1. Linear Equations and Lines
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2. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines

3. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines
We view this solution as the address of a position on a line
and label it to produce
a "picture“ of the answer:

4. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines
We view this solution as the address of a position on a line
and label it to produce
a "picture“ of the answer:
0
2
x
The picture of x = 2

5. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines
We view this solution as the address of a position on a line
and label it to produce
a "picture“ of the answer:
0
2
x
If we have a two–variable 1st degree equation such as
2x + y = 5
then we are free to select x and y.
The picture of x = 2

6. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines
We view this solution as the address of a position on a line
and label it to produce
a "picture“ of the answer:
0
2
x
If we have a two–variable 1st degree equation such as
2x + y = 5
then we are free to select x and y.
For instance x = 2 and y = 1 make the equation true.
The picture of x = 2

7. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines
We view this solution as the address of a position on a line
and label it to produce
a "picture“ of the answer:
0
2
x
If we have a two–variable 1st degree equation such as
2x + y = 5
then we are free to select x and y.
For instance x = 2 and y = 1 make the equation true.
By viewing (2, 1) as the coordinate
of a position in the xy-coordinate system,
we have a picture of this solution.
The picture of x = 2

8. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines
We view this solution as the address of a position on a line
and label it to produce
a "picture“ of the answer:
0
2
x
If we have a two–variable 1st degree equation such as
2x + y = 5
then we are free to select x and y.
For instance x = 2 and y = 1 make the equation true.
By viewing (2, 1) as the coordinate
of a position in the xy-coordinate system,
we have a picture of this solution.
(2, 1)
The picture of x = 2
The picture of
(x = 2, y = 1)

9. We solved 1st degree (linear) equations such as 2x + 1 = 5,
which has a single variable x, to obtain its solution x = 2.
Linear Equations and Lines
We view this solution as the address of a position on a line
and label it to produce
a "picture“ of the answer:
0
2
x
If we have a two–variable 1st degree equation such as
2x + y = 5
then we are free to select x and y.
For instance x = 2 and y = 1 make the equation true.
By viewing (2, 1) as the coordinate
of a position in the xy-coordinate system,
we have a picture of this solution.
(2, 1)
The picture of x = 2
Having the liberty of choosing two numbers
means there are many pairs of solutions,
thus more solution-points can be plotted.
These points form the graph of the equation. The picture of
(x = 2, y = 1)

10. In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points.
Linear Equations and Lines

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

12. In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematics 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.
Linear Equations and Lines

13. In the rectangular coordinate system, ordered pairs (x, y)’s
correspond to locations of points. Collections of points may be
specified by the mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

14. Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

15. Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

16. Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

17. Linear Equations and Lines
Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

18. Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

19. Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

20. Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

21. Example A. 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 mathematics 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 mathematics relation, make a table of points
that fit the description and plot them.
Linear Equations and Lines

22. Example B. Graph the points (x, y) where y = x.
Linear Equations and Lines

23. Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x.
Linear Equations and Lines

24. Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate.
Linear Equations and Lines

25. Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Linear Equations and Lines

26. x y
-1
0
1
2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Linear Equations and Lines

27. x y
-1 -1
0
1
2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Linear Equations and Lines

28. x y
-1 -1
0 0
1
2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Linear Equations and Lines

29. x y
-1 -1
0 0
1 1
2 2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Linear Equations and Lines

30. x y
-1 -1
0 0
1 1
2 2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Graph the points (x, y) where y = x
Linear Equations and Lines

31. x y
-1 -1
0 0
1 1
2 2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Graph the points (x, y) where y = x
Linear Equations and Lines

32. x y
-1 -1
0 0
1 1
2 2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Graph the points (x, y) where y = x
Linear Equations and Lines

33. x y
-1 -1
0 0
1 1
2 2
Example B. Graph the points (x, y) where y = x.
Make a table of points that fit the description y = x. To find
one such point, we set one of the coordinates to be a
number, any number, than use the relation to find the other
coordinate. Repeat this a few times.
Graph the points (x, y) where y = x

34. 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.
Linear Equations and Lines

35. 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.
Linear Equations and Lines

36. 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.
Linear Equations and Lines

37. 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.
Linear Equations and Lines

38. 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,
Linear Equations and Lines

39. 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
Linear Equations and Lines

40. 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).
Linear Equations and Lines

41. 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.
Linear Equations and Lines

42. 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 C.
Graph the following linear equations.
a. y = 2x – 5
Linear Equations and Lines

43. 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 C.
Graph the following linear equations.
a. y = 2x – 5
Make a table by selecting a few numbers for x.
Linear Equations and Lines

44. 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 C.
Graph the following linear equations.
a. y = 2x – 5
Make a table by selecting a few numbers for x. For easy
caluation we set x = -1, 0, 1, and 2.
Linear Equations and Lines

45. 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 C.
Graph the following linear equations.
a. y = 2x – 5
Make a table by selecting a few numbers for x. For easy
caluation we set x = -1, 0, 1, and 2. Plug each of these value
into x and find its corresponding y to form an ordered pair.
Linear Equations and Lines

46. For y = 2x – 5:
x y
-1
0
1
2
Linear Equations and Lines

47. For y = 2x – 5:
x y
-1
0
1
2
If x = -1, then
y = 2(-1) – 5
Linear Equations and Lines

48. For y = 2x – 5:
x y
-1 -7
0
1
2
If x = -1, then
y = 2(-1) – 5 = -7
Linear Equations and Lines

49. 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
Linear Equations and Lines

50. 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
Linear Equations and Lines

51. 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
Linear Equations and Lines

52. 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
Linear Equations and Lines

53. 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
Linear Equations and Lines

54. 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
Linear Equations and Lines

55. 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
Linear Equations and Lines

56. 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
Linear Equations and Lines

57. b. -3y = 12
Linear Equations and Lines

58. b. -3y = 12
Simplify as y = -4
Make a table by
selecting a few
numbers for x.
Linear Equations and Lines

59. b. -3y = 12
Simplify as y = -4
Make a table by
selecting a few
numbers for x.
x y
-3
0
3
6
Linear Equations and Lines

60. b. -3y = 12
Simplify as y = -4
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Linear Equations and Lines

61. b. -3y = 12
Simplify as y = -4
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Linear Equations and Lines

62. b. -3y = 12
Simplify as y = -4
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Linear Equations and Lines

63. b. -3y = 12
Simplify as y = -4
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Linear Equations and Lines

64. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Linear Equations and Lines

65. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6.
Linear Equations and Lines

66. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selction for
x is x = 6
Linear Equations and Lines

67. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selction for
x is x = 6
x y
6
6
6
6
Linear Equations and Lines

68. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selction for
x is x = 6 and y
could be any
number.
x y
6 0
6 2
6 4
6 6
Linear Equations and Lines

69. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selction for
x is x = 6 and y
could be any
number.
x y
6 0
6 2
6 4
6 6
Linear Equations and Lines

70. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selction for
x is x = 6 and y
could be any
number.
x y
6 0
6 2
6 4
6 6
Linear Equations and Lines

71. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selction for
x is x = 6 and y
could be any
number.
x y
6 0
6 2
6 4
6 6
Linear Equations and Lines

72. b. -3y = 12
Simplify as y = -4
c. 2x = 12
Make a table by
selecting a few
numbers for x.
However, y = -4
is always.
x y
-3 -4
0 -4
3 -4
6 -4
Simplify as x = 6
Make a table.
However the
only selction for
x is x = 6 and y
could be any
number.
x y
6 0
6 2
6 4
6 6
Linear Equations and Lines

73. Summary of the graphs of linear equations:
Linear Equations and Lines

74. a. y = 2x – 5
Summary of the graphs of linear equations:
Linear Equations and Lines

75. 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:
Linear Equations and Lines

76. 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:
Linear Equations and Lines

77. 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:
Linear Equations and Lines

78. 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:
Linear Equations and Lines

79. 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:
Linear Equations and Lines

80. 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:
Linear Equations and Lines

81. 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:
If the equation has
only x (no y), the
graph is a
vertical line.
Linear Equations and Lines

82. 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:
If the equation has
only x (no y), the
graph is a
vertical line.
Linear Equations and Lines

83. The x-Intercepts is where the line crosses the x-axis;
Linear Equations and Lines

84. The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
Linear Equations and Lines

85. The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis;
Linear Equations and Lines

86. The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
Linear Equations and Lines

87. The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
Since two points determine a line, an easy method to
graph linear equations is the intercept method,
Linear Equations and Lines

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

89. The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
Example C. 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. plot the
x-intercept and the y intercept and the graph is the line that
passes through them.
Linear Equations and Lines

90. x y
0
0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and the graph is the line that
passes through them.
Linear Equations and Lines

91. x y
0
0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
Linear Equations and Lines

92. x y
0 -4
0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and the graph is the line that
passes through them.
If x = 0, we get
2(0) – 3y = 12
so y = -4
Linear Equations and Lines

93. x y
0 -4
0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and 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
Linear Equations and Lines

94. x y
0 -4
6 0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and 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
Linear Equations and Lines

95. x y
0 -4
6 0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and 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
Linear Equations and Lines

96. x y
0 -4
6 0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and 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
Linear Equations and Lines

97. x y
0 -4
6 0
The x-Intercepts is where the line crosses the x-axis. We set
y = 0 in the equation to find the x-intercept.
The y-Intercepts is where the line crosses the y-axis. We set
x = 0 in the equation to find the y-intercept.
y-int
x-int
Example C. 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. plot the
x-intercept and the y intercept and 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
Linear Equations and Lines

98. Exercise. A. Solve the indicated variable for each equation with
the given assigned value.
1. x + y = 3 and x = –1, find y.
2. x – y = 3 and y = –1, find x.
3. 2x = 6 and y = –1, find x.
4. –y = 3 and x = 2, find y.
5. 2y = 3 – x and x = –2 , find y.
6. y = –x + 4 and x = –4, find y.
7. 2x – 3y = 1 and y = 3, find x.
8. 2x = 6 – 2y and y = –2, find x.
9. 3y – 2 = 3x and x = 2, find y.
10. 2x + 3y = 3 and x = 0, find y.
11. 2x + 3y = 3 and y = 0, find x.
12. 3x – 4y = 12 and x = 0, find y.
13. 3x – 4y = 12 and y = 0, find x.
14. 6 = 3x – 4y and y = –3, find x.
Linear Equations and Lines

99. B. a. Complete the tables for each equation with given values.
b. Plot the points from the table. c. Graph the line.
15. x + y = 3 16. 2y = 6
x y
-3
0
3
x y
1
0
–1
17. x = –6
x y
0
–1
– 2
18. y = x – 3
x y
2
1
0
19. 2x – y = 2 20. 3y = 6 + 2x
x y
2
0
–1
x y
1
0
–1
21. y = –6
x y
0
–1
– 2
22. 3y + 4x =12
x y
0
0
1
Linear Equations and Lines

100. C. Make a table for each equation with at least 3 ordered pairs.
(remember that you get to select one entry in each row as
shown in the tables above) then graph the line.
23. x – y = 3 24. 2x = 6 25. –y – 7= 0
26. 0 = 8 – 2x 27. y = –x + 4 28. 2x – 3 = 6
29. 2x = 6 – 2y 30. 4y – 12 = 3x 31. 2x + 3y = 3
32. –6 = 3x – 2y 33.
35. For problems 29, 30, 31 and 32, use the
intercept-tables as shown to graph the lines.
x y
0
0
intercept-table
36. Why can’t we use the above intercept method
to graph the lines for problems 25, 26 or 33?
37. By inspection identify which equations give
horizontal lines, which give vertical lines and
which give tilted lines.
3x = 4y 34. 5x + 2y = –10
Linear Equations and Lines