MATH 105: Finite Mathematics 1-2: Pairs of Lines Prof. Jonathan - - PowerPoint PPT Presentation

▶
math 105 finite mathematics 1 2 pairs of lines
SMART_READER_LITE
LIVE PREVIEW

MATH 105: Finite Mathematics 1-2: Pairs of Lines Prof. Jonathan - - PowerPoint PPT Presentation

Nov 12, 2023 30 likes •383 views

Pairs of Lines Perpendicular Lines Conclusion MATH 105: Finite Mathematics 1-2: Pairs of Lines Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Pairs of Lines Perpendicular Lines Conclusion Outline Pairs of Lines 1

slide-1
SLIDE 1

Pairs of Lines Perpendicular Lines Conclusion

MATH 105: Finite Mathematics 1-2: Pairs of Lines

  • Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

slide-2
SLIDE 2

Pairs of Lines Perpendicular Lines Conclusion

Outline

1

Pairs of Lines

2

Perpendicular Lines

3

Conclusion

slide-3
SLIDE 3

Pairs of Lines Perpendicular Lines Conclusion

Outline

1

Pairs of Lines

2

Perpendicular Lines

3

Conclusion

slide-4
SLIDE 4

Pairs of Lines Perpendicular Lines Conclusion

Relationships Between Lines

There are several ways in which two lines can interact with each

  • ther. In this section we will examine those possibilities and find
  • ut how to determine which relationship a pair of lines has.

Pairs of Lines Let L and M be two lines in a plane. Then L and M must be:

1 intersecting 2 parallel 3 coincidental

slide-5
SLIDE 5

Pairs of Lines Perpendicular Lines Conclusion

Relationships Between Lines

There are several ways in which two lines can interact with each

  • ther. In this section we will examine those possibilities and find
  • ut how to determine which relationship a pair of lines has.

Pairs of Lines Let L and M be two lines in a plane. Then L and M must be:

1 intersecting 2 parallel 3 coincidental

slide-6
SLIDE 6

Pairs of Lines Perpendicular Lines Conclusion

Relationships Between Lines

There are several ways in which two lines can interact with each

  • ther. In this section we will examine those possibilities and find
  • ut how to determine which relationship a pair of lines has.

Pairs of Lines Let L and M be two lines in a plane. Then L and M must be:

1 intersecting 2 parallel 3 coincidental

L M

slide-7
SLIDE 7

Pairs of Lines Perpendicular Lines Conclusion

Relationships Between Lines

There are several ways in which two lines can interact with each

  • ther. In this section we will examine those possibilities and find
  • ut how to determine which relationship a pair of lines has.

Pairs of Lines Let L and M be two lines in a plane. Then L and M must be:

1 intersecting 2 parallel 3 coincidental

L M L M

slide-8
SLIDE 8

Pairs of Lines Perpendicular Lines Conclusion

Relationships Between Lines

There are several ways in which two lines can interact with each

  • ther. In this section we will examine those possibilities and find
  • ut how to determine which relationship a pair of lines has.

Pairs of Lines Let L and M be two lines in a plane. Then L and M must be:

1 intersecting 2 parallel 3 coincidental

L M L M L M

slide-9
SLIDE 9

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

While we can certainly identify how two lines are related by graphing them, it is often better to compare equations. Comparing Lines A pair of lines can be classified as intersecting, parallel, or coincident based on the lines slope and y-intercept.

1 Intersecting lines have different slopes. 2 Parallel lines have the same slope and different intercepts. 3 Coincident lines have the same slope and the same intercepts.

slide-10
SLIDE 10

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

While we can certainly identify how two lines are related by graphing them, it is often better to compare equations. Comparing Lines A pair of lines can be classified as intersecting, parallel, or coincident based on the lines slope and y-intercept.

1 Intersecting lines have different slopes. 2 Parallel lines have the same slope and different intercepts. 3 Coincident lines have the same slope and the same intercepts.

slide-11
SLIDE 11

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

While we can certainly identify how two lines are related by graphing them, it is often better to compare equations. Comparing Lines A pair of lines can be classified as intersecting, parallel, or coincident based on the lines slope and y-intercept.

1 Intersecting lines have different slopes. 2 Parallel lines have the same slope and different intercepts. 3 Coincident lines have the same slope and the same intercepts.

slide-12
SLIDE 12

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

While we can certainly identify how two lines are related by graphing them, it is often better to compare equations. Comparing Lines A pair of lines can be classified as intersecting, parallel, or coincident based on the lines slope and y-intercept.

1 Intersecting lines have different slopes. 2 Parallel lines have the same slope and different intercepts. 3 Coincident lines have the same slope and the same intercepts.

slide-13
SLIDE 13

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

While we can certainly identify how two lines are related by graphing them, it is often better to compare equations. Comparing Lines A pair of lines can be classified as intersecting, parallel, or coincident based on the lines slope and y-intercept.

1 Intersecting lines have different slopes. 2 Parallel lines have the same slope and different intercepts. 3 Coincident lines have the same slope and the same intercepts.

slide-14
SLIDE 14

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

Identifying Lines Using the slope and y-intercept, identify each pair of lines as intersecting, parallel, or coincident.

1 The lines 3x + 5y = 15 and 6x + 10y = 30. 2 The lines 7x − 2y = 14 and −14x + 4y = 28. 3 The lines 4x − 6y = 12 and 6x + 4y = −8.

slide-15
SLIDE 15

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

Identifying Lines Using the slope and y-intercept, identify each pair of lines as intersecting, parallel, or coincident.

1 The lines 3x + 5y = 15 and 6x + 10y = 30. 2 The lines 7x − 2y = 14 and −14x + 4y = 28. 3 The lines 4x − 6y = 12 and 6x + 4y = −8.

slide-16
SLIDE 16

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

Identifying Lines Using the slope and y-intercept, identify each pair of lines as intersecting, parallel, or coincident.

1 The lines 3x + 5y = 15 and 6x + 10y = 30. (coincident)

y = −3 5x + 3 y = −3 5x + 3

2 The lines 7x − 2y = 14 and −14x + 4y = 28. 3 The lines 4x − 6y = 12 and 6x + 4y = −8.

slide-17
SLIDE 17

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

Identifying Lines Using the slope and y-intercept, identify each pair of lines as intersecting, parallel, or coincident.

1 The lines 3x + 5y = 15 and 6x + 10y = 30. (coincident)

y = −3 5x + 3 y = −3 5x + 3

2 The lines 7x − 2y = 14 and −14x + 4y = 28. 3 The lines 4x − 6y = 12 and 6x + 4y = −8.

slide-18
SLIDE 18

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

Identifying Lines Using the slope and y-intercept, identify each pair of lines as intersecting, parallel, or coincident.

1 The lines 3x + 5y = 15 and 6x + 10y = 30. (coincident)

y = −3 5x + 3 y = −3 5x + 3

2 The lines 7x − 2y = 14 and −14x + 4y = 28. (parallel)

y = 7 2x − 7 y = 7 2x + 7

3 The lines 4x − 6y = 12 and 6x + 4y = −8.

slide-19
SLIDE 19

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

Identifying Lines Using the slope and y-intercept, identify each pair of lines as intersecting, parallel, or coincident.

1 The lines 3x + 5y = 15 and 6x + 10y = 30. (coincident)

y = −3 5x + 3 y = −3 5x + 3

2 The lines 7x − 2y = 14 and −14x + 4y = 28. (parallel)

y = 7 2x − 7 y = 7 2x + 7

3 The lines 4x − 6y = 12 and 6x + 4y = −8.

slide-20
SLIDE 20

Pairs of Lines Perpendicular Lines Conclusion

Identifying Pairs of Lines

Identifying Lines Using the slope and y-intercept, identify each pair of lines as intersecting, parallel, or coincident.

1 The lines 3x + 5y = 15 and 6x + 10y = 30. (coincident)

y = −3 5x + 3 y = −3 5x + 3

2 The lines 7x − 2y = 14 and −14x + 4y = 28. (parallel)

y = 7 2x − 7 y = 7 2x + 7

3 The lines 4x − 6y = 12 and 6x + 4y = −8. (intersecting)

y = 2 3x − 2 y = −3 2x − 2

slide-21
SLIDE 21

Pairs of Lines Perpendicular Lines Conclusion

Outline

1

Pairs of Lines

2

Perpendicular Lines

3

Conclusion

slide-22
SLIDE 22

Pairs of Lines Perpendicular Lines Conclusion

Perpendicular Lines

In the last example, not only were the lines intersecting, but they intersected each other at right angles. Perpendicular Lines Limes L1 and L2 with slopes m1 and m2 are perpendicular if m1 · m2 = −1 In practice, the slopes of perpendicular lines are negative reciprocals

  • f each other. This makes it easy to check for perpendicular lines,

and to construct a line perpendicular to a given line.

slide-23
SLIDE 23

Pairs of Lines Perpendicular Lines Conclusion

Examples

Examples Use the line 2x − 10y = 20 to perform the following tasks.

1 Find the equation of a line parallel to this line through the

point (1, 2).

2 Find the equation of a line perpendicular to this line through

the point (1, 2).

3 Graph all three lines.

slide-24
SLIDE 24

Pairs of Lines Perpendicular Lines Conclusion

Examples

Examples Use the line 2x − 10y = 20 to perform the following tasks.

1 Find the equation of a line parallel to this line through the

point (1, 2).

2 Find the equation of a line perpendicular to this line through

the point (1, 2).

3 Graph all three lines.

slide-25
SLIDE 25

Pairs of Lines Perpendicular Lines Conclusion

Examples

Examples Use the line 2x − 10y = 20 to perform the following tasks.

1 Find the equation of a line parallel to this line through the

point (1, 2). x − 5y = −9

2 Find the equation of a line perpendicular to this line through

the point (1, 2).

3 Graph all three lines.

slide-26
SLIDE 26

Pairs of Lines Perpendicular Lines Conclusion

Examples

Examples Use the line 2x − 10y = 20 to perform the following tasks.

1 Find the equation of a line parallel to this line through the

point (1, 2). x − 5y = −9

2 Find the equation of a line perpendicular to this line through

the point (1, 2).

3 Graph all three lines.

slide-27
SLIDE 27

Pairs of Lines Perpendicular Lines Conclusion

Examples

Examples Use the line 2x − 10y = 20 to perform the following tasks.

1 Find the equation of a line parallel to this line through the

point (1, 2). x − 5y = −9

2 Find the equation of a line perpendicular to this line through

the point (1, 2). 5x + y = 7

3 Graph all three lines.

slide-28
SLIDE 28

Pairs of Lines Perpendicular Lines Conclusion

Examples

Examples Use the line 2x − 10y = 20 to perform the following tasks.

1 Find the equation of a line parallel to this line through the

point (1, 2). x − 5y = −9

2 Find the equation of a line perpendicular to this line through

the point (1, 2). 5x + y = 7

3 Graph all three lines.

slide-29
SLIDE 29

Pairs of Lines Perpendicular Lines Conclusion

Outline

1

Pairs of Lines

2

Perpendicular Lines

3

Conclusion

slide-30
SLIDE 30

Pairs of Lines Perpendicular Lines Conclusion

Important Concepts

Things to Remember from Section 1-2

1 Three relationships between a pair of lines. 2 Parallel lines have the same slope. 3 Perpendicular lines have negative reciprocals slopes.

slide-31
SLIDE 31

Pairs of Lines Perpendicular Lines Conclusion

Important Concepts

Things to Remember from Section 1-2

1 Three relationships between a pair of lines. 2 Parallel lines have the same slope. 3 Perpendicular lines have negative reciprocals slopes.

slide-32
SLIDE 32

Pairs of Lines Perpendicular Lines Conclusion

Important Concepts

Things to Remember from Section 1-2

1 Three relationships between a pair of lines. 2 Parallel lines have the same slope. 3 Perpendicular lines have negative reciprocals slopes.

slide-33
SLIDE 33

Pairs of Lines Perpendicular Lines Conclusion

Important Concepts

Things to Remember from Section 1-2

1 Three relationships between a pair of lines. 2 Parallel lines have the same slope. 3 Perpendicular lines have negative reciprocals slopes.

slide-34
SLIDE 34

Pairs of Lines Perpendicular Lines Conclusion

Next Time. . .

Next time we will begin chapter 2, in which we look at solving two

  • r more equations (usually for lines) simultaneously.

For next time Review sections 2-1

slide-35
SLIDE 35

Pairs of Lines Perpendicular Lines Conclusion

Next Time. . .

Next time we will begin chapter 2, in which we look at solving two

  • r more equations (usually for lines) simultaneously.

For next time Review sections 2-1