counter-clockwisely denoted
as quadrants I, II, III, and IV.
The Rectangular Coordinate System

32. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
The axes divide the plane
into four parts and they are
counter-clockwisely denoted
as quadrants I, II, III, and IV.
Respectively, the signs of
the coordinates of each
quadrant are shown.
III
III IV
The Rectangular Coordinate System

33. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
The axes divide the plane
into four parts and they are
counter-clockwisely denoted
as quadrants I, II, III, and IV.
Respectively, the signs of
the coordinates of each
quadrant are shown.
III
III IV
(+,+)(–,+)
(–,–) (+,–)
The Rectangular Coordinate System

34. When the x-coordinate of the
point (x, y) is changed to its
opposite as (–x , y), the new
point is the reflection
across the y-axis.
(5,4)
The Rectangular Coordinate System

35. When the x-coordinate of the
point (x, y) is changed to its
opposite as (–x , y), the new
point is the reflection
across the y-axis.
(5,4)(–5,4)
The Rectangular Coordinate System

36. When the x-coordinate of the
point (x, y) is changed to its
opposite as (–x , y), the new
point is the reflection
across the y-axis.
When the y-coordinate of
the point (x, y) is changed to
its opposite as (x , –y), the
new point is the reflection
across the x-axis.
(5,4)(–5,4)
The Rectangular Coordinate System

37. When the x-coordinate of the
point (x, y) is changed to its
opposite as (–x , y), the new
point is the reflection
across the y-axis.
When the y-coordinate of
the point (x, y) is changed to
its opposite as (x , –y), the
new point is the reflection
across the x-axis.
(5,4)(–5,4)
(5, –4)
The Rectangular Coordinate System

38. When the x-coordinate of the
point (x, y) is changed to its
opposite as (–x , y), the new
point is the reflection
across the y-axis.
When the y-coordinate of
the point (x, y) is changed to
its opposite as (x , –y), the
new point is the reflection
across the x-axis.
(5,4)(–5,4)
(5, –4)
(–x, –y) is the reflection of
(x, y) across the origan.
The Rectangular Coordinate System

39. When the x-coordinate of the
point (x, y) is changed to its
opposite as (–x , y), the new
point is the reflection
across the y-axis.
When the y-coordinate of
the point (x, y) is changed to
its opposite as (x , –y), the
new point is the reflection
across the x-axis.
(5,4)(–5,4)
(5, –4)
(–x, –y) is the reflection of
(x, y) across the origan.
(–5, –4)
The Rectangular Coordinate System

40. Graphs of Lines

41. Graphs of Lines
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points.

42. Graphs of Lines
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate.

43. Graphs of Lines
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation.

44. Graphs of Lines
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

45. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

46. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
(y can be anything).
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

47. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
(y can be anything).
Make a table of
ordered pairs of
points that fit the
description
x = –4.
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

48. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
(y can be anything).
x y
–4
–4
–4
–4
Make a table of
ordered pairs of
points that fit the
description
x = –4.
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

49. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
(y can be anything).
x y
–4 0
–4
–4
–4
Make a table of
ordered pairs of
points that fit the
description
x = –4.
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

50. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
(y can be anything).
x y
–4 0
–4 2
–4
–4
Make a table of
ordered pairs of
points that fit the
description
x = –4.
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

51. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
(y can be anything).
x y
–4 0
–4 2
–4 4
–4 6
Make a table of
ordered pairs of
points that fit the
description
x = –4.
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

52. Graphs of Lines
Example C. Graph the points (x, y) where x = –4
(y can be anything).
x y
–4 0
–4 2
–4 4
–4 6
Graph of x = –4
Make a table of
ordered pairs of
points that fit the
description
x = –4.
In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematical relations between the
x-coordinate and the y coordinate. The plot of points that fit a
given relation is called the graph of that relation. To make a
graph of a given mathematical relation, make a table of points
that fit the description and plot them.

53. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers.

54. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations.

55. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines.

56. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation.

57. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x,

58. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x, plug it into the equation and solve for the y

59. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x, plug it into the equation and solve for the y
(or assign a value to y and solve for the x).

60. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x, plug it into the equation and solve for the y
(or assign a value to y and solve for the x). For graphing lines,
find at least two ordered pairs.

61. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x, plug it into the equation and solve for the y
(or assign a value to y and solve for the x). For graphing lines,
find at least two ordered pairs.
Example D.
Graph the following linear equations.
a. y = 2x – 5

62. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x, plug it into the equation and solve for the y
(or assign a value to y and solve for the x). For graphing lines,
find at least two ordered pairs.
Example D.
Graph the following linear equations.
a. y = 2x – 5
Make a table by selecting a few numbers for x.

63. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x, plug it into the equation and solve for the y
(or assign a value to y and solve for the x). For graphing lines,
find at least two ordered pairs.
Example D.
Graph the following linear equations.
a. y = 2x – 5
Make a table by selecting a few numbers for x. For easy
calculations we set x = -1, 0, 1, and 2.

64. Graphs of Lines
First degree equation in the variables x and y are equations
that may be put into the form Ax + By = C where A, B, C are
numbers. First degree equations are the same as linear
equations. They are called linear because their graphs are
straight lines. To graph a linear equation, find a few ordered
pairs that fit the equation. To find one such ordered pair, assign
a value to x, plug it into the equation and solve for the y
(or assign a value to y and solve for the x). For graphing lines,
find at least two ordered pairs.
Example D.
Graph the following linear equations.
a. y = 2x – 5
Make a table by selecting a few numbers for x. For easy
calculations we set x = -1, 0, 1, and 2. Plug each of these
values into x and find its corresponding y to form an ordered
pair.

65. Graphs of Lines
For y = 2x – 5:
x y
-1
0
1
2

66. Graphs of Lines
For y = 2x – 5:
x y
-1
0
1
2
If x = -1, then
y = 2(-1) – 5

67. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0
1
2
If x = -1, then
y = 2(-1) – 5 = -7

68. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1
2
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5

69. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1
2
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5

70. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1 -3
2 -1
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5
If x = 1, then
y = 2(1) – 5 = -3
If x = 2, then
y = 2(2) – 5 = -1

71. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1 -3
2 -1
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5
If x = 1, then
y = 2(1) – 5 = -3
If x = 2, then
y = 2(2) – 5 = -1

72. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1 -3
2 -1
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5
If x = 1, then
y = 2(1) – 5 = -3
If x = 2, then
y = 2(2) – 5 = -1

73. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1 -3
2 -1
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5
If x = 1, then
y = 2(1) – 5 = -3
If x = 2, then
y = 2(2) – 5 = -1

74. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1 -3
2 -1
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5
If x = 1, then
y = 2(1) – 5 = -3
If x = 2, then
y = 2(2) – 5 = -1

75. Graphs of Lines
For y = 2x – 5:
x y
-1 -7
0 -5
1 -3
2 -1
If x = -1, then
y = 2(-1) – 5 = -7
If x = 0, then
y = 2(0) – 5 = -5
If x = 1, then
y = 2(1) – 5 = -3
If x = 2, then
y = 2(2) – 5 = -1

76. b. -3y = 12
Graphs of Lines

77. b. -3y = 12
Simplify as y = -4
Graphs of Lines
Make a table by
selecting a few
numbers for x.

78. b. -3y = 12
Simplify as y = -4
Graphs of Lines
Make a table by
selecting a few
numbers for x.
x y
-3
0
3
6

79. b. -3y = 12
Simplify as y = -4
Graphs of Lines
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4

80. b. -3y = 12
Simplify as y = -4
Graphs of Lines
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4

81. b. -3y = 12
Simplify as y = -4
Graphs of Lines
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4

82. b. -3y = 12
Simplify as y = -4
Graphs of Lines
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4

83. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4

84. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6.

85. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selection
for x is x = 6

86. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selection
for x is x = 6
x y
6
6
6
6

87. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selection
for x is x = 6 and
y could be any
number.
x y
6 0
6 2
6 4
6 6

88. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selection
for x is x = 6 and
y could be any
number.
x y
6 0
6 2
6 4
6 6

89. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selection
for x is x = 6 and
y could be any
number.
x y
6 0
6 2
6 4
6 6

90. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selection
for x is x = 6 and
y could be any
number.
x y
6 0
6 2
6 4
6 6

91. b. -3y = 12
Simplify as y = -4
Graphs of Lines
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However y is
always -4 .
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selection
for x is x = 6 and
y could be any
number.
x y
6 0
6 2
6 4
6 6

92. Summary of the graphs of linear equations:
Graphs of Lines

93. a. y = 2x – 5
Summary of the graphs of linear equations:
Graphs of Lines

94. a. y = 2x – 5
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
Summary of the graphs of linear equations:
Graphs of Lines

95. a. y = 2x – 5
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
Summary of the graphs of linear equations:
Graphs of Lines

96. a. y = 2x – 5 b. -3y = 12
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
Summary of the graphs of linear equations:
Graphs of Lines

97. a. y = 2x – 5 b. -3y = 12
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
If the equation has
only y (no x), the
graph is a
horizontal line.
Summary of the graphs of linear equations:
Graphs of Lines

98. a. y = 2x – 5 b. -3y = 12
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
If the equation has
only y (no x), the
graph is a
horizontal line.
Summary of the graphs of linear equations:
Graphs of Lines

99. a. y = 2x – 5 b. -3y = 12 c. 2x = 12
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
If the equation has
only y (no x), the
graph is a
horizontal line.
Summary of the graphs of linear equations:
Graphs of Lines

100. a. y = 2x – 5 b. -3y = 12 c. 2x = 12
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
If the equation has
only y (no x), the
graph is a
horizontal line.
Summary of the graphs of linear equations:
Graphs of Lines
If the equation has
only x (no y), the
graph is a
vertical line.

101. a. y = 2x – 5 b. -3y = 12 c. 2x = 12
If both variables
x and y are
present in the
equation, the
graph is a
tilted line.
If the equation has
only y (no x), the
graph is a
horizontal line.
Summary of the graphs of linear equations:
Graphs of Lines
If the equation has
only x (no y), the
graph is a
vertical line.

102. x-Intercepts is where the line crosses the x-axis;
Graphs of Lines

103. x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
Graphs of Lines

104. x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis;
Graphs of Lines

105. x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
Graphs of Lines

106. x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
Graphs of Lines
Since two points determine a line, an easy method to
graph linear equations is the intercept method,

107. x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
Graphs of Lines
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.

108. x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.

109. x y
0
0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.

110. x y
0
0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12

111. x y
0 -4
0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
so y = -4

112. x y
0 -4
0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
so y = -4
If y = 0, we get
2x – 3(0) = 12

113. x y
0 -4
6 0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
so y = -4
If y = 0, we get
2x – 3(0) = 12
so x = 6

114. x y
0 -4
6 0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
so y = -4
If y = 0, we get
2x – 3(0) = 12
so x = 6

115. x y
0 -4
6 0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
so y = -4
If y = 0, we get
2x – 3(0) = 12
so x = 6

116. x y
0 -4
6 0
x-Intercepts is where the line crosses the x-axis; set y = 0 in
the equation to find the x-intercept.
y-Intercepts is where the line crosses the y-axis; set x = 0 in
the equation to find the y-intercept.
y-int
x-int
Graphs of Lines
Example E. Graph 2x – 3y = 12
by the intercept method.
Since two points determine a line, an easy method to
graph linear equations is the intercept method, i.e. we plot
the x-intercept and the y intercept, the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
so y = -4
If y = 0, we get
2x – 3(0) = 12
so x = 6

117. Exercise B. Graph the following equations. Identify the vertical
and the horizontal lines first. Then use the intercept method if
the method feasible.
9. x – y = 3 10. 2x = 6 11. –y – 7= 0
12. 0 = 8 – 2x 13. y = –x + 4 14. 2x – 3 = 6
15. 2x = 6 – 2y 16. 4y – 12 = 3x 17. –2x + 3y = 3
18. –6 = 3x – 2y 19. 3x + 2 = 4y + 3x 20. 5x + 2y = –10
The Rectangular Coordinate System and Lines
Exercise A. Starting at (3, 4), find the coordinate of the point if
we move. Draw the points.
1. 2 units up, 2 units right. 2. 2 units up, 2 units left.
3. 2 units down, 2 units right. 4. 2 units down, 2 units left.
Starting at (–3, 4), find the coordinate of the point if we move.
Draw the points.
6. 7 units up, 9 units right.5. 12 units up, 4 units left.
7. 7 units down, 6 units right. 8. 11 units down, 7 units left.