This document provides an overview of linear equations and graphs. It introduces the Cartesian coordinate system and defines linear equations in two variables. It discusses using intercepts and graphing calculators to graph lines. Special cases of vertical and horizontal lines are covered. The concepts of slope, slope-intercept form, and point-slope form of a line are explained. An example of using a linear equation to model the depreciation of office equipment is provided. Finally, the relationship between supply and demand is discussed and an example of finding the equilibrium point using supply and demand equations is worked through.
Chapter 5: Determinant
Covered Topics:
5.1 Definition of Determinant
5.2 Expansion of Determinant of order 2X3
5.3 Crammer’s rule to solve simultaneous equations in 3 unknowns
Youtube Link: https://youtu.be/C2qctvyjG7U
Document:
Our Blog Link: http://jjratnani.wordpress.com/
Chapter 5: Determinant
Covered Topics:
5.1 Definition of Determinant
5.2 Expansion of Determinant of order 2X3
5.3 Crammer’s rule to solve simultaneous equations in 3 unknowns
Youtube Link: https://youtu.be/C2qctvyjG7U
Document:
Our Blog Link: http://jjratnani.wordpress.com/
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
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
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
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
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
Friends if you found this helpful please click the like button. and share it :) thanks for watching
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
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.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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.
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.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
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.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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
2. Learning Objectives for Section 1.2
Graphs and Lines
The student will be able to identify and work with the
Cartesian coordinate system.
The student will be able to draw graphs for equations of the
form Ax + By = C.
The student will be able to calculate the slope of a line.
The student will be able to graph special forms of equations
of lines.
The student will be able to solve applications of linear
equations.
Barnett/Ziegler/Byleen College Mathematics 12e 2
3. The Cartesian Coordinate System
The Cartesian coordinate system was named after René Descartes. It
consists of two real number lines, the horizontal axis (x-axis) and the
vertical axis (y-axis) which meet in a right angle at a point called the
origin. The two number lines divide the plane into four areas called
quadrants.
The quadrants are numbered using Roman numerals as shown on the
next slide. Each point in the plane corresponds to one and only one
ordered pair of numbers (x,y). Two ordered pairs are shown.
Barnett/Ziegler/Byleen College Mathematics 12e 3
4. The Cartesian Coordinate System
(continued)
x
II I
(3,1)
(–1,–1)
III IV
y
Two points, (–1,–1)
and (3,1), are plotted.
Four quadrants are as
labeled.
Barnett/Ziegler/Byleen College Mathematics 12e 4
5. Linear Equations in Two Variables
A linear equation in two variables is an equation that can
be written in the standard form Ax + By = C, where A, B,
and C are constants (A and B not both 0), and x and y are
variables.
A solution of an equation in two variables is an ordered
pair of real numbers that satisfy the equation. For example,
(4,3) is a solution of 3x - 2y = 6.
The solution set of an equation in two variables is the set
of all solutions of the equation.
The graph of an equation is the graph of its solution set.
Barnett/Ziegler/Byleen College Mathematics 12e 5
6. Linear Equations in Two Variables
(continued)
If A is not equal to zero and B is not equal to zero, then
Ax + By = C can be written as
This is known as
y = - A x + C = mx +
b
slope-intercept form.
If A = 0 and B is not equal to zero,
then the graph is a horizontal line
If A is not equal to zero and B = 0,
then the graph is a vertical line
B B
y C
B
=
x C
A
=
Barnett/Ziegler/Byleen College Mathematics 12e 6
7. Using Intercepts to Graph a Line
Graph 2x – 6y = 12.
Barnett/Ziegler/Byleen College Mathematics 12e 7
8. Using Intercepts to Graph a Line
Graph 2x – 6y = 12.
x y
0 –2 y-intercept
6 0 x-intercept
3 –1 check point
Barnett/Ziegler/Byleen College Mathematics 12e 8
9. Using a Graphing Calculator
Graph 2x – 6y = 12 on a graphing calculator and find the intercepts.
Barnett/Ziegler/Byleen College Mathematics 12e 9
10. Using a Graphing Calculator
Graph 2x – 6y = 12 on a graphing calculator and find the intercepts.
Solution: First, we solve the equation for y.
2x – 6y = 12 Subtract 2x from each side.
–6y = –2x + 12 Divide both sides by –6
y = (1/3)x – 2
Now we enter the right side of this equation in a calculator, enter
values for the window variables, and graph the line.
Barnett/Ziegler/Byleen College Mathematics 12e 10
11. Special Cases
The graph of x = k is the graph of a vertical line k units from
the y-axis.
The graph of y = k is the graph of the horizontal line k units
from the x-axis.
Examples:
1. Graph x = –7
2. Graph y = 3
Barnett/Ziegler/Byleen College Mathematics 12e 11
12. Solutions
x = –7
y = 4
Barnett/Ziegler/Byleen College Mathematics 12e 12
13. Slope of a Line
Slope of a line:
rise
m = y -
y 2 1
=
x x
2 1
run
o
rise
o ( ) 2 2 x , y
( ) 1 1 x , y
run
-
Barnett/Ziegler/Byleen College Mathematics 12e 13
14. Slope-Intercept Form
The equation
y = mx+b
is called the slope-intercept form of an equation of a line.
The letter m represents the slope and b represents the
y-intercept.
Barnett/Ziegler/Byleen College Mathematics 12e 14
15. Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line
whose equation is 5x – 2y = 10.
Barnett/Ziegler/Byleen College Mathematics 12e 15
16. Find the Slope and Intercept
from the Equation of a Line
Example: Find the slope and y intercept of the line
whose equation is 5x – 2y = 10.
Solution: Solve the equation
for y in terms of x. Identify the
x - y
=
coefficient of x as the slope
y x
and the y intercept as the
constant term.
Therefore: the slope is 5/2 and
the y intercept is –5.
5 2 10
2 5 10
5 10 5 5
2 2 2
- = - +
= - + = -
- -
y x x
Barnett/Ziegler/Byleen College Mathematics 12e 16
17. Point-Slope Form
The point-slope form of the equation of a line is
1 1 y - y = m(x - x )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y - y =
m
x -
x
2 1
2 1
Cross-multiply and
substitute the more
general x for x2
Barnett/Ziegler/Byleen College Mathematics 12e 17
18. Example
Find the equation of the line through the points (–5, 7) and (4, 16).
Barnett/Ziegler/Byleen College Mathematics 12e 18
19. Example
Find the equation of the line through the points (–5, 7) and (4, 16).
Solution:
1
m = -
16 7 = 9
=
- -
9
4 ( 5)
Now use the point-slope form with m = 1 and (x1, x2) = (4, 16).
(We could just as well have used (–5, 7)).
y x
- = -
y x x
16 1( 4)
= - + = +
4 16 12
Barnett/Ziegler/Byleen College Mathematics 12e 19
20. Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years:
Barnett/Ziegler/Byleen College Mathematics 12e 20
21. Application
Office equipment was purchased for $20,000 and will have a
scrap value of $2,000 after 10 years. If its value is depreciated
linearly, find the linear equation that relates value (V) in dollars
to time (t) in years:
Solution: When t = 0, V = 20,000 and when t = 10, V = 2,000.
Thus, we have two ordered pairs (0, 20,000) and (10, 2000).
We find the slope of the line using the slope formula.
The y intercept is already known (when t = 0, V = 20,000, so
the y intercept is 20,000).
The slope is (2000 – 20,000)/(10 – 0) = –1,800.
Therefore, our equation is V(t) = –1,800t + 20,000.
Barnett/Ziegler/Byleen College Mathematics 12e 21
22. Supply and Demand
In a free competitive market, the price of a product is
determined by the relationship between supply and
demand. The price tends to stabilize at the point of
intersection of the demand and supply equations.
This point of intersection is called the equilibrium point.
The corresponding price is called the equilibrium price.
The common value of supply and demand is called the
equilibrium quantity.
Barnett/Ziegler/Byleen College Mathematics 12e 22
23. Supply and Demand
Example
Use the barley market data in the following table to find:
(a) A linear supply equation of the form p = mx + b
(b) A linear demand equation of the form p = mx + b
(c) The equilibrium point.
Year Supply
Mil bu
Demand
Mil bu
Price
$/bu
2002 340 270 2.22
2003 370 250 2.72
Barnett/Ziegler/Byleen College Mathematics 12e 23
24. Supply and Demand
Example (continued)
(a) To find a supply equation in the form p = mx + b, we
must first find two points of the form (x, p) on the supply line.
From the table, (340, 2.22) and (370, 2.72) are two such
points. The slope of the line is
m = - = =
2.72 2.22 0.5 0.0167
370 -
340 30
Now use the point-slope form to find the equation of the line:
p – p1 = m(x – x1)
p – 2.22 = 0.0167(x – 340)
p – 2.22 = 0.0167x – 5.678
p = 0.0167x – 3.458 Price-supply equation.
Barnett/Ziegler/Byleen College Mathematics 12e 24
25. Supply and Demand
Example (continued)
(b) From the table, (270, 2.22) and (250, 2.72) are two points
on the demand equation. The slope is
m = - = = -
2.72 2.22 .5 0.025
250 - 270 -
20
p – p1 = m(x – x1)
p – 2.22 = –0.025(x – 270)
p – 2.22 = –0.025x + 6.75
p = –0.025x + 8.97 Price-demand equation
Barnett/Ziegler/Byleen College Mathematics 12e 25
26. Supply and Demand
Example (continued)
(c) If we graph the two equations on a graphing
calculator, set the window as shown, then use the
intersect operation, we obtain:
The equilibrium point is approximately (298, 1.52).
This means that the common value of supply and
demand is 298 million bushels when the price is $1.52.
Barnett/Ziegler/Byleen College Mathematics 12e 26