The document discusses slopes and how they are measured and calculated. It defines slope as the ratio of the rise over the run between two points on a line. The slope of a line can be calculated using the formula: m = (y2-y1)/(x2-x1). This measures the steepness of the line. Examples are given to demonstrate calculating the slope between two points and interpreting it geometrically as the ratio of rise over run. Slopes apply to lines, streets, roofs, and any surface with an incline or grade.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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Straight-line depreciation is the simplest method for calculating depreciation over time. Under this method, the same amount of depreciation is deducted from the value of an asset for every year of its useful life.
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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3. Slopes of Lines
a Seattle trolleybus climbing
an 18%-grade street (Wikipedia)
The steepness of a street
is measured in “grade”.
For example:
4. Slopes of Lines
a Seattle trolleybus climbing
an 18%-grade street (Wikipedia)
The steepness of a street
is measured in “grade”.
For example:
The 18%-grade means the
ratio of 18 to100 as shown here:
18 ft
100 ft
5. Slopes of Lines
a Seattle trolleybus climbing
an 18%-grade street (Wikipedia)
The steepness of a street
is measured in “grade”.
For example:
The 18%-grade means the
ratio of 18 to100 as shown here:
18 ft
100 ft
The steepness of a roof is
measured in “pitch”.
6. Slopes of Lines
a Seattle trolleybus climbing
an 18%-grade street (Wikipedia)
The steepness of a street
is measured in “grade”.
For example:
The 18%-grade means the
ratio of 18 to100 as shown here:
18 ft
100 ft
The steepness of a roof is
measured in “pitch”.
For example:
(12”)
7. Slopes of Lines
a Seattle trolleybus climbing
an 18%-grade street (Wikipedia)
The steepness of a street
is measured in “grade”.
For example:
The 18%-grade means the
ratio of 18 to100 as shown here:
18 ft
100 ft
Here is outline of a roof with
a pitch of 4:12 or 1/3.
(12”)
The steepness of a roof is
measured in “pitch”.
For example:
8. Slopes of Lines
a Seattle trolleybus climbing
an 18%-grade street (Wikipedia)
The steepness of a street
is measured in “grade”.
For example:
The 18%-grade means the
ratio of 18 to100 as shown here:
18 ft
100 ft
Here is outline of a roof with
a pitch of 4:12 or 1/3.
(12”)
The steepness of a roof is
measured in “pitch”.
For example:
In mathematics, these measurements are called “slopes”.
10. Slopes of Lines
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
11. Slopes of Lines
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
12. Slopes of Lines
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
Definition of Slope
13. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
14. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
15. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
16. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
17. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
18. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4,
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
19. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4,
then Δ x = x2 – x1
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
20. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4,
then Δ x = x2 – x1 = –4 – 7 = –11
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
22. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
Slopes of Lines
(x1, y1)
(x2, y2)
23. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δxm =
Slopes of Lines
(x1, y1)
(x2, y2)
24. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
25. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
Geometry of Slope
(x1, y1)
(x2, y2)
26. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
27. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
28. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
29. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m = Δy
Δx
easy to
memorize
30. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m = Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
31. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run
=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m = Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
the geometric
meaning
32. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run
=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m = Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
the geometric
meaning
Note that
y2 – y1
x2 – x1
y1 – y2
x1 – x2
=
so the numbering of the
two points is not relevant–
we get the same answer.
33. Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
34. Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
35. Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
36. (–2 , 8)
( 3 , –2)
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
37. (–2 , 8)
( 3 , –2)
–5 , 10
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
38. Δy
(–2 , 8)
( 3 , –2)
–5 , 10
Δx
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
39. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
40. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
41. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
42. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
43. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
(–2, 5)
( 3, 5)
44. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
Δy
(–2, 5)
( 3, 5)
–5, 0
Δx
45. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
Δy
(–2, 5)
( 3, 5)
–5, 0
Δx
So the slope is
Δx
Δy
m =
46. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
Δy
(–2, 5)
( 3, 5)
–5, 0
Δx
So the slope is
Δx
Δy 0
–5m = = = 0
47. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
48. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
49. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
50. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
51. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
52. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
is undefined!
53. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
is undefined!
Hence the slope m
of a horizontal line is m = 0.
54. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
is undefined!
Hence the slope m of a vertical line is undefined (UDF).
of a horizontal line is m = 0.
55. As shown in example C, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
is undefined!
Hence the slope m of a vertical line is undefined (UDF).
of a horizontal line is m = 0.
of a tilted line is a non–zero number.
56. Exercise A. Identify the vertical and the horizontal lines by
inspection first. Find their slopes or if it’s undefined, state so.
Fine the slopes of the other ones by solving for the y.
1. x – y = 3 2. 2x = 6 3. –y – 7= 0
4. 0 = 8 – 2x 5. y = –x + 4 6. 2x/3 – 3 = 6/5
7. 2x = 6 – 2y 8. 4y/5 – 12 = 3x/4 9. 2x + 3y = 3
10. –6 = 3x – 2y 11. 3x + 2 = 4y + 3x 12. 5x/4 + 2y/3 = 2
Exercise B.
13–18. Select two points and estimate the slope of each line.
13. 14. 15.
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57. 16. 17. 18.
Exercise C. Draw and find the slope of the line that passes
through the given two points. Identify the vertical line and the
horizontal lines by inspection first.
19. (0, –1), (–2, 1) 20. (1, –2), (–2, 0) 21. (1, –2), (–2, –1)
22. (3, –1), (3, 1) 23. (1, –2), (–2, 3) 24. (2, –1), (3, –1)
25. (4, –2), (–3, 1) 26. (4, –2), (4, 0) 27. (7, –2), (–2, –6)
28. (3/2, –1), (3/2, 1) 29. (3/2, –1), (1, –3/2)
30. (–5/2, –1/2), (1/2, 1) 31. (3/2, 1/3), (1/3, 1/3)
32. (–2/3, –1/4), (1/2, 2/3) 33. (3/4, –1/3), (1/3, 3/2)
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58. Exercise D.
34. Identify which lines are parallel and which one are
perpendicular.
A. The line that passes through (0, 1), (1, –2)
D. 2x – 4y = 1
B. C.
E. The line that’s perpendicular to 3y = x
F. The line with the x–intercept at 3 and y intercept at 6.
Find the slope, if possible of each of the following lines.
35. The line passes with the x intercept at x = 2,
and y–intercept at y = –5.
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59. 36. The equation of the line is 3x = –5y+7
37. The equation of the line is 0 = –5y+7
38. The equation of the line is 3x = 7
39. The line is parallel to 2y = 5 – 6x
40. the line is perpendicular to 2y = 5 – 6x
41. The line is parallel to the line in problem 30.
42. the line is perpendicular to line in problem 31.
43. The line is parallel to the line in problem 33.
44. the line is perpendicular to line in problem 34.
More on Slopes
Find the slope, if possible of each of the following lines
60. Summary of Slope
The slope of the line that passes through (x1, y1) and (x2, y2) is
Horizontal line
Slope = 0
Vertical line
Slope is UDF.
Tilted line
Slope = –2 0
rise
run
=m =
Δy
Δx
y2 – y1
x2 – x1
=
61. Exercise A.
Select two points and estimate the slope of each line.
1. 2. 3. 4.
Slopes of Lines
5. 6. 7. 8.
62. Exercise B. Draw and find the slope of the line that passes
through the given two points. Identify the vertical line and the
horizontal lines by inspection first.
9. (0, –1), (–2, 1) 10. (1, –2), (–2, 0) 11. (1, –2), (–2, –1)
12. (3, –1), (3, 1) 13. (1, –2), (–2, 3) 14. (2, –1), (3, –1)
15. (4, –2), (–3, 1) 16. (4, –2), (4, 0) 17. (7, –2), (–2, –6)
18. (3/2, –1), (3/2, 1) 19. (3/2, –1), (1, –3/2)
20. (–5/2, –1/2), (1/2, 1) 21. (3/2, 1/3), (1/3, 1/3)
22. (–2/3, –1/4), (1/2, 2/3) 23. (3/4, –1/3), (1/3, 3/2)
Slopes of Lines