Accounting Case Study Essay Example | Topics and Well Written Essays - 500 words - 1

Accounting Case Study - Essay Example There is a belief that poor branding identity is the root cause of declining market fortunes. In addition, its recent methods of retailing have been overcome by recent market shifts. Looking at the giant entrants as if the Wal-Mart and Canadian Tire depicts a total shift in retailing strategies. Perhaps the failure of the manufacturer to respond to market trends has largely caused the problem that is being currently mitigated. More importantly, the larger manufacturers have adopted the use of modern information technology and sophisticated inventory management systems. It is certain that deliberate steps need to be undertaken immediately. Apart from a shift of the marketing trends towards information technology, the company has failed to build a strong brand as the marketing forces become fierce. Today, the category suppliers have grown significantly that they command specific designs for manufacturers to produce. Evidently, the recent times has seen category supply, demand particular sizes and colors of goods being supplied. While Clearwater remains stagnant in the traditional retail systems, the market forces have overwhelmingly become unbearable. Besides, the problem has arisen because of untimely management decisions. In business, the management team should be vibrant and creative. Acting timely to respond to changing market patterns is critical in ensuring that a company is not forced out of the market. With a timely and a continuous product adjustment, the company would not be experiencing the current marketing doldrums. In addition, it is certain that the problem is solely a marketing and strategic step. One of the problem would have been solved before it became evident was branding, as a business grows that need to maintain a highly competitive manageable is crucial, management decisions should explore product and study market patterns with a view of making necessary

Soviet Participation Essay Example for Free

Soviet Participation Essay When the games were reinstated in 1920 and again in 1948, [9]the Soviets initially declined to take part. The primary goal of â€Å"Socialist sports† was to benefit the masses, not the elite athletes. Marxist-Leninist ideology intended all citizens to participate to enhance their own strength as well as the strength of the state. They believed that recreation and the training of the body allowed workers to remain strong and healthy as well as productive. Health and productivity would also lead to building moral character and social responsibility. Since sports were intended for the masses, rather than for an athletic elite, the Soviet rulers saw no use for participation in the Olympic games. Soon after the 1917 revolution ended, Bolshevik leaders established a state-run sports system, which consisted of hundreds of sports clubs as well as two large sports societies run by the secret police and the Army (Rosellini n.pag.). Unfortunately, as quickly as these well- intended clubs were formed, their original purpose began to disintegrate. As early as 1926, the sports clubs were accused of ignoring â€Å"the vast mass of young people† and focusing on the athletic elite, because the sports facilities were straying from their initial goal of supporting the masses and focusing on the athletically gifted (Rosselini n.pag.). Between the World Wars the Soviets remained separate from the games. But away from the rest of the world, they were devising a plan to make their athletes rank among the best in the world. Finally, in 1948, the Soviets and their Satellites decided to challenge the West in the 1952 Olympics. The ruling Soviet party demanded that government officials â€Å"raise the level of skill, so that Soviet sportsmen might win the world supremacy in the major sports in the immediate future† (Rosellini n.pag.). To do this children were evaluated and if found suitable, were taken to live in â€Å"sports factories.† There, children were trained many hours a day and were secretly dosed with steroids. Child athletes were usually between the ages of six and eighteen in these training facilities and could have been taken from their parents as early as the age of three. Each athlete had his or her own trainer, doctor, masseuse, physiologist, and sports conceptualizer to plan an individual program. The outside world remained unaware of the Soviet design for sports dominance. Not until members of the Soviet Union entered Olympic competition did steroid use affect athletic achievement. The Soviet factories greatly threatened the ideals of the International Olympic Committee. Competitive fair play and international cooperation were jeopardized by the Soviet’s desire for political superiority. The factories were a symptom of one of the greatest threats to the Olympic ideals: countries seeking political influence and international recognition. The Soviets used these â€Å"factories† as a way to improve the international standing of their country. If they came out on top in sports they believed they would appear to validate the communist political system to the rest of the world. But, Soviet athletes physically looked different. When swimmer Raik Hannemann was seventeen, his trainer approached him and said, Were going to try something secret, keep it to yourself. It will add two percent to your time and bring you to that extra level of excellence (Rosselini n.pag). Hannemann knew the blue tablets had to do something with steroids, but he didn’t know what the side effects were and trusted his trainer. He was even told to keep the tablets a secret from his parents. Once he took the performance-enhancing drugs his speed immediately increased by six seconds (Rosselini n.pag). Athletes competing in a variety of sports were treated with performance-enhancing drugs, but the swimmers especially boasted abnormal musculature. The judges and spectators demanded explanations, of how these athletes grew so grotesquely strong. Although no medical tests for sports enhancing drugs existed yet, in 1976, performance-enhancing drugs were placed on the International Olympic Committee’s banned substances list in response to the unproven Soviet’s steroid use in the recent Olympic games (Chidlovski n.pag.).[10] A decade later the Olympic committee introduced newly designed doping tests to detect if athletes used the drugs in their training period rather than during competition. Some Olympic events, such as the Hungarian defeat of the Soviet water polo team in 1956 took a large symbolism significance (Sterngass pg. 37).  Drug Use by Other Countries  Of course, the U.S.S.R. was not the only country to seek international recognition through Olympic success. Many other countries have defied the Olympic code by using performance-enhancing drugs as well. â€Å"Sports became a propaganda tool and athletic success was closely tied to nationalism and patriotism† (Sterngass pg.37). [11] As medal counts became more important, the use of performance-enhancing drugs also became more prevalent. Steroids first threatened the Olympic ideals by countries seeking political superiority. Suspicion of steroid use began as early as 1968, but the drugs did not become common until the 1972 Olympic games (â€Å"Steroid Abuse in Sports-Steroid Abuse.com† n.pag.)[12] The East German’s joined in the use of performance-enhancing drugs early on as well. â€Å"In 1968 East Germany’s chief medical officer submitted a report to the government recommending the total and collective administration of steroids to all East German athletes† (â€Å"Steroid Abuse in Sports- Steroid Abuse.com† n.pag.). In the twenty years preceding this recommendation, Eastern Germany dominated nearly every international sporting competition. The East German drug use was yet another effort to prove their own superiority over the West, just as the Soviets had done just a few years earlier. Many East-German athletes were told that they were taking vitamins, rather than steroids. So, not only was the East German government practicing the use of unfair drugs, but they were also lying to their own athletes and jeopardizing the long-term health of the individuals. Since then, as more drugs have been developed, drug tests and methods for testing have also expanded (Benagh n.pag.).[13] In recent years, many American athletes have been accused and have tested positively for drug use. This kind of cheating is highly unethical and defies the Olympic code of fair play and good sportsmanship. Performance-enhancing drugs add a more political aspect to the games, causing less focus on the actual competitive athletics. As more and more athletes defy the Olympic ideals, rules must be regulated even more closely. These changes in rules also affect the spirit of the Olympics, which fosters an international feeling of community and competition. As more athletes break the rules, more stringent oversight ensues. This strict regulation takes much of the integrity out of the games, diminishing the Olympic spirit. In the 2008 Beijing games alone, 4,500 athletes were tested; only eight of those tested positive and were banned from competing (â€Å"OLYMPIC GAMESà ¢â‚¬  n.pag.).[14] But, many more athletes also may have been guilty, perhaps they were not caught because the tests were not rigorous enough. Canadian sprinter, Ben Johnson, was just one athlete who was found guilty after winning an Olympic medal. In 1988, Johnson won the gold medal in the one hundred meter final in the Seoul games, setting a world record and was considered the best one hundred meter runner of his time. However, just three days after winning Olympic gold, Johnson’s urine sample tested positive for performance-enhancing drugs and his medal and records were stripped. Johnson was suspended from competition until 1991, but re-entered the indoor track scene and qualified for the 1992 games, where he placed last. Just one year later, Johnson came close to a fifty-meter record, but again failed a drug test (Whooley n.pag). [15]Marion Jones, a former track star who won five medals in the 2000 Sydney games, was convicted of steroid usage and all of her medals were stripped. Years later, evidence and testimony supported a conviction, sentencing Jones to six months in prison (Mulero n. pag.).[16]

Autism is a developmental disorder of the brain

Autism is a developmental disorder of the brain What is Autism? Autism is a developmental disorder of the brain. People with autism have problems communicating or interacting socially with society. They also may have unusual patterns of behavior, interests and activities. There are five kinds of autism, which is why doctors use the term autism spectrum disorder (ASD). The three main types of autism include: Classic autism, Aspergers syndrome, Nonspecific pervasive developmental disorder (PDD-NOS) A group of children who dont quite fit the criteria for the other types. There are also two rare autism disorders: Rett syndrome a neurodevelopment disorder that affects mostly girls; it includes problems with movement and speech, along with autistic features. Childhood disintegrative disorder a severe type in which the child loses more physical, language and social skills than in classic autism. Autism used to be the term used for anyone with that particular condition. Today, there are several different sub categories for different levels of disability or function. No two children diagnosed will be the same, but there will be many things that they do have in common. With some high functioning autistics most people may not even be aware that they have autism, while others need assistance in almost any part of life, and they are obviously living a very different life than other children. More is being discovered about each of these different autism types as time goes on (Evans). Aspergers Disorder is a type of autism that you hear more and more about. These children are often misdiagnosed at first, and are thought to have Obsessive-Compulsive Disorder, or perhaps Attention Deficit Disorder. These children are very unskilled with social interactions and have problems with communicating. They have repetitive motions, and are fixated on patterns of all types. They can have above average language skills, though they dont use them well in social situations. They are often clumsy, as motor skills are under developed. Those with Aspergers are thought to have a talent that they focus on almost exclusively, and are considered to be highly intelligent. Recent findings indicate that Albert Einstein may have had this condition (Evans). Kanners Syndrome is a particular type of autism that was named after Dr. Kanner. He described and studied it in the 1930s and into the 1940s. This is the well-known type of this condition that is very common. Those with Kanners have very limited emotional connection with anyone, and they are very into their own little world. They want everything to be the same all of the time, and this includes routine (sometimes down to the exact minute) clothing, food, and television shows or movies. They can be deeply affected by noises, bright lights and smells. They are generally considered to be low functioning, but how well their mind works is largely unknown because of extremely poor social and communication skills (Evans). Pervasive Developmental DisorderNot Otherwise Specified (PDD-NOS) is used to describe children who have most of the same symptoms as classic autism. They will need the same interventions and help that autistic children require. The differences between PDD-NOS and autism are minor and usually only obvious to researchers and doctors (Evans). Retts Syndrome is a rare and relatively little-known type of autism, and it seems to only happen in girls. This branch of autism was first described by Dr. Rett. These patients often have problems with muscle atrophy, and tend to do repetitive hand motions. They are almost always mentally retarded to some degree. These girls are very low functioning and will need care for most of their lives. This particular type of autism has been diagnosed since the sixties, but in the late 1990s a gene that might cause this condition was found (Evans). Childhood Disintegrative Disorder is also rare and something that strikes children who appear to have normal development from birth. Usually between two and four years of age this changes. These children begin to regress, and often do not potty train. They will lose the will and the ability to interact with other children, and will lose an interest in playing. They will also have problem with the motor skills that were something they at one time had mastered. They will stop talking, or their communication skills will regress to some degree (Evans). Signs of autism spectrum disorder (ASD) are typically first seen in toddlers before the age of three, but only half of the children with it are diagnosed before kindergarten. Rett syndrome is caused by a mutation on a gene, while the cause of the other types of autism remain unknown. Some studies suggest that other forms of autism may be inherited (genetic), while other evidence points to infection or the effects of an environmental toxin (poison). Some doctors believe autism may result from a brain injury or brain abnormality that occurred during development in the womb or in early infancy. Others have reported evidence that the disorder is a result of abnormal levels of chemicals called neurotransmitters, such as dopamine and serotonin, which send messages between cells in the brain and nerves. ASD affects about 2 to 6 out of 1,000 children, from all racial, ethnic and social backgrounds. It is three to four times more common in boys than in girls, with Rett syndrome being the exce ption (Wiki). Some of the diagnostic tests performed to see whether a child is autistic or not are: Behavioral assessments. Various guidelines and questionnaires are used to help a doctor determine the specific type of developmental delay a child has. These include: Medical history. During the medical history interview, a doctor asks general questions about a childs development, such as whether a child shows parents things by pointing to objects. Young children with autism often point to items they want, but do not point to show parents an item and then check to see if parents are looking at the item being pointed out. Diagnostic guidelines for autism. The American Association of Childhood and Adolescent Psychiatry (AACAP) has established guidelines for diagnosing autism.2 The criteria are designed so a doctor can assess a childs behavior relating to core symptoms of autism. The criteria are designed for children age 3 and older. Other behavioral questionnaires. Additional diagnostic tests focus on children younger than age 3. Clinical observations. A doctor may want to observe the developmentally delayed child in different situations. The parents may be asked to interpret whether certain behaviors are usual for the child in those circumstances. Developmental and intelligence tests. The AACAP also recommends that tests be given to evaluate whether a childs developmental delays affect his or her ability to think and make decisions (WebMD). Some parents believe that the MMR vaccine (an immunization shot against measles, mumps, and rubella) children receive may cause autism. This theory was based on two facts. First, the incidence of autism has increased steadily since around the same time the MMR vaccine was introduced. Second, children with the regressive form of autism (a type of autism that develops after a period of normal development) tend to start to show symptoms around the time the MMR vaccine is given. Several major studies have found no connection between the vaccine and autism. The American Academy of Pediatrics and the Center for Disease Control and Prevention report that there is no proven link between autism and the MMR vaccine, or any other vaccine. Some doctors believe the increased incidence in autism is due to newer definitions of autism. The term autism now includes a wider spectrum of children. For example, a child who is diagnosed with high-functioning autism today may have been thought to simply be odd or strange 30 years ago (Oasis). An early, intensive, appropriate treatment program will greatly improve the outlook for most young children with autism. Most programs will build on the interests of the child in a highly structured schedule of constructive activities. Visual aids are often helpful. Treatment is most successful when it is geared toward the childs particular needs. An experienced specialist or team should design the program for the individual child. A variety of therapies are available, including: applied behavior analysis (ABA), medications, occupational therapy, physical therapy, and speech-language therapy. Sensory integration and vision therapy are also common, but there is little research supporting their effectiveness. The best treatment plan may use a combination of techniques (Oasis). Autism remains a challenging condition for children and their families, but the outlook today is much better than it was a generation ago. At that time, most people with autism were placed in institutions. Today, with the right therapy, many of the symptoms of autism can be improved, though most people will have some symptoms throughout their lives. Most people with autism are able to live with their families or in the community. The outlook depends on the severity of the autism and the level of therapy the person receives.

World Trade Organizations or developed Countries Organization? :: Essays Papers

Organizations or Developed Countries Organization? World Trade Organizations or developed Countries Organization? In the 16th century, England had a lot of colonies, which were located in Africa. At that time, the primary function of colonies were to supply raw materials to England, and England can sold all the finished products to the colonies in order to make profits. This story is the beginning of international trade. In the 21st century, international trade is more busy than ever. According to comparative advantage theory, each country should specialize and produce those products if the country has a comparative advantage on those products, and use those products to trade with other countries in order to achieve specialization and exchange theory. However, during the trade process, it may have a lot of problems coming out, so a world organization were established in order to solve those problems and try to make trade into a smooth process. The World Trade Organization (WTO), was established in 1st, January, 1995, which was created during the Uruguay Round Negotiations. There are 146 countries as a member in the WTO. The main functions of a WTO are administering WTO trade agreements, as a forum for trade negotiations, handling trade disputes and monitoring national trade policies. One of the main function of the WTO are to enforce GATS (General Agreement on Trade in Services) and TRIPS (Trade-Related Intellectual Property). But what is TRIPS? It is agreements, which protect invent innovation and design around the world. In other words TRIPS is used to protect copyrights, trademarks which developed countries already, owns most of the shares on these. According to WTO, although TRIPS will bring a short term cost to developing countries, and only short term benefits for developed countries. In the long term it can encourage innovation, discovery in the developing countries. Once the developing country reaches certain levels on protection copyrights or trademark, this TRIPS agreement will benefits to every single person. To explain the short-term benefits to the developed countries, as I have mentioned before, most of the patents, copyrights owned by developed countries, if developing countries want to produce a product that have patents on it, the develop ing countries need to pay a royalty to developed countries or the corporation owns the right. In this TRIPS’ agreement, my argument is this agreement really helps the developing countries to become more innovation, discover, or is just the industrial countries want more money from the poor countries.

McMurphy as Christ in Ken Keseys One Flew Over the Cuckoos Nest :: One Flew Over Cuckoos Nest

McMurphy as Christ in One Flew Over The Cuckoo's Nest In "One Flew Over The Cuckoo's Nest," McMurphy is successfully perceived as a heroic Christ figure. Kesey uses foreshadowing and images, the fishing trip, actions and feelings of other characters to develop this character. Foreshadowing clues and images are used to contribute to McMurphy as a figure of Christ. In the beginning of the novel McMurphy is baptized with a shower before entering the ward. The reader is also introduced to Ellis, a character who spends the entire novel in a cross position "nailed against the wall, arms out," (page 20). Another clue to McMurphy's developing character is presented during the electroshock therapy. McMurphy willingly lies down on a cross shaped table, ending up in the same position Ellis foreshadowed. McMurphy also asks for his crown of thorns. Before the therapy a schizophrenic patient approaches him and says "I wash my hands of the whole deal", as Pontius Pilate said to Jesus before sentencing him to death. Jesus was also friends with a prostitute named Mary, just like McMurphy was friends with prostitutes. The development of McMurphy as a Christ figure deepends, when he leads the patients on a fishing trip. McMurphy takes the "twelve of us [patients] towards the ocean," (page 203) just like Jesus' 12 disciples, to test and strengthen their faith in him and empower them. Fish have also been an important religious Christian symbol, as the fishing trip is an important symbol of the novel. When the trip is over, the Chief describes the sense of change that most of the patients had and even claims that they "weren't the same bunch of weak-knees from a nuthouse anymore." (Page 215). This really shows the way McMurphy is starting to guide and lead the patients, just as Jesus lead his disciples. Finally the actions and feelings of the other characters successfully shows the development of McMurphy as a Christ figure and hero. Clearly smiliarities can be drawn between McMurphy and Jesus' healing. Jesus, made blind men see and mute men speak. McMurphy is the one who prompted the Chief to speak for the first time in years, when he says "Thank-you." (Page 184) and eventually, McMurphy "heals" Chief of his `deafness' and `dumbness'.

Act II Analysis & Character Development :: English Literature

Act II Analysis & Character Development At the start of Act II, John Proctor returns from the fields and sits down to dinner with his wife, Elizabeth. She has cooked up a rabbit, which apparently walked into the house and sat itself in the corner. Proctor seems out to please Elizabeth throughout this scene, kissing her and complimenting her on her cooking. Their small talk continues for a page or so, until the atmosphere abruptly changes, as Proctor enquires, â€Å"I think you’re sad again aren’t you?† Elizabeth responds by saying that he had returned so late that she thought he had gone to Salem. When Elizabeth mentions that Mary Warren is currently in Salem, Proctor becomes angered, demanding why Elizabeth did not stop her. Elizabeth suggests that he himself, go to Salem to testify that the accusations of witchcraft are false. Proctor says that he cannot prove his allegation because Abigail told him this information while they were alone at Parris’ house. Elizabeth is greatly dismayed upon learning that he and Abigail were alone together. Proctor demands that she stop judging him. He says that he feels as though his home is a courtroom, but Elizabeth responds that the real court is in his own heart. This is implied by the line: â€Å"I do not judge you. The magistrate sits in your heart that judges you.† This also suggests that regardless of whether Elizabeth forgives Proctor, he still cannot forgive himself. When Mary Warren returns home, the mood of the scene changes dramatically. As soon as Mary enters the room, Proctor goes directly to her and grabs her by the cloak, furious. â€Å"How do you go to Salem when I forbid it? Do you mock me? [shaking her.] I’ll whip you if you dare leave this house again! Mary responds by saying she is sick and gives Elizabeth a doll that she sewed in court, saying that it is a gift. She reports that thirty-nine people now stand accused. John and Mary argue over whether Mary can continue attending the trials. Elizabeth’s name was apparently mentioned in the accusations (Mary will not name the accuser), but Mary spoke out in Elizabeth’s defense. Proctor instructs Mary to go to bed, but she demands that he stop ordering her around. Elizabeth, meanwhile, is convinced that it was Abigail who accused her of witchcraft, in order to take her place in the Proctor household. Overall, this is a very important Act in terms of the relationship between Proctor and Elizabeth. It brings to light a number of crucial issues such as deceit, dishonesty, unfaithfulness and a growing sense of mistrust. Throughout the scene, Proctor seems motivated by feelings

Patterns Within Systems of Linear Equations

Jasmine Chai Grade 10 196298501 Patterns within systems of linear equations Systems of linear equations are a collection of linear equations that are related by having one solution, no solution or many solutions. A solution is the point of intersection between the two or more lines that are described by the linear equation. Consider the following equations: x + 2y = 3 and 2x – y = -4. These equations are an example of a 2Ãâ€"2 system due to the two unknown variables (x and y) it has. In one of the patterns, by multiplying the coefficient of the y variable by 2 then subtract the coefficient of x from it you will be given the constant.As a word equation it can be written like so with the coefficient of x as A and coefficient of y as B and the constant as C, 2B – Ax = C. This can be applied to the first equation (x + 2y = 3) as 2(2) – 1 = 3. To the second equation (2x – y = -4), it is -1(2) – 2 = -4. By using matrices or graphs, we can solve this syst em. Regarding other systems that also has such as pattern, it should also have the same solution as the two examples displayed. For instance, 3x + 4y = 5 and x -2y = -5, another system, also displays the same pattern as the first set and has a solution of (-1, 2).Essentially, this pattern is indicating an arithmetic progression sequence. Arithmetic progression is described as common difference between sequences of numbers. In a specific sequence, each number accordingly is labelled as an. the subscript n is referring to the term number, for instance the 3rd term is known as a3. The formula, an = a1 + (n – 1) d, can be used to find an, the unknown number in the sequence. The variable d represents the common difference between the numbers in the sequence. In the first equation (x + 2y = 3) given, the common differences between the constants c – B and B – A is 1.Variable A is the coefficient of x and variable b represents the coefficient of y, lastly, c represents the constant. The common difference of the second equation (2x – y = -4) is -3 because each number is decreasing by 3. In order to solve for the values x and y, you could isolate a certain variable in one of the equations and substitute it into the other equation. x + 2y = 3 2x – y = -4 x + 2y = 3 * x = 3 – 2y * 2(3 – 2y) – y = -4 * 6 – 4y – y = -4 * 6 – 5y = -4 * -5y = -10 * y = 2 Now that the value of y is found, you can substitute 2 in as y in any of the equations to solve for x. x + 2y = 3 x + 2(2) = 3 * x + 4 = 3 * x = 3 – 4 * x = -1 Solution: (-1, 2) Even though the solution has already been found, there are many different ways to solve it, such as graphically solving it. By graphing the two linear lines, you can interpolate or extrapolate if necessary to find the point where the two lines intersect. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Graph 1 Graph 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Just from the equations given, it is not in a format where it can be easily graphed. By changing it into y=mx + b form, the first equation will result as y = – (1/2) x + 3/2 or y = -0. 5x + 1. 5 and the second equation will result as y = 2x + 4. The significance of the solution is that it is equal to the point of intersection as shown on Graph 1. This can then allow the conclusion that the solution of the two linear equations is also the point of intersection when graphed. According to this arithmetic progression sequence, it could be applied to other similar systems.For instance, the examples below demonstrates how alike 2Ãâ€"2 systems to the previous one will display a similarity. Example 1: In the first equation the common difference between (3, 4 and 5) is 1. In the second equation, the common differen ce is -3. The common differences in these equations are exact to the previous example. 3x + 4y = 5 x – 2y = -5 x – 2y = -5 * x = 2y – 5 (Substitution) 3x + 4y = 5 * 3(2y – 5) + 4y = 5 * 6y – 15 + 4y = 5 * 10y – 15 = 5 * 10y = 20 * y = 2 (Substituting y) x – 2y = -5 * x – 2(2) = -5 * x – 4 = -5 * x = -5 +4 * x = -1 Solution: (-1, 2)Example 2: In the first equation below, it has a common difference of 18 for (2, 20 and 38). For the second equation, in (15, -5 and -25), it has a common difference of -20. In this example, the system is solved graphically. 2x + 20y = 38 15x – 5 y = -25 Solution: (-1, 2) | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Graph 2 Graph 2 | | |From the examples given above that are very similar to the first system, we can conclude that there is something common between them, that is the point of intersection or the values of x and y. That would imply that the x and y values and the point of intersection will always be (-1, 2) for all systems that follow arithmetic progression sequences. Due to that similarity, an equation that can be applied to these types of equations can be made. If the first coefficient of the first equation is identified as A and the common difference is c, an equation such as, Ax + (A + c) y = A + 2c, is made.This equation is so, because it is describes an arithmetic sequence, where the coefficients and constant are increasing by one in response to the coefficient before. In the second equation of the system, another equation can be made relatively the same to the first, with exceptions of different variables used. If B is used to represent the first coefficient of the second equation and d is used as the common difference, the equation, Bx + (B + d) y = B + 2d is created. With 2 equations, we have now created a system; to solve the system we can use the elimination method.This method is used to eliminate certain variables in order to find the value of another variable. After doing so, you could substitute in the value for the found variable and solve for the other(s). Ax + (A + c) y = A + 2c Bx + (B + d) y = B + 2d In order to use the elimination method, you must make the coefficient of x or y the same depending on which one you would like to eliminate. In this case, we will start by eliminating x. To proceed to do so, we must first multiply the first equation by B and the second equation by A: ABx + (AB + Bc) y = AB + 2Bc ABx + (AB + Bd) y = AB + 2BdAfter we have made the coefficient of x the same for both equations, we can now subtract the equations from one another: ABx + ABy + Bcy = AB + 2Bc ABx + ABy + Bdy = AB + 2Bd * Bcy – Bdy = 2Bc – 2Bd To find the val ue of y, we must isolate the variable y. Bcy – Bdy = 2Bc – 2Bd * y(Bc – Bd) = 2(Bc – Bd) * y = 2 Now that the value of y is found, to find the value of x is to substitute the value of y, which is 2, into any equation that includes that variable x and y. Bx + (B + d) y = B + 2d * Bx + (B + d) 2 = B + 2d * Bx + 2B + 2d = B + 2d * Bx + 2B – B = 2d – 2d * Bx + B = 0 * Bx = -B * x = -1To conclude the results of the equations above, it is making thee statement that all 2Ãâ€"2 systems that display an arithmetic progression sequence, which has a common difference between the coefficients and constant, it will have a result, point of intersection, of (-1, 2). To confirm that this is correct, the example systems below will demonstrate this property: Equation 1 (common difference of 8): 2x + 10y = 18 Equation 2 (common difference of 3): x + 4y = 7 Substitution Method x + 4y = 7 * x = 7 – 4y Substitute 2x + 10y = 18 * 2 (7 – 4y) + 10y = 1 8 * 14 – 8y +10y = 18 * 14 + 2y = 18 2y = 18 – 14 * 2y = 4 * y = 2 Substitute x + 4y = 7 * x + 4(2) = 7 * x + 8 = 7 * x = 7 – 8 * x = -1 Solution: (-1, 2) Once again from the example above, it displays that the solution or the point of intersection is identified as (-1, 2). From previous examples, all have a common difference that is different from the other equation involved in that system. In the following example, it will experiment whether having the same common difference will make a difference in the result. Equation 1 (common difference of 3): 2x + 5y = 8 Equation 2 (common difference of 3): x + 3y = 6 Graph 3 Graph 3As you can see on the graph, it shows that the two lines do not intersect at (-1, 2) even though it is a 2Ãâ€"2 system that has a common difference in both equations, meaning that the intersection at (-1, 2) can only be applied to systems that has 2 different common differences. To conclude, all 2Ãâ€"2 systems that follow arithmetic progres sion sequence with different common difference have a solution of (-1, 2). Furthermore, now that it is known that there is a certain pattern for a specific type of system, if this property is applied to a 3Ãâ€"3 system, with 3 different variables can it still work?Consider the following 3Ãâ€"3 system, (x + 2y + 3z = 4), (5x + 7y + 9z = 11) and (2x + 5y + 8z = 11). In this system, it has similar patterns to the 2Ãâ€"2 systems above due to its arithmetic progression. In the first equation, it has a common difference of 1 and the second equation has a common difference of 2 and lastly, the third equation has a common difference of 3. To solve this system, we can solve it using the method of elimination or matrices. Equation 1 (common difference: 1): x + 2y + 3z = 4 Equation 2 (common difference: 2): 5x + 7y + 9z = 11Equation 3 (common difference: 3): 2x + 5y + 8z = 11 Elimination Method To eliminate the variable x, we must first start by making the coefficients of x in two equations the same. We can do so by finding the lowest common multiple of the two coefficients and multiplying the whole equation by it. Equation 1: x + 2y + 3z = 4 * 2(x + 2y + 3z = 4) * 2x + 4y + 6z = 8 We can eliminate the variable x now that the coefficients of x in both equations are the same. To eliminate x, we can subtract equation 3 from equation 1. Equation 1 and 3: 2x + 4y + 6z = 8 2x + 5y + 8z = 11 -y -2z = -3 After eliminating x from two equations to form another equation that does not involve x (-y -2z = -3), another equation that does not involve x must be made to further eliminate another variable such as y or z. Equation 1: x + 2y + 3z = 4 * 5(x + 2y + 3z = 4) * 5x + 10y + 15z = 20 We can eliminate the variable x now that the coefficients of x in both equations are the same. To eliminate x, we can subtract equation 2 from equation 1. Equation 1 and 2: 5x + 10y + 15z = 20 – 5x + 7y + 9z = 11 3y + 6z = 9Now that two different equations that do not involve x ((-y -2z = -3 ) and (3y + 6z = 9)) are created, we can find the common coefficient of y and eliminate it to find the value of the variable z. Let (-y -2z = -3) to be known as equation A and (3y + 6z = 9) will be known as equation B. Equation A: -y -2z = -3 * 3(-y -2z = -3) * -3y -6z = -9 Equation A and B: -3y -6z = -9 + 3y + 6z = 9 0 = 0 As you can see from the result, 0 = 0, this is indicating that the system either has many solutions, meaning a collinear line or no solution, where all the lines do not intersect together at a specific point.Even if you attempt to isolate a different variable it will still have the same result. For instance, using the same equations above, you eliminate the variable y first as displayed below. Equation 1 (common difference: 1): x + 2y + 3z = 4 Equation 2 (common difference: 2): 5x + 7y + 9z = 11 Equation 3 (common difference: 3): 2x + 5y + 8z = 11 Elimination Method Equation 1: x + 2y + 3z = 4 * 7(x + 2y + 3z = 4) * 7x +14y + 21z = 28 Equation 2: 5x + 7y + 9z = 1 1 * 2(5x + 7y + 9z = 11) * 10x + 14y + 18z = 22 Equation 1 and 2: 7x +14y + 21z = 28 – 10x + 14y + 18z = 22 3x + 3z = 6 Equation 1: x + 2y + 3z = 4 * 5(x + 2y + 3z = 4) * 5x +10y + 15z = 20 Equation 3: 2x + 5y + 8z = 11 * 2(2x + 5y + 8z = 11) * 4x + 10y +16z = 22 Equation 1 and 3: 5x +10y + 15z = 20 – 4x + 10y +16z = 22 x – z = -2 Two equations have been made that has already eliminated the variable y. Let (-3x + 3z = 6) be equation A and let (x – z = -2) be equation B. Doing this, is in attempt to solve for variable x. Equation A: -3x + 3z = 6 Equation B: x – z = -2 * 3(x – z = -2) * 3x – 3z = -6 Equation A and B: -3x + 3z = 6 + 3x – 3z = -6 0 = 0As you can see the result, it is the same even if you try to solve another variable, from that we can confirm that this system has either no solution or infinite solutions, meaning that they are collinear lines. Furthermore, because this is a 3Ãâ€"3 system, meaning that it has three different variables, such as x, y and z, graphing it will also be very different from a graph of a 2Ãâ€"2 system. In a 3Ãâ€"3 system, the graph would be a surface chart, where the variable z allows the graph to become 3D. From this, we can conclude 3Ãâ€"3 systems that follow an arithmetic progression will always have either no solution or infinite solutions.This is saying that all linear equations do not intersect together in one point or they do not intersect. A way to prove this is through finding the determinant. The determinant is a single number that describes the solvability of the system. To find the determinant of all 3Ãâ€"3 systems that possesses arithmetic progression, we can start by creating a formula. Allow the first coefficient of the first equation be A and the second equation’s first coefficient be B and lastly, the first coefficient of the third equation be C.The common difference of equation one will be c, the common difference of equation two will be d, and the common difference of equation e will be e. This can be described through the following equations: 1. Ax + (A + c) y + (A + 2c) z = (A + 3c) 2. Bx + (B + d) y + (B + 2d) z = (B + 3d) 3. Cx + (C + e) y + (C + 2e) z = (C + 3e) When developing a matrix to find the determinant, you must have a square matrix. In this case, we do not have a square matrix. A square matrix is where the number of rows and columns are equal, for example, it could be a 2Ãâ€"2, 3Ãâ€"3, or 4Ãâ€"4. Looking at the equations, it is a 3Ãâ€"4 matrix; as a result it must be rearranged.Below is the rearranged matrix of the equations above. x A (A + c) (A + 2c) (A + 3c) y B (B + d) (B + 2d) = (B + 3d) z C (C + e) (C + 2e) (C + 3e) To find the determinant, you must find 4 values from the 3Ãâ€"3 matrix that helps find the determinant of A, B and C. In this case, if you were to find the values for A, you would cover the values that are in the same row and column as A, like so, A (A + c) (A + 2c) B (B + d) (B + 2d)C (C + e) (C + 2e) You would be left with four separate values that can be labelled as A, B, C and D. Respectively to the model below: a b c d In order to find the determinant you must find the four values for A, (A + c) and (A +2c). To find the determinant the equation ad – cb is used. The equation in this situation would be like the one below: A[(B + d)(C + 2e) – (C + e)(B + 2d)] – (A + c)[B(C + 2e) – C(B + 2d)] + (A +2c)[B(C + 2e) – C(B + 2d)] Expand * = A(BC – BC + Cd – 2Cd + 2Be – Be + 2de – 2de) – (A + c)(BC – BC + 2Be – 2Cd) + (A + 2c)(BC – BC + 2Be – 2Cd) Simplify 2ABe – 2ABe + 2ACd – 2ACd + 2Ccd – 2Ccd + 2Bce – 2Bce * = 2ABe – 2ABe + 2ACd – 2ACd + 2Ccd – 2Ccd + 2Bce – 2Bce * = 0 As it is visible, above it shows that the determinant found in this type of matrix is zero. If it is zero, it means that there are infinite an swers or no answer at all. Using technology, a graphing calculator, once entering a 3Ãâ€"3 matrix that exhibits arithmetic progression, it states that it is an error and states that it is a singular matrix. This may mean that there is no solution. To conclude, there is no solution or infinite solution to 3Ãâ€"3 systems that exhibit the pattern of arithmetic sequencing.This can be proved when the sample 3Ãâ€"3 system is graphed and results as a 3D collinear segment. As well as the results from above when a determinant is found to be zero proves that 3Ãâ€"3 systems that pertains an arithmetic sequence. Arithmetic sequences within systems of linear equations are one pattern of systems. Regarding other patterns, it is questionable if geometric sequences can be applied to systems of linear equations. Consider the following equations, x + 2y = 4 and 5x – y = 1/5. It is clear that the coefficients and constants have a certain relation through multiplication.In the first equation (x + 2y = 4), it has the relation where it has a common ratio of 2 between numbers 1, 2 and 4. For the second equation (5x – y = 1/5), it has a common ratio of -1/5 between 5, -1 and 1/5. The common ratio is determined through the multiplicative succession from the previous number in the order of the numbers. When the equations are rearranged into the form y=mx+b, as y = – ? x + 2 and y = 5x – 1/5, there is a visible pattern. Between the two equations they both possess the pattern of the constant, where constant a is the negative inverse of constant b and vice versa.This would infer that if they are multiplied together, as follows (-1/2 x 2 = -1 and 5 x -1/5 = -1), it will result as -1. With equations that are also similar to these, such as the following, y = 2x – 1/2, y = -2x + 1/2, y = 1/5x – 5 or y = -1/5x +5. Displayed below, is a linear graph that shows linear equations that are very similar to the ones above. Graph 4 Graph 4 From the graph a bove, you can see that the equations that are the same with exceptions of negatives and positives, they reflect over the axis and displays the same slope.For instance, the linear equations y = 2x -1/2 and y=-2x +1/2 are essentially the same but reflected as it shows in the graph below. Also, all equations have geometric sequencing, which means that they are multiplied by a common ratio. Secondly, the points of intersection between similar lines are always on the x-axis. Graph 5 Graph 5 Point of intersection: (0. 25, 0) Point of intersection: (0. 25, 0) To solve a general 2Ãâ€"2 system that incorporates this pattern, a formula must be developed. In order to do so, something that should be kept in mind is that it must contain geometric sequencing in regards to the coefficients and constants.An equation such as, Ax + (Ar) y = Ar2 with A representing the coefficients and r representing the common ratio. The second equation of the system could be as follows, Bx + (Bs) y = Bs2 with B as the coefficient and s as the common ratio. As a general formula of these systems, they can be simplified through the method of elimination to find the values of x and y. Ax + (Ar) y = Ar2 Bx + (Bs) y = Bs2 Elimination Method B (Ax + (Ar) y = Ar2) * BAx + BAry = BAr2 A (Bx + (Bs) y = Bs2) * ABx + ABsy = ABs2 Eliminate BAx + BAry = BAr2 – ABx + ABsy = ABs2 BAry – ABsy = BAr2 – ABs2 ABy (r – s) = AB (r2 – s2) * y = (r + s) Finding value of x by inputting y into an equation ABx + ABsy = ABs2 * ABx + ABs(r + s) = ABs2 * ABx = ABs2 – ABs(r +s) * x = s2 – s(r +s) * x = s2 – s2 – rs * x = rs To confirm that the formula is correct, we can apply the equation into the formula and solve for x and y and compare it to the results of graph 4. The equations that we will be comparing will be y = 5x – 1/5 and y = -1/5x + 5. The point of intersection, (1, 4. 8) of these equations is shown graphically on graph 4 and 6. The common rat io (r) of the first equation is -0. and the common ratio, also known as s in the equation of the second equation is 5. X = – (-0. 2 x 5) = 1 Y = (-0. 2 + 5) = 4. 8 As you can see, above, the equations are correctly matching the point of intersection as shown on the graphs. Due to such as result, it is known that it can now be applied to any equations that display geometric sequencing. Graph 6 Graph 6 Resources: 1. Wolfram MathWorld. Singular Matrix. Retrieved N/A, from http://mathworld. wolfram. com/SingularMatrix. html 2. Math Words. Noninvertible Matrix. Retrieved March 24, 2011 from, http://www. mathwords. com/s/singular_matrix. htm