The document discusses the point-slope formula for writing the equation of a line given a point and slope. It provides examples of using the formula, such as writing the equation of the line through point (3, -2) with slope 5. It also discusses that horizontal lines have a slope of 0 and the equation y=b, since the y-coordinate remains constant while the x-coordinate changes. The slope of a horizontal line is 0 because when calculating slope using two points, the change in y-values is 0.
Ellipsoidal Representations about correlations (2011-11, Tsukuba, Kakenhi-Sym...Toshiyuki Shimono
A fundamental theory in statistics, possibly applicable to data mining, machine learning, as well as epistemology. The principia mathematica of mine, 2nd version.
Ellipsoidal Representations about correlations (2011-11, Tsukuba, Kakenhi-Sym...Toshiyuki Shimono
A fundamental theory in statistics, possibly applicable to data mining, machine learning, as well as epistemology. The principia mathematica of mine, 2nd version.
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
Distinguish equations representing the circles and the conics; use the properties of a particular geometry to sketch the graph in using the rectangular or the polar coordinate system. Furthermore, to be able to write the equation and to solve application problems involving a particular geometry.
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
Distinguish equations representing the circles and the conics; use the properties of a particular geometry to sketch the graph in using the rectangular or the polar coordinate system. Furthermore, to be able to write the equation and to solve application problems involving a particular geometry.
Mobile Attention Management Identifying and Harnessing Factors that Determine...Veljko Pejovic
Presentation at the Free University of Bolzano, Italy, September 2016.
Accompanying Android intelligent notification scheduling library: bitbucket.org/veljkop/intelligenttrigger
Android machine learning library: github.com/vpejovic/MachineLearningToolkit
Contact: veljko.pejovic@fri.uni-lj.si
Object-oriented software engineering: Example (for teaching purposes) of a refactoring case study based on a very simple Java example of a Local Area Network. Used as part of the software engineering and software evolution courses of the University of Mons, taught by Prof. Tom Mens, Software Engineering Lab.
Dealing with Uncertainty: What the reverend Bayes can teach us.OReillyStrata
By Jurgen Van Gael - http://jvangael.github.io/ - @jvangael
As data scientists and decision makers, uncertainty is all around us: data is noisy, missing, wrong or inherently uncertain. Statistics offers a wide set of theories and tools to deal with this uncertainty, yet most people are unaware of a unifying theory of uncertainty. In this talk I want to introduce the audience to a branch of statistics called Bayesian reasoning which is a unifying, consistent, logical and most importantly successful way of dealing with uncertainty.
Over the past two centuries there have been many proposals for dealing with uncertainty (e.g. frequentist probabilities, fuzzy logic, ...). Under the influence of early 20th century statisticians, the Bayesian formalism was somewhat pushed into the background of the statistical scene. More recently though, some to the credit of computer science, Bayesian thinking has seen a revival. So what and how much should a data scientist or decision maker know about Bayesian thinking?
My talk will consist of four different parts. In the first part, I will explain the central dogma of Bayesian thinking: Bayes Rule. This simple equation (4 variables, one multiplication and one division!) describes how we should update our beliefs about the world in light of new data. I will discuss evidence from neuroscience and psychology that the brain uses Bayesian mechanism to reason about the world. Unfortunately, sometimes the brain fails miserably at taking all the variables of Bayes rule into account.
This leads to the second part of the talk where I will illustrate Bayes rule as a tool for decision makers to reason about uncertainty.
In the third part of the talk I will give an example of how we can build machine learning systems around Bayes rule. The key idea here is that Bayes rule allows us to keep track of uncertainty about the world. In this part I will illustrate one a Bayesian machine learning system in action.
In the final part of the talk I will introduce the concept of “Probabilistic Programming”. Probabilistic programming is a new embryonic programming paradigm that introduces “uncertain variables” as a first class citizen of a programming language and then uses Bayes rule to execute the programs.
When we look at machine learning conferences in the last few years, the Bayesian framework has been prominent. In this talk I want to help the audience understand how the Bayesian framework can help them in their data mining and decision making processes. If people leave the talk thinking Bayes rule is the E=MC^2 of data science, I will consider the presentation a success.
Slope of A line Slope of A line Slope of A line Slope of A line Slope of A line Slope of A line Slope of A line Slope of A lineSlope of A line Slope of A line Slope of A line
* Find the slope of a line.
* Use slopes to identify parallel and perpendicular lines.
* Write the equation of a line through a given point
- parallel to a given line
- perpendicular to a given line
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
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.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
5. POINT-SLOPE FORMULA
• Alternatemethod to find the equation of a line with the slope and
one point.
y − y1 = m ( x − x1 )
• The x and y stay the same. Never substitute a value for these.
6. POINT-SLOPE FORMULA
• Alternatemethod to find the equation of a line with the slope and
one point.
y − y1 = m ( x − x1 )
• The x and y stay the same. Never substitute a value for these.
• The x1 and y1 represent the given point. This is where you substitute
the given point.
8. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5.
9. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5.
y − y1 = m ( x − x1 )
10. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5. • Label your given point.
y − y1 = m ( x − x1 )
11. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5. • Label your given point.
( x1, y1 )
y − y1 = m ( x − x1 )
12. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 )
13. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 )
y − ( −2 ) = 5 ( x − 3)
14. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 ) • When the question asks for the
equation in point-slope form, the
y − ( −2 ) = 5 ( x − 3) only simplifying done is to change
any subtracting negatives to
addition.
15. USING POINT-SLOPE FORMULA
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(3, -2) with a slope of 5. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 ) • When the question asks for the
equation in point-slope form, the
y − ( −2 ) = 5 ( x − 3) only simplifying done is to change
any subtracting negatives to
addition.
y + 2 = 5 ( x − 3)
16. YOUR TURN...
Write the point-slope form of
the equation passing through
(7, -3) with a slope of -2.
17. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2.
18. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2.
y − y1 = m ( x − x1 )
19. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2. • Label your given point.
y − y1 = m ( x − x1 )
20. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2. • Label your given point.
( x1, y1 )
y − y1 = m ( x − x1 )
21. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 )
22. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 )
y − ( −3) = −2 ( x − 7 )
23. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 ) • When the question asks for the
equation in point-slope form, the
y − ( −3) = −2 ( x − 7 ) only simplifying is change to
addition any subtraction
negatives.
24. YOUR TURN...
Write the point-slope form of • Write the general formula for
the equation passing through point-slope form.
(7, -3) with a slope of -2. • Label your given point.
( x1, y1 )
• Substitute the slope and point.
y − y1 = m ( x − x1 ) • When the question asks for the
equation in point-slope form, the
y − ( −3) = −2 ( x − 7 ) only simplifying is change to
addition any subtraction
negatives.
y + 3 = −2 ( x − 7 )
26. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
27. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
28. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
29. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
30. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
• Why is the slope zero?
31. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
• Why is the slope zero?
• Suppose you have points (2, 5) and (3, 5). You know they are horizontal
because the x-coordinate changes.
32. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
• Why is the slope zero?
• Suppose you have points (2, 5) and (3, 5). You know they are horizontal
because the x-coordinate changes.
• Substitute into the slope formula.
y2 − y1
m=
x2 − x1
33. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
• Why is the slope zero?
• Suppose you have points (2, 5) and (3, 5). You know they are horizontal
because the x-coordinate changes.
• Substitute into the slope formula.
y2 − y1 5 − 5
m= =
x2 − x1 2 − 3
34. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
• Why is the slope zero?
• Suppose you have points (2, 5) and (3, 5). You know they are horizontal
because the x-coordinate changes.
• Substitute into the slope formula.
y2 − y1 5 − 5 0
m= = =
x2 − x1 2 − 3 −1
35. HORIZONTAL LINES
• Look at the horizon at the right.
• The direction of the horizon is from left to
right. This is how all horizontal lines
appear on a graph.
• The x-coordinate changes but the y-coordinate remains constant.
• Horizontal lines have the equation y = b.
• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
• Why is the slope zero?
• Suppose you have points (2, 5) and (3, 5). You know they are horizontal
because the x-coordinate changes.
• Substitute into the slope formula.
y2 − y1 5 − 5 0
m= = =
x2 − x1 2 − 3 −1
• Zero divided by anything is 0. Therefore, the slope is 0.
37. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
38. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
39. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
• The slope of all vertical lines is undefined.
40. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
• The slope of all vertical lines is undefined.
• Why is the slope undefined?
41. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
• The slope of all vertical lines is undefined.
• Why is the slope undefined?
• Suppose you have points (1, 5) and (1, 3). You know they are
vertical because they have the same x-coordinate.
42. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
• The slope of all vertical lines is undefined.
• Why is the slope undefined?
• Suppose you have points (1, 5) and (1, 3). You know they are
vertical because they have the same x-coordinate.
• Substitute into the slope formula.
y2 − y1
m=
x2 − x1
43. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
• The slope of all vertical lines is undefined.
• Why is the slope undefined?
• Suppose you have points (1, 5) and (1, 3). You know they are
vertical because they have the same x-coordinate.
• Substitute into the slope formula.
y2 − y1 5 − 3
m= =
x2 − x1 1 − 1
44. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
• The slope of all vertical lines is undefined.
• Why is the slope undefined?
• Suppose you have points (1, 5) and (1, 3). You know they are
vertical because they have the same x-coordinate.
• Substitute into the slope formula.
y2 − y1 5 − 3 2
m= = =
x2 − x1 1 − 1 0
45. VERTICAL LINES
• Vertical lines go up and down.
• The x-value stays the same but the y-value changes.
• Vertical lines have the equation x = point equation crosses x-axis
• The slope of all vertical lines is undefined.
• Why is the slope undefined?
• Suppose you have points (1, 5) and (1, 3). You know they are
vertical because they have the same x-coordinate.
• Substitute into the slope formula.
y2 − y1 5 − 3 2
m= = =
x2 − x1 1 − 1 0
• Can’t have division by 0. Therefore, the slope is undefined.
47. WRITE THE EQUATIONS
( 2, −5 )
• Write the equation for the vertical line that goes through the above
point.
48. WRITE THE EQUATIONS
( 2, −5 )
• Write the equation for the vertical line that goes through the above
point.
• Because the x-coordinate never changes, the equation is x = 2.
49. WRITE THE EQUATIONS
( 2, −5 )
• Write the equation for the vertical line that goes through the above
point.
• Because the x-coordinate never changes, the equation is x = 2.
• Writethe equation for the horizontal line that goes through the
above point.
50. WRITE THE EQUATIONS
( 2, −5 )
• Write the equation for the vertical line that goes through the above
point.
• Because the x-coordinate never changes, the equation is x = 2.
• Writethe equation for the horizontal line that goes through the
above point.
• Because the y-coordinate never changes, the equation is y = -5.
52. YOU TRY...
( −7, 9 )
• Write the equation for the vertical line that goes through the above
point.
53. YOU TRY...
( −7, 9 )
• Write the equation for the vertical line that goes through the above
point.
• Because the x-coordinate never changes, the equation is x = -7.
54. YOU TRY...
( −7, 9 )
• Write the equation for the vertical line that goes through the above
point.
• Because the x-coordinate never changes, the equation is x = -7.
• Writethe equation for the horizontal line that goes through the
above point.
55. YOU TRY...
( −7, 9 )
• Write the equation for the vertical line that goes through the above
point.
• Because the x-coordinate never changes, the equation is x = -7.
• Writethe equation for the horizontal line that goes through the
above point.
• Because the y-coordinate never changes, the equation is y = 9.
58. STANDARD FORM
• An equation written in the form Ax + By = C
• Rules for standard form:
59. STANDARD FORM
• An equation written in the form Ax + By = C
• Rules for standard form:
• A, B, and C must be Integers.
60. STANDARD FORM
• An equation written in the form Ax + By = C
• Rules for standard form:
• A, B, and C must be Integers.
•A must be positive.
61. STANDARD FORM
• An equation written in the form Ax + By = C
• Rules for standard form:
• A, B, and C must be Integers.
•A must be positive.
• Either A OR B can be 0. Both can NOT be 0.
62. STANDARD FORM
• An equation written in the form Ax + By = C
• Rules for standard form:
• A, B, and C must be Integers.
•A must be positive.
• Either A OR B can be 0. Both can NOT be 0.
• As long as A ≠ 0, Ax must be the first term.
63. STANDARD FORM
• An equation written in the form Ax + By = C
• Rules for standard form: Examples
• A, B, and C must be Integers. 2x − y = 5
•A must be positive.
5y = −3
• Either A OR B can be 0. Both can NOT be 0.
x + 2y = 4
• As long as A ≠ 0, Ax must be the first term.
64. STANDARD FORM
• An equation written in the form Ax + By = C
• Rules for standard form: Examples
• A, B, and C must be Integers. 2x − y = 5
•A must be positive.
5y = −3
• Either A OR B can be 0. Both can NOT be 0.
x + 2y = 4
• As long as A ≠ 0, Ax must be the first term.
Non Examples
−2x + y = 5 4
2x + 3y =
0.5y = −3.4 7
65. WRITING AN EQUATION IN STANDARD FORM
Write the equation
2
y − 1 = ( x + 3)
3
in standard form.
66. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3)
3
in standard form.
67. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3)
3
in standard form.
2 2
y −1= x + ⋅3
3 3
68. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form.
2 2
y −1= x + ⋅3
3 3
69. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form.
2 2
y −1= x + ⋅3
3 3
2
y −1= x + 2
3
70. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form. • Use the properties of equality to get
2 2 the constants on the right and the
y −1= x + ⋅3 variables on the left.
3 3
2
y −1= x + 2
3
71. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form. • Use the properties of equality to get
2 2 the constants on the right and the
y −1= x + ⋅3 variables on the left.
3 3
2
y −1= x + 2
+1 3 +1
72. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form. • Use the properties of equality to get
2 2 the constants on the right and the
y −1= x + ⋅3 variables on the left.
3 3
2
y −1= x + 2
+1 3 +1
2 2
− x − x
3 3
73. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form. • Use the properties of equality to get
2 2 the constants on the right and the
y −1= x + ⋅3 variables on the left.
3 3
2
y −1= x + 2
+1 3 +1
2 2
− x − x
3 3
2
− x+y=3
3
74. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form. • Use the properties of equality to get
2 2 the constants on the right and the
y −1= x + ⋅3 variables on the left.
3 3
2 • Multiplyby -3 so A is positive and
y −1= x + 2 no fractions exist.
+1 3 +1
2 2
− x − x
3 3
2
− x+y=3
3
75. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form. • Use the properties of equality to get
2 2 the constants on the right and the
y −1= x + ⋅3 variables on the left.
3 3
2 • Multiplyby -3 so A is positive and
y −1= x + 2 no fractions exist.
+1 3 +1
2 2 ⎛ 2 ⎞
− x − x −3 ⎜ − x ⎟ + −3 ⋅ y = −3 ⋅ 3
3 3 ⎝ 3 ⎠
2
− x+y=3
3
76. WRITING AN EQUATION IN STANDARD FORM
Write the equation • Distribute first.
2
y − 1 = ( x + 3) • Simplify multiplication.
3
in standard form. • Use the properties of equality to get
2 2 the constants on the right and the
y −1= x + ⋅3 variables on the left.
3 3
2 • Multiplyby -3 so A is positive and
y −1= x + 2 no fractions exist.
+1 3 +1
2 2 ⎛ 2 ⎞
− x − x −3 ⎜ − x ⎟ + −3 ⋅ y = −3 ⋅ 3
3 3 ⎝ 3 ⎠
2
− x+y=3 2x − 3y = −9
3
77. PARALLEL
• Parallel lines are 2 lines that never
touch. The distance between them remains
constant forever.
78. PARALLEL
• Parallel lines are 2 lines that never
touch. The distance between them remains
constant forever.
• Symbol used to represent: ∕∕
79. PARALLEL
• Parallel lines are 2 lines that never
touch. The distance between them remains
constant forever.
• Symbol used to represent: ∕∕
• Real life examples...
80. PARALLEL
• Parallel lines are 2 lines that never
touch. The distance between them remains
constant forever.
• Symbol used to represent: ∕∕
• Real life examples...
• The top of the Berlin Wall and ground in
Berlin, Germany.
81. PARALLEL
• Parallel lines are 2 lines that never
touch. The distance between them remains
constant forever.
• Symbol used to represent: ∕∕
• Real life examples...
• The top of the Berlin Wall and ground in
Berlin, Germany.
82. PARALLEL
• Parallel lines are 2 lines that never
touch. The distance between them remains
constant forever.
• Symbol used to represent: ∕∕
• Real life examples...
• The top of the Berlin Wall and ground in
Berlin, Germany.
• The towers of the Tower Bridge in
London, England.
83. PARALLEL
• Parallel lines are 2 lines that never
touch. The distance between them remains
constant forever.
• Symbol used to represent: ∕∕
• Real life examples...
• The top of the Berlin Wall and ground in
Berlin, Germany.
• The towers of the Tower Bridge in
London, England.
86. PERPENDICULAR
• Perpendicular lines are 2 lines
that intersect at a 90 degree angle.
• Symbol used to represent: ⊥
• Real life examples...
87. PERPENDICULAR
• Perpendicular lines are 2 lines
that intersect at a 90 degree angle.
• Symbol used to represent: ⊥
• Real life examples...
• The adjacent sides of a picture
frame located in the Louvre
Museum in Paris, France.
88. PERPENDICULAR
• Perpendicular lines are 2 lines
that intersect at a 90 degree angle.
• Symbol used to represent: ⊥
• Real life examples...
• The adjacent sides of a picture
frame located in the Louvre
Museum in Paris, France.
89. PERPENDICULAR
• Perpendicular lines are 2 lines
that intersect at a 90 degree angle.
• Symbol used to represent: ⊥
• Real life examples...
• The adjacent sides of a picture
frame located in the Louvre
Museum in Paris, France.
• The frame of a doorway in the
ruins of Pompeii, Italy.
90. PERPENDICULAR
• Perpendicular lines are 2 lines
that intersect at a 90 degree angle.
• Symbol used to represent: ⊥
• Real life examples...
• The adjacent sides of a picture
frame located in the Louvre
Museum in Paris, France.
• The frame of a doorway in the
ruins of Pompeii, Italy.
92. EXPLORATION ON GRAPHING CALCULATOR
• Parallel lines and Perpendicular lines can be VERY
deceiving on a graphing calculator.
93. EXPLORATION ON GRAPHING CALCULATOR
• Parallel lines and Perpendicular lines can be VERY
deceiving on a graphing calculator.
• Enter
these 2 equations in y= and graph on a standard window
(Zoom - 6:ZStandard)
• y1 = .1x - 3
• y2 = .11x + 3
94. EXPLORATION ON GRAPHING CALCULATOR
• Parallel lines and Perpendicular lines can be VERY
deceiving on a graphing calculator.
• Enter
these 2 equations in y= and graph on a standard window
(Zoom - 6:ZStandard)
• y1 = .1x - 3
• y2 = .11x + 3
• Are these parallel, perpendicular or neither based on the screen?
95. EXPLORATION ON GRAPHING CALCULATOR
• Parallel lines and Perpendicular lines can be VERY
deceiving on a graphing calculator.
• Enter
these 2 equations in y= and graph on a standard window
(Zoom - 6:ZStandard)
• y1 = .1x - 3
• y2 = .11x + 3
• Are these parallel, perpendicular or neither based on the screen?
• Do the same with these equations.
• y1 = 1.9x - 3
• y2 = -1.2x - 3
96. EXPLORATION ON GRAPHING CALCULATOR
• Parallel lines and Perpendicular lines can be VERY
deceiving on a graphing calculator.
• Enter
these 2 equations in y= and graph on a standard window
(Zoom - 6:ZStandard)
• y1 = .1x - 3
• y2 = .11x + 3
• Are these parallel, perpendicular or neither based on the screen?
• Do the same with these equations.
• y1 = 1.9x - 3
• y2 = -1.2x - 3
• Keep your answers because you will need them shortly!
97. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
98. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
• To determine if 2 lines are parallel, check the slope.
99. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
• To determine if 2 lines are parallel, check the slope.
• Same slope, different y-intercept = parallel lines
100. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
• To determine if 2 lines are parallel, check the slope.
• Same slope, different y-intercept = parallel lines
Parallel
y = 2x − 5
y = 2x + 5
101. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
• To determine if 2 lines are parallel, check the slope.
• Same slope, different y-intercept = parallel lines
• Same slope, same y-intercept = SAME line
Parallel
y = 2x − 5
y = 2x + 5
102. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
• To determine if 2 lines are parallel, check the slope.
• Same slope, different y-intercept = parallel lines
• Same slope, same y-intercept = SAME line
Same line
Parallel (÷ second by 5)
y = 2x − 5 y = −x + 3
y = 2x + 5 5y = −5x + 15
103. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
• To determine if 2 lines are parallel, check the slope.
• Same slope, different y-intercept = parallel lines
• Same slope, same y-intercept = SAME line
• Different slope, same or different y-intercept = intersecting lines
Same line
Parallel (÷ second by 5)
y = 2x − 5 y = −x + 3
y = 2x + 5 5y = −5x + 15
104. PARALLEL LINES
• Becausethe distance never changes between parallel lines, their
slopes must remain constant or eventually they will cross.
• To determine if 2 lines are parallel, check the slope.
• Same slope, different y-intercept = parallel lines
• Same slope, same y-intercept = SAME line
• Different slope, same or different y-intercept = intersecting lines
Same line
Parallel (÷ second by 5) NOT parallel
y = 2x − 5 y = −x + 3 y = 4x + 7
y = 2x + 5 5y = −5x + 15 y = 3x + 7
105. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
y2 = .11x + 3
106. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
• Do you agree with your original answer?
y2 = .11x + 3
107. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
• Do you agree with your original answer?
y2 = .11x + 3
• Why do you still agree or why did you change you mind?
108. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
• Do you agree with your original answer?
y2 = .11x + 3
• Why do you still agree or why did you change you mind?
• These equations look parallel on the graphing calculator but what do
you notice about the slopes?
109. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
• Do you agree with your original answer?
y2 = .11x + 3
• Why do you still agree or why did you change you mind?
• These equations look parallel on the graphing calculator but what do
you notice about the slopes?
• The slopes are different. They are very close, which is why they appear
parallel on the graphing calculator.
110. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
• Do you agree with your original answer?
y2 = .11x + 3
• Why do you still agree or why did you change you mind?
• These equations look parallel on the graphing calculator but what do
you notice about the slopes?
• The slopes are different. They are very close, which is why they appear
parallel on the graphing calculator.
• Change your window settings to Xmin=-1100, Xmax=5, Ymin=-100,
Ymax=5 and graph the 2 equations again.
111. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
• Do you agree with your original answer?
y2 = .11x + 3
• Why do you still agree or why did you change you mind?
• These equations look parallel on the graphing calculator but what do
you notice about the slopes?
• The slopes are different. They are very close, which is why they appear
parallel on the graphing calculator.
• Change your window settings to Xmin=-1100, Xmax=5, Ymin=-100,
Ymax=5 and graph the 2 equations again.
• Can you see the lines intersect now?
112. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel,
Perpendicular, or Neither? y1 = .1x −3
• Do you agree with your original answer?
y2 = .11x + 3
• Why do you still agree or why did you change you mind?
• These equations look parallel on the graphing calculator but what do
you notice about the slopes?
• The slopes are different. They are very close, which is why they appear
parallel on the graphing calculator.
• Change your window settings to Xmin=-1100, Xmax=5, Ymin=-100,
Ymax=5 and graph the 2 equations again.
• Can you see the lines intersect now?
• The Window setting is crucial to “seeing” if the equations are parallel. It is
easier to determine parallel lines by comparing the slopes.
113. PERPENDICULAR LINES
• Stand up nice and tall. What do you notice about your body in
relation to the floor? Are you parallel or perpendicular with the floor?
114. PERPENDICULAR LINES
• Stand up nice and tall. What do you notice about your body in
relation to the floor? Are you parallel or perpendicular with the floor?
• Foryou to be parallel, you must lay your body on the floor. (This
change is called a “rotation” in math.) Standing you meet the floor at
a 90 degree angle so the “rotation” would be 90 degrees for you to
be parallel to the floor.
115. PERPENDICULAR LINES
• Stand up nice and tall. What do you notice about your body in
relation to the floor? Are you parallel or perpendicular with the floor?
• Foryou to be parallel, you must lay your body on the floor. (This
change is called a “rotation” in math.) Standing you meet the floor at
a 90 degree angle so the “rotation” would be 90 degrees for you to
be parallel to the floor.
• Draw a coordinate plane on your paper. Place 2 pencils on the graph
so they cross with a 90 degree angle. Don’t place them vertical and
horizontal on the coordinate plane because these are a special case
but do move them around maintaining the 90 degree angle.
116. PERPENDICULAR LINES
• Stand up nice and tall. What do you notice about your body in
relation to the floor? Are you parallel or perpendicular with the floor?
• Foryou to be parallel, you must lay your body on the floor. (This
change is called a “rotation” in math.) Standing you meet the floor at
a 90 degree angle so the “rotation” would be 90 degrees for you to
be parallel to the floor.
• Draw a coordinate plane on your paper. Place 2 pencils on the graph
so they cross with a 90 degree angle. Don’t place them vertical and
horizontal on the coordinate plane because these are a special case
but do move them around maintaining the 90 degree angle.
• What did you notice about the slopes of the pencils? Both positive?
Both negative? One of each? ...
118. PERPENDICULAR LINES (CONTINUED)
• Noticeone is always positive and one is
always negative.
• Perpendicular
lines have slopes that are
negative reciprocals.
119. PERPENDICULAR LINES (CONTINUED)
• Noticeone is always positive and one is
always negative.
• Perpendicular
lines have slopes that are
negative reciprocals.
• Negative reciprocals mean the slopes
have opposite signs and the number is flipped.
120. PERPENDICULAR LINES (CONTINUED)
• Noticeone is always positive and one is
always negative.
Perpendicular
y = 2x − 3
• Perpendicular
lines have slopes that are
negative reciprocals. 1
y=− x+5
• Negative reciprocals mean the slopes 2
have opposite signs and the number is flipped.
• Such as -1/2 and 2 are negative reciprocals.
121. PERPENDICULAR LINES (CONTINUED)
• Noticeone is always positive and one is
always negative.
Perpendicular
y = 2x − 3
• Perpendicular
lines have slopes that are
negative reciprocals. 1
y=− x+5
• Negative reciprocals mean the slopes 2
have opposite signs and the number is flipped.
• Such as -1/2 and 2 are negative reciprocals.
•3 and -3 are opposite
but NOT reciprocals. NOT ⊥
y = 3x + 7
y = −3x + 7
122. PERPENDICULAR LINES (CONTINUED)
• Noticeone is always positive and one is
always negative.
Perpendicular
y = 2x − 3
• Perpendicular
lines have slopes that are
negative reciprocals. 1
y=− x+5
• Negative reciprocals mean the slopes 2
have opposite signs and the number is flipped.
• Such as -1/2 and 2 are negative reciprocals.
Perpendicular
•3 and -3 are opposite 3
but NOT reciprocals. NOT ⊥ y = x −1
4
• 3/4and -4/3 are negative y = 3x + 7
reciprocals. Can have 4
y = −3x + 7 y = − x −1
the same y-intercept. 3
123. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
y2 = −1.2x − 3
124. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
• Do you agree with your answer? You may want to change
these to fractions to make a better determination. y2 = −1.2x − 3
125. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
• Do you agree with your answer? You may want to change
these to fractions to make a better determination. y2 = −1.2x − 3
• Why do you still agree or why did you change you mind?
126. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
• Do you agree with your answer? You may want to change
these to fractions to make a better determination. y2 = −1.2x − 3
• Why do you still agree or why did you change you mind?
• These equations look perpendicular on the graphing calculator but what do you
notice about the slopes?
127. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
• Do you agree with your answer? You may want to change
these to fractions to make a better determination. y2 = −1.2x − 3
• Why do you still agree or why did you change you mind?
• These equations look perpendicular on the graphing calculator but what do you
notice about the slopes?
• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They
are very close, which is why they appear perpendicular on the graphing calculator.
128. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
• Do you agree with your answer? You may want to change
these to fractions to make a better determination. y2 = −1.2x − 3
• Why do you still agree or why did you change you mind?
• These equations look perpendicular on the graphing calculator but what do you
notice about the slopes?
• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They
are very close, which is why they appear perpendicular on the graphing calculator.
• Change your window settings to Xmin=-5, Xmax=5 and graph the 2 equations
again.
129. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
• Do you agree with your answer? You may want to change
these to fractions to make a better determination. y2 = −1.2x − 3
• Why do you still agree or why did you change you mind?
• These equations look perpendicular on the graphing calculator but what do you
notice about the slopes?
• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They
are very close, which is why they appear perpendicular on the graphing calculator.
• Change your window settings to Xmin=-5, Xmax=5 and graph the 2 equations
again.
• Do they appear perpendicular?
130. BACK TO THE EXPLORATION
• What did you say about the graph of these equations? Parallel, Perpendicular, or
Neither?
y1 = 1.9x − 3
• Do you agree with your answer? You may want to change
these to fractions to make a better determination. y2 = −1.2x − 3
• Why do you still agree or why did you change you mind?
• These equations look perpendicular on the graphing calculator but what do you
notice about the slopes?
• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They
are very close, which is why they appear perpendicular on the graphing calculator.
• Change your window settings to Xmin=-5, Xmax=5 and graph the 2 equations
again.
• Do they appear perpendicular?
• Like for parallel lines, the Window setting is crucial to “seeing” if the equations are
perpendicular. It is easier to determine perpendicular lines by comparing the slopes.
132. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
before determining the slope of
the given line.
133. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
134. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
135. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
−1 −1
136. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
−1 −1
y = 2x − 5
137. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
• Identify the slope.
−1 −1
y = 2x − 5
138. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
• Identify the slope.
−1 −1
y = 2x − 5
m=2
139. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
• Identify the slope.
−1 −1
• Parallel slopes are the same.
y = 2x − 5
m=2
140. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
• Identify the slope.
−1 −1
• Parallel slopes are the same.
y = 2x − 5
m=2
Parallel slope
m=2
141. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
• Identify the slope.
−1 −1
• Parallel slopes are the same.
y = 2x − 5
m=2 • Perpendicular slopes are negative
reciprocals.
Parallel slope
m=2
142. FINDING ∕∕ OR ⊥ SLOPE
• Always put equation in slope-
2x − y = 5 intercept form (y = mx + b)
−2x −2x before determining the slope of
the given line.
−y = −2x + 5
• Identify the slope.
−1 −1
• Parallel slopes are the same.
y = 2x − 5
m=2 • Perpendicular slopes are negative
reciprocals.
Perpendicular slope
Parallel slope
1
m=2 m=−
2
144. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
before determining the slope of
the given line.
145. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
146. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
2y = 3x + 7
147. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
2y = 3x + 7
2 2
148. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
2y = 3x + 7
2 2
3 7
y= x+
2 2
149. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
2y = 3x + 7
2 2 • Identify the slope.
3 7
y= x+
2 2
150. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
2y = 3x + 7
2 2 • Identify the slope.
3 7
y= x+
2 2
3
m=
2
151. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
2y = 3x + 7
2 2 • Identify the slope.
3 7 • Parallel slopes are the same.
y= x+
2 2
3
m=
2
Parallel slope
3
m=
2
152. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...
−3x + 2y = 7 • Always put equation in slope-
intercept form (y = mx + b)
+3x +3x before determining the slope of
the given line.
2y = 3x + 7
2 2 • Identify the slope.
3 7 • Parallel slopes are the same.
y= x+
2 2
• Perpendicular slopes are negative
3
m= reciprocals.
2
Perpendicular slope
Parallel slope
3
2
m= m=−
2 3
153. WRITING ∕∕ EQUATION
Write the slope-intercept form of the
equation parallel to y = -3x + 4,
which passes through the point (2, -5).
154. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
155. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3
156. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
157. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
• Use point-slope formula to find parallel
equation.
158. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
equation.
159. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
equation.
• Substitute given point and parallel slope.
160. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − ( −5 ) = −3( x − 2 ) equation.
• Substitute given point and parallel slope.
161. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − ( −5 ) = −3( x − 2 ) equation.
• Substitute given point and parallel slope.
• Put equation in slope-intercept form.
162. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − ( −5 ) = −3( x − 2 ) equation.
y + 5 = −3x + 6 • Substitute given point and parallel slope.
• Put equation in slope-intercept form.
163. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − ( −5 ) = −3( x − 2 ) equation.
y + 5 = −3x + 6 • Substitute given point and parallel slope.
−5 −5
• Put equation in slope-intercept form.
164. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − ( −5 ) = −3( x − 2 ) equation.
y + 5 = −3x + 6 • Substitute given point and parallel slope.
−5 −5
• Put equation in slope-intercept form.
y = −3x + 1
165. WRITING ∕∕ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation parallel to y = -3x + 4, equation. (May need to put in slope-
which passes through the point (2, -5). intercept form.)
m = −3 • Parallel slopes are the same so use m =
-3.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − ( −5 ) = −3( x − 2 ) equation.
y + 5 = −3x + 6 • Substitute given point and parallel slope.
−5 −5
• Put equation in slope-intercept form.
• Always check your equation to ensure it
y = −3x + 1 makes sense. The lines are parallel so
the slopes must be the same (they are)
and y-intercepts different (they are).
166. YOUR TURN...
Write the slope-intercept form of the
equation parallel to y = 4x + 7, which
passes through the point (-3, 8).
167. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
168. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4
169. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
170. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
• Use point-slope formula to find parallel
equation.
171. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
equation.
172. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
equation.
• Substitute given point and parallel slope.
173. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − 8 = 4 ( x − ( −3)) equation.
• Substitute given point and parallel slope.
174. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − 8 = 4 ( x − ( −3)) equation.
• Substitute given point and parallel slope.
• Put equation in slope-intercept form.
175. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − 8 = 4 ( x − ( −3)) equation.
y − 8 = 4 ( x + 3) • Substitute given point and parallel slope.
• Put equation in slope-intercept form.
176. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − 8 = 4 ( x − ( −3)) equation.
y − 8 = 4 ( x + 3) • Substitute given point and parallel slope.
y − 8 = 4x + 12 • Put equation in slope-intercept form.
177. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − 8 = 4 ( x − ( −3)) equation.
y − 8 = 4 ( x + 3) • Substitute given point and parallel slope.
y − 8 = 4x + 12 • Put equation in slope-intercept form.
+8 +8
178. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − 8 = 4 ( x − ( −3)) equation.
y − 8 = 4 ( x + 3) • Substitute given point and parallel slope.
y − 8 = 4x + 12 • Put equation in slope-intercept form.
+8 +8
y = 4x + 20
179. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation parallel to y = 4x + 7, which equation. (No need to put in slope-
passes through the point (-3, 8). intercept form here.)
m=4 • Parallel slopes are the same so use m =
4.
y − y1 = m ( x − x1 )
• Use point-slope formula to find parallel
y − 8 = 4 ( x − ( −3)) equation.
y − 8 = 4 ( x + 3) • Substitute given point and parallel slope.
y − 8 = 4x + 12 • Put equation in slope-intercept form.
+8 +8 • Check that equation makes sense. The
lines are parallel so the slopes must be
the same (they are) and y-intercepts
y = 4x + 20 different (they are).
180. WRITING ⊥ EQUATION
Write the slope-intercept form of the
equation perpendicular to y = -5x + 2,
which passes through the point (10, 3).
181. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
182. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5
183. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 • Perpendicular slopes are negative
reciprocals.
184. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
185. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
• Use point-slope formula to find
perpendicular equation.
186. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
perpendicular equation.
187. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
perpendicular equation.
• Substitute given point and perpendicular
slope.
188. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
1 perpendicular equation.
y − 3 = ( x − 10 )
5 • Substitute given point and perpendicular
slope.
189. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
1 perpendicular equation.
y − 3 = ( x − 10 )
5 • Substitute given point and perpendicular
slope.
• Put equation in slope-intercept form.
190. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
1 perpendicular equation.
y − 3 = ( x − 10 )
5 • Substitute given point and perpendicular
1 slope.
y−3= x−2
5 • Put equation in slope-intercept form.
191. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
1 perpendicular equation.
y − 3 = ( x − 10 )
5 • Substitute given point and perpendicular
1 slope.
y−3= x−2
+3 5 +3 • Put equation in slope-intercept form.
192. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
1 perpendicular equation.
y − 3 = ( x − 10 )
5 • Substitute given point and perpendicular
1 slope.
y−3= x−2
+3 5 +3 • Put equation in slope-intercept form.
1
y = x +1
5
193. WRITING ⊥ EQUATION
Write the slope-intercept form of the • Always determine the slope of the given
equation perpendicular to y = -5x + 2, equation. (May need to put in slope-
which passes through the point (10, 3). intercept form.)
m = −5 1 • Perpendicular slopes are negative
m⊥ =
5 reciprocals.
y − y1 = m ( x − x1 ) • Use point-slope formula to find
1 perpendicular equation.
y − 3 = ( x − 10 )
5 • Substitute given point and perpendicular
1 slope.
y−3= x−2
+3 5 +3 • Put equation in slope-intercept form.
• Always check your equation to ensure it
1 makes sense. The lines are perpendicular
y = x +1 so the slopes must be negative
5 reciprocals (they are).
194. YOUR TURN...
Write the slope-intercept form of the
equation perpendicular to y = 3x - 1,
which passes through the point (6, 9).
195. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
196. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3
197. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 • Perpendicular slopes are negative
reciprocals.
198. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
199. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
• Use point-slope formula to find
perpendicular equation.
200. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
perpendicular equation.
201. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
perpendicular equation.
• Substitute given point and
perpendicular slope.
202. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
1 perpendicular equation.
y − 9 = − ( x − 6)
3 • Substitute given point and
perpendicular slope.
203. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
1 perpendicular equation.
y − 9 = − ( x − 6)
3 • Substitute given point and
perpendicular slope.
• Put equation in slope-intercept form.
204. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
1 perpendicular equation.
y − 9 = − ( x − 6)
3 • Substitute given point and
1 perpendicular slope.
y−3= − x+2
3 • Put equation in slope-intercept form.
205. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
1 perpendicular equation.
y − 9 = − ( x − 6)
3 • Substitute given point and
1 perpendicular slope.
y−3= − x+2
+3 3 +3 • Put equation in slope-intercept form.
206. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
1 perpendicular equation.
y − 9 = − ( x − 6)
3 • Substitute given point and
1 perpendicular slope.
y−3= − x+2
+3 3 +3 • Put equation in slope-intercept form.
1
y=− x+5
3
207. YOUR TURN...
Write the slope-intercept form of the • Determine the slope of the given
equation perpendicular to y = 3x - 1, equation. (No need to put in slope-
which passes through the point (6, 9). intercept form here.)
m=3 1
m⊥ = − • Perpendicular slopes are negative
3 reciprocals.
y − y1 = m ( x − x1 )
• Use point-slope formula to find
1 perpendicular equation.
y − 9 = − ( x − 6)
3 • Substitute given point and
1 perpendicular slope.
y−3= − x+2
+3 3 +3 • Put equation in slope-intercept form.
1 • Check that equation makes sense. The
y=− x+5 lines are perpendicular so the slopes
must be negative reciprocals (they are).
3