7.7: Exchange, Value, and Consumption
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Learning Objectives
By the end of this section, you will be able to:
- Outline four types of exchange.
- Define the concept of reciprocity.
- Define the concepts of money and market exchange.
- Describe how money expresses conflicted notions of morality.
Before moving ahead to discuss the last of the four major subsistence methods, it’s worth reviewing the ways in which goods circulate in societies in accordance with each mode of subsistence. The four subsistence strategies are defined primarily by their techniques of production—that is, the way people use materials from their environments to make the things they need, such as food, clothing, shelter, and medicines. Previous sections have described how each production strategy entails its own distinctive methods of allocating those needful things to individuals and groups within the community. This section details the various methods of circulating things through social groups.
Most societies rely on one primary strategy for making a living, though they very often combine it with one or more others in flexible ways over time. If key foods become impossible to find, gatherer-hunters may take up farming for a few seasons. Many herding groups regularly hunt and sometimes plant crops along their nomadic routes, returning the next season to harvest the crops. Many farmers also keep domesticated animals. So it is with modes of exchange. Most societies practice not just one strategy but a combination of many, dominated by the form of exchange that dovetails with the main subsistence strategy.
Forms of Exchange
Recall the importance of egalitarian sharing in gatherer-hunter societies. When hunters return to camp bearing large game, they divide it equally among members of the band.  Everyone is expected and required to share with everyone else. Generalized reciprocity is the anthropological term to describe how people share things with no regard for their value or interest in compensation. This form of exchange doesn’t look like exchange at all; it looks much more like altruism. But when rigorously practiced by a group, with social sanctions used to punish laziness and stinginess, the result of generalized reciprocity over time is more or less the equal exchange of goods among all members of the group.
Outside of gathering-hunting societies, generalized reciprocity is also common in many close relationships, such as family relationships and friendships. When you’re staying in your parents’ house, does it occur to you to pay them when you grab a soft drink from the fridge? If a friend wants to borrow a pair of boots, do you charge them a rental fee? Probably not. However, the logic of exchange changes as the intensity of the relationship decreases and value of the object increases. Your parents might give you a car if you needed one, but you would not expect a friend to do so without some sort of compensation.
In gathering-hunting and horticultural societies, another form of reciprocal exchange is common among individuals. Among the Dobe Ju/’hoansi and other San groups in southern Africa, people develop relationships with one another based on a gift-giving practice called hxaro (Barnard 2018). The relationship begins when one person asks another person, often someone in another band, to give them a particular item, such as a digging stick or a cooking pot. This request may be rejected or accepted. If accepted, the two enter into an ongoing relationship of exchange, which may last forever or be broken off at some future point. After an unspecified period of time, the receiver makes a return gift, often of somewhat of equal or slightly greater value. The value of the items is never discussed; nor is the time between episodes of gift giving. All is made to seem natural and spontaneous. This form of exchange is known as balanced reciprocity . These relationships come with many advantages—for instance, the right to hunt and gather in the band of your hxaro partner. For this reason, many people maintain as many as 10 to 12 ongoing hxaro relationships. The main point of balanced reciprocity is not to gain resources and opportunities. Rather, the whole point of these serial exchanges of things is to establish and affirm relationships among people. Some degree of unspoken calculation is involved in choosing gifts that affirm and intensify the relationship over time, with givers slowly raising the value of gifts to deepen the relationship. These special relationships based on reciprocal gift giving are found in many other horticultural and agricultural societies as well.
While such gift-giving relationships seem to be governed by a sense of mutual goodwill, a more fierce and competitive form of balanced reciprocity developed among the Indigenous peoples of the Pacific Northwest coast of Canada and the United States (High 2018). Among groups such as the Haida, Kwakiutl, and Tlingit, chiefs sponsored great feasts called potlatches to commemorate births, weddings, deaths, and other important events. At these potlatch feasts, the chief of the host community would present an abundance of gifts to the chief of an invited community. Such gifts included blankets, animal skins, copper plaques, and preserved food. Sometimes, these items were deliberately burned in spectacles of extravagant waste. By foisting this abundance of gifts upon a guest chief, the host chief demonstrated their wealth and power and levied a challenge to the guest chief to counter with an even more lavish feast and greater trove of gifts. Power among neighboring communities was established and reinforced through this competitive feasting, not by acquiring wealth but by giving it away. More recent interpretations of potlatch suggest that such ceremonies not only operated as forms of reciprocity but also helped distribute specific goods found in one community to surrounding areas where those goods might be impossible to find.
The role of extended-family leaders in the practice of potlatch is an example of the tendency for leaders to gain control of community wealth and use it for distribution as well as prestige. This practice is particularly pronounced in agricultural societies that have chiefs, such as the peoples of the Hawaiian Islands in the precolonial era. Before contact with Europeans, the Hawaiian Islands were ruled by a multilevel system of chiefs who controlled land, natural resources, and trade. Commoners were required to pay tribute to their chiefs in the form of labor, food, and other products. For farmers, this meant that a portion of their agricultural surplus was relayed to local chiefs. These local chiefs then relayed a portion of the tribute they received to regional chiefs, and so on up the pyramid to the great chief. This tribute supported government at each level, including royal courts, political advisers, priests, military strategists, guards, and entertainers. In this way, political leaders became centers of the concentration of wealth, which was then used to provide communities with the benefits of government, such as social order, conflict resolution, military protection, trade coordination, and the construction of public works such as fishponds, water channels, and temples. In this hierarchical system, tribute flowed up to elites, while government goods and services flowed down to commoners. This two-way flow is called redistribution , and it’s a very common feature of chiefdoms, as will be discussed in the next chapter. Tribute was used by leaders to finance monument building, warfare, trade, and ceremonial feasting as well as the chief’s own lavish regalia and large retinue of assistants, bureaucrats, and servants.
Redistribution is practiced in all state societies. Consider the roads in your neighborhood, the postal service, the public schools, the libraries, government-funded scientific research, the courts, the prisons, the police—all are paid for by taxation, the form of redistribution conducted by states. While some see taxation as a predatory fee extracted by unproductive elites, taxation makes possible the social order, the economy, and the well-being of state citizens. It’s important to recognize that redistribution is not a way for individuals to purchase goods and services from the state but rather a system of allocating resources for the well-being of society as a whole.
Read this first-hand experience of the author of this chapter, Jennifer Hasty.
First-hand experience of Jennifer Hasty
Imagine that you go to take a shower and discover that you’re out of soap. How can you solve this problem? Gift exchange? A government program? Surely not. In contemporary Western society, forms of reciprocity and redistribution have become increasingly sidelined by the other main form of economic exchange: markets . A market is an institution that makes it possible for buyers and sellers of goods to meet for the purposes of exchange. In the most concrete sense, a market is an actual place. If you need a bar of soap (or shampoo, or a towel, or a bathtub), you go the market and buy one. In fact, I never pack soap when I travel to Ghana, as one of the first thing I do when I get there is head for the nearest market.
 At big markets such as Makola and Kaneshie, you can find almost anything you might want, from large appliances to clothes, school supplies, fresh spices, and produce. The air is infused with the shifting aromas of fried plantain, “stinky” fish, and freshly baked bofrot, a kind of Ghanaian doughnut. Music blasts from radios posted at kiosks here and there. Mobile vendors ply the crowded paths, their goods carefully draped on their bodies or stacked on their heads. Customers from all walks of life browse the rows upon rows of seated vendors, everyone chatting and socializing, buyers and sellers haggling the price of goods. Early on, I learned that the value of a product is not fixed but contingent on many factors, such as time of day, amount of stock, and the perceived identity of the buyer. Just buying a few bars of soap can be a complex social interaction combined with a rich sensory experience.
More recently, Western-style grocery stores have opened up in Ghana. In contrast to the intense sensory experience of markets, these stores are quiet and, to me, a bit underwhelming. A small number of shoppers silently push their carts up and down the aisles, avoiding eye contact with one another. At checkout, a bored clerk rings up your items and informs you of the total. It does not matter who you are—rich or poor, the total is the same. This is a routinized, predictable experience. In the United States, automatic checkout is becoming increasingly common in stores, eliminating the off chance that you might have any sort of meaningful human contact in the course of your market transaction. With online purchasing, the market is no longer a place at all but a virtual site on a computer screen that absolutely precludes any possibility of direct human interaction. Consumers have responded to the desocialization of online market relations by embracing the highly expressive and interactive realm of consumer reviews. And even in the brick-and-mortar shops, people resist the boring antisocial regimen of modern shopping by talking on their cell phones, enjoying food samples, and looking for romantic partners.
All of these forms of exchange can be found in contemporary capitalist societies. Generalized reciprocity is practiced among family members and very, very close friends. Balanced reciprocity is the unspoken logic guiding most exchanges among friends and acquaintances. If you ask your neighbor to collect your mail while you’re out of town, you might expect your neighbor to ask you for a similar favor in the future. Recently, while I was out of town for a week, I asked the parents of my daughter’s school friend if they could take her to and from school. They kindly obliged. A month later, I had to leave town again for a week. In that intervening month, I had been unable to reciprocate for the favor of shuttling my child around, so I hesitated to ask those same parents to do it again. Instead, I hired someone, the friend of a friend but a stranger to me.
Among strangers, market exchange is the most common form of transaction. In capitalist societies, market exchange is the default setting; if all else fails, pay for it. Market transactions are quick and easy, and the participants walk away relatively unencumbered by future obligation. If this is the advantage of market exchange, it can also be a big disadvantage. Without the relations of mutuality and trust established by forms of reciprocity, the participants in market exchange are motivated by the desire to get more than they give. A society dominated by market exchange is therefore dominated by the logic of self-interest and greed rather than cooperation and social well-being.
Money
In the midst of the COVID-19 pandemic, many shops witnessed a scarcity of hard currency, prompting them to put up signs requesting people to use credit or debit cards or mobile payment apps to make purchases. This episode is part of a larger shift over the past several decades away from coins and bills and toward more abstract forms of payment such as chip cards and “fintech,” the mobile debit apps accessed through smartphones. And even more abstractly, now there are even virtual currencies such as bitcoin and other cryptocurrencies. Bitcoin is a currency generated by a computer dedicated to solving complex mathematical problems. How is that even money?
What is money ? In the formulation of classical philosophy, money is defined by three functions: it serves as a medium of exchange, a unit of account, and a store of value. Imagine that two friends from neighboring groups, one a pastoralist group and the other a horticultural group, meet in town. The pastoralist has a freshly slaughtered goat slung over their shoulder. The horticulturalist is carrying a small sack of vegetables. They decide they’d like to trade. The farmer wants all of the meat, but the herder wants only a small portion of the vegetables. Each person wants the trade to be equal; that is, they both want to give and receive the same value. How can they conduct this transaction? How do they know the value of the things they want to trade?
It seems natural to imagine these two trader friends attempting to negotiate some sort of barter. The swapping of goods on the spot, however, was never a dominant form of exchange in any culture in the past. Instead, many anthropologists argue that precapitalist peoples relied more on gift exchange, redistribution, and debt to circulate goods through society. So it’s more likely that the pastoralist would make a good-faith gift of the whole goat to their gardening friend, knowing that both would remember the gardener’s obligation to return the favor with more vegetables (or something else of fairly equal value) in the future. If this seems complicated, it probably was. Individuals would have been involved in many such relationships simultaneously – whole communities of people all mutually entwined in relations of credit and debt.
The other possible solution is money. If these two traders live in a society that uses some arbitrary other thing to enumerate value, they would know that all of the meat has the value of 50 units (or shekels, cowrie shells, tally sticks, bones, animal skins, brass rods, gold coins, bank notes, or any one of the myriad other objects used as money in the past). A small portion of vegetables might have the value of only 10 units. If these two have come with their wallets, they can use money to make two separate transactions for items of different value rather than trying to negotiate one swap. They can make the exchanges and walk away without entanglement.
There are two kinds of money, general purpose and special purpose. The transaction described above is an example of general-purpose money —that is, money that can be exchanged for a wide variety of goods and services. Dollars, euros, pesos, yen, and bitcoin are all forms of general-purpose money. General-purpose money is portable, divisible, and easily available. Special-purpose money is currency that is used to purchase one particular kind of thing. In some pastoral societies of West African, cattle have been used as forms of bridewealth, or the payment made by a groom to the family of his prospective bride. The Tinputz people of Papua New Guinea had two forms of special-purpose money, strings of flying fox teeth and strings of shell disks. These were used for marriages and other socially important occasions. Special-purpose money is generally more difficult to obtain, transport, and/or measure precisely. In American society, many grocery stores now offer special-use “points” for loyal shoppers that can be used to buy gas at particular gas stations. Credit card companies, airlines, and other businesses offer similar forms of special-purpose points. Such special-use currency illustrates the arbitrary nature of money.


LEARNING OBJECTIVES
By the end of this section, you will be able to:
- Describe different types of possible stressors
- Explain the importance of life changes as potential stressors
- Describe the Social Readjustment Rating Scale
- Understand the concepts of job strain and job burnout
For an individual to experience stress , he must first encounter a potential stressor . In general, stressors can be placed into one of two broad categories: chronic and acute. Chronic stressors include events that persist over an extended period of time, such as caring for a parent with dementia, long-term unemployment, or imprisonment. Acute stressors involve brief focal events that sometimes continue to be experienced as overwhelming well after the event has ended, such as falling on an icy sidewalk and breaking your leg (Cohen, Janicki-Deverts, & Miller, 2007). Whether chronic or acute, potential stressors come in many shapes and sizes. They can include major traumatic events, significant life changes, daily hassles, as well as other situations in which a person is regularly exposed to threat, challenge, or danger.
TRAUMATIC EVENTS
Some stressors involve traumatic events or situations in which a person is exposed to actual or threatened death or serious injury. Stressors in this category include exposure to military combat, threatened or actual physical assaults (e.g., physical attacks, sexual assault, robbery, childhood abuse), terrorist attacks, natural disasters (e.g., earthquakes, floods, hurricanes), and automobile accidents. Men, non-Whites, and individuals in lower socioeconomic status (SES) groups report experiencing a greater number of traumatic events than do women, Whites, and individuals in higher SES groups (Hatch & Dohrenwend, 2007). Some individuals who are exposed to stressors of extreme magnitude develop post-traumatic stress disorder (PTSD): a chronic stress reaction characterized by experiences and behaviors that may include intrusive and painful memories of the stressor event, jumpiness, persistent negative emotional states, detachment from others, angry outbursts, and avoidance of reminders of the event (American Psychiatric Association [APA], 2013).
LIFE CHANGES
Most stressors that we encounter are not nearly as intense as the ones described above. Many potential stressors we face involve events or situations that require us to make changes in our ongoing lives and require time as we adjust to those changes. Examples include death of a close family member, marriage, divorce, and moving ( [link] ).
In the 1960s, psychiatrists Thomas Holmes and Richard Rahe wanted to examine the link between life stressors and physical illness, based on the hypothesis that life events requiring significant changes in a person’s normal life routines are stressful, whether these events are desirable or undesirable. They developed the Social Readjustment Rating Scale (SRRS), consisting of 43 life events that require varying degrees of personal readjustment (Holmes & Rahe, 1967). Many life events that most people would consider pleasant (e.g., holidays, retirement, marriage) are among those listed on the SRRS; these are examples of eustress. Holmes and Rahe also proposed that life events can add up over time, and that experiencing a cluster of stressful events increases one’s risk of developing physical illnesses.
In developing their scale, Holmes and Rahe asked 394 participants to provide a numerical estimate for each of the 43 items; each estimate corresponded to how much readjustment participants felt each event would require. These estimates resulted in mean value scores for each event—often called life change units (LCUs) (Rahe, McKeen, & Arthur, 1967). The numerical scores ranged from 11 to 100, representing the perceived magnitude of life change each event entails. Death of a spouse ranked highest on the scale with 100 LCUs, and divorce ranked second highest with 73 LCUs. In addition, personal injury or illness, marriage, and job termination also ranked highly on the scale with 53, 50, and 47 LCUs, respectively. Conversely, change in residence (20 LCUs), change in eating habits (15 LCUs), and vacation (13 LCUs) ranked low on the scale ( [link] ). Minor violations of the law ranked the lowest with 11 LCUs. To complete the scale, participants checked yes for events experienced within the last 12 months. LCUs for each checked item are totaled for a score quantifying the amount of life change. Agreement on the amount of adjustment required by the various life events on the SRRS is highly consistent, even cross-culturally (Holmes & Masuda, 1974).
| Life event | Life change units |
|---|---|
| Death of a close family member | 63 |
| Personal injury or illness | 53 |
| Dismissal from work | 47 |
| Change in financial state | 38 |
| Change to different line of work | 36 |
| Outstanding personal achievement | 28 |
| Beginning or ending school | 26 |
| Change in living conditions | 25 |
| Change in working hours or conditions | 20 |
| Change in residence | 20 |
| Change in schools | 20 |
| Change in social activities | 18 |
| Change in sleeping habits | 16 |
| Change in eating habits | 15 |
| Minor violation of the law | 11 |
Extensive research has demonstrated that accumulating a high number of life change units within a brief period of time (one or two years) is related to a wide range of physical illnesses (even accidents and athletic injuries) and mental health problems (Monat & Lazarus, 1991; Scully, Tosi, & Banning, 2000). In an early demonstration, researchers obtained LCU scores for U.S. and Norwegian Navy personnel who were about to embark on a six-month voyage. A later examination of medical records revealed positive (but small) correlations between LCU scores prior to the voyage and subsequent illness symptoms during the ensuing six-month journey (Rahe, 1974). In addition, people tend to experience more physical symptoms, such as backache, upset stomach, diarrhea, and acne, on specific days in which self-reported LCU values are considerably higher than normal, such as the day of a family member’s wedding (Holmes & Holmes, 1970).
The Social Readjustment Rating Scale (SRRS) provides researchers a simple, easy-to-administer way of assessing the amount of stress in people’s lives, and it has been used in hundreds of studies (Thoits, 2010). Despite its widespread use, the scale has been subject to criticism. First, many of the items on the SRRS are vague; for example, death of a close friend could involve the death of a long-absent childhood friend that requires little social readjustment (Dohrenwend, 2006). In addition, some have challenged its assumption that undesirable life events are no more stressful than desirable ones (Derogatis & Coons, 1993). However, most of the available evidence suggests that, at least as far as mental health is concerned, undesirable or negative events are more strongly associated with poor outcomes (such as depression) than are desirable, positive events (Hatch & Dohrenwend, 2007). Perhaps the most serious criticism is that the scale does not take into consideration respondents’ appraisals of the life events it contains. As you recall, appraisal of a stressor is a key element in the conceptualization and overall experience of stress. Being fired from work may be devastating to some but a welcome opportunity to obtain a better job for others. The SRRS remains one of the most well-known instruments in the study of stress, and it is a useful tool for identifying potential stress-related health outcomes (Scully et al., 2000).
Link to Learning
Go to this site to complete the SRRS scale and determine the total number of LCUs you have experienced over the last year.
Connect the Concepts: Correlational Research
The Holmes and Rahe Social Readjustment Rating Scale (SRRS) uses the correlational research method to identify the connection between stress and health. That is, respondents’ LCU scores are correlated with the number or frequency of self-reported symptoms indicating health problems. These correlations are typically positive—as LCU scores increase, the number of symptoms increase. Consider all the thousands of studies that have used this scale to correlate stress and illness symptoms: If you were to assign an average correlation coefficient to this body of research, what would be your best guess? How strong do you think the correlation coefficient would be? Why can’t the SRRS show a causal relationship between stress and illness? If it were possible to show causation, do you think stress causes illness or illness causes stress?


10.1: Introduction
As we wrap up, let’s recall our definition of mindfulness—awareness of what is going on inside and around us in the present moment. Awareness of inside and around, has intertwined like a thread through Fingerspelling: A Mindful Approach.
As I write this last chapter, it feels metaphoric to reference the very historic time that surrounds this present moment—the COVID-19 pandemic. For the last three months, education and learning has been relegated to online. I’m sitting outside my home in Tahoe, California on my laptop, with a view of Mt. Tallac. Tufts of clouds float by while I contemplate the immediacy of our final topic—Engagement. Perhaps it’s being on stay-at-home orders, but…engagement seems the most vital of topics so far! The old adage, “Practice makes perfect,” is crucial to apply, and yet, in this world of pandemic, the ability to practice our skills has been challenged. I must say, even as a certified ASL interpreter, fluent in ASL, after over three months of social distancing, my skills have suffered. I’m aware, I feel it in the slight hesitancy to recall a sign or perhaps a bumble of a fingerspelled word. With mindfulness, I can notice the slipups and begin to formally set time aside to practice. Learning ASL takes discipline! If we think back to the introduction of this book we remember—fingerspelling is often the first skill you learn and the last skill you perfect. Another way to say this might be, fingerspelling is a skill in state of continuous flux. Studies show that to master any new skill, it needs to be practiced 10,000 hours. So, these 10 lessons are only the beginning.
Most students who find themselves in a fingerspelling class have found that they know just enough to be dangerous. By dangerous I mean that we have discovered a tool (fingerspelling) that enables us to express something we do not know how to sign, so we therefore spell it. However, the ideas and concepts tumbling around our heads, go far beyond a few manually spelled words. Fingerspelling is only a smidge of the visual ASL communication equation. So, what do we do? Firstly, take an American Sign Language class! Fingerspelling is in no way a substitute for signing. Then, having a few classes under your belt, you can be on the search for Deaf friends who will entertain your beginning skills. Engagement gives purpose to our learning and satisfies our innate desires to be effective communicators. This brings us to our last tip:
Mindful Tip 15: To improve our fingerspelling skills (especially our receptive skills) we must engage.
In the time of the pandemic, engagement means Facetime, ZOOM or maintaining social distancing by fingerspelling 6 feet away, no problem! There are also incredible fingerspelling websites out there, YouTube videos and even the ability to record yourself very effortlessly on your device and then come back a few days later and watch, seeing if you can catch your fingerspelled words. There’s an App called, Marco Polo, that uses visual recording texts as a kind of walkie-talkie to your friends. But none of these, replace signing with a native signer, a Deaf person. One of the best uses of time in your ASL journey is to engage with the Deaf community and invest in friends and acquaintances. Resourcing the Deaf community is the best insurance for growing and mastering your ASL skills, beyond any class, beyond any manual.
I sincerely hope, Fingerspelling: A Mindful Approach provided you with practices that built a strong foundational platform for fingerspelling skill development, and most importantly offered tools to be aware and present for the treasured ones in your life.


7.4: Homework- u-substitution
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As a review of the chain rule from derivatives, find \(\frac{d}{dx} e^{-x^2 + 1}\).
\(2x e^{x^2 + 1}\)ans
- Read through section 6A again and then read through section 6B.
- Watch the Khan Academy video u-substitution .
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Compute the following indefinite integral using the method of \(u\)-substitution.
\(\int (5x^4 + 2x) e^{x^5 + x^2}dx\)
\(e^{x^5 + x^2} + C\)ans - Watch another example of \(u\)-substitution: u-substitution 2.
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Compute the following indefinite integral using the method of \(u\)-substitution.
\(\int \frac{2x}{x^2 - 5}dx\)
\(\ln(x^2 - 5) + C\)ans -
Reread the part about the chain rule shortcut for \(u\)-substitution in chapter 6 of the online notes, and reread Example 6B.2. Then try the following problems.
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\(\int e^{-3x}dx\).
\(\frac{1}{-3} e^{-3x} + C\)ans
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\(\int \left( \frac{1}{2}x - 1 \right)^4dx\)
\(.4 \left( \frac{1}{2} x - 1 \right)^5 + C\)ans
.
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\(\int_0^2 (5x - 3)^3dx\).
\(116\)ans
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\(\int \frac{1}{7x - 2}dx\)
\(\frac{1}{7} \ln(7x - 2) + C\)ans
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\(\int \frac{1}{\sqrt{0.5x - 1}}dx\)
\(4 \sqrt{0.5x - 1} + C\)ans
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\(\int e^{-3x}dx\).
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Try some more \(u\)-substitution integrals.
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\(\int (8x^3) (x^4 + 1)^2dx\)
- We see that \(u = x^4 + 1\) in this case.
- We see \(\frac{du}{dx} = 4x^3\), so \(dx = \frac{1}{4x^3} du\).
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We have
\[\begin{align*} & = \int (8x^3) u^2 \frac{1}{4x^3} du \\ & = \int 8 u^2 \frac{1}{4}du \\ & = 2 \int u^2du \end{align*}\]
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Integrating we have
\(2 \int u^2du = 2 \left( \frac{1}{3} u^3 \right ) + C = \frac{2}{3} u^3 + C\)
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Substitution of \(u = x^4 + 1\) yields
\(\frac{2}{3} (x^4 + 1)^3 + C\)
ans -
\(\int (x^4 + 1)^2 (8x^3)dx\)
This is exactly the same as the previous problem, just written a different way. No need to redo work.ans
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\(\int (3x^2 - 1) e^{x^3 - x}dx\)
- We see that \(u = x^3 - x\) in this case.
- We see \(\frac{du}{dx} = 3x^2 - 1\), so \(dx = \frac{1}{3x^2 - 1} du\).
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We have
\[\begin{align*} & = \int (3x^2 - 1) e^u \frac{1}{3x^2 - 1} du \\ & = \int e^udu \end{align*}\]
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Integrating we have
\(\int e^udu = e^u + C\)
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Substitution of \(u = x^3 - x\) yields
\(e^{x^3 - x} + C\)
ans -
\(\int (x^2 - \frac{1}{3}) e^{x^3 - x}dx\)
\(\frac{1}{3} e^{x^3 - x} + C\)ans
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\(\int (e^x+ x)^5 (e^x+1)dx\)
\(\frac{1}{6} (e^x + x)^6 + C\)ans
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\(\int \frac{2x}{x^2 - 1}dx\)
\(\ln(x^2 -1) + C\)ans
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\(\int e^x \sqrt{e^x + 1}dx\)
ans
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\(\int \frac{\sin(x)}{\cos(x)}dx\)
\(-\ln(\cos(x)) + C\)ans
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\(\int \frac{\sqrt{ \ln(x) }}{x}dx\)
ans
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\(\int \frac{x}{(x^2 - 5)^3}dx\)
- We see that \(u = x^2 - 5\) in this case.
- We see \(\frac{du}{dx} = 2x\), so \(dx = \frac{1}{2x} du\).
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We have
\[\begin{align*} & = \int \frac{x}{u^3}\frac{1}{2x}du \\ & = \int \frac{1}{2u^3}du \\ & = \int \frac{1}{2}u^{-3}du \end{align*}\]
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Integrating we have
\(\int \frac{1}{2}u^{-3}du = - \frac{1}{4} u^{-2} + C\)
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Substitution of \(u = x^2 - 5\) yields
\(- \frac{1}{4} (x^2 - 5)^{-2} + C\)
ans -
\(\int \frac{\cos(x)}{\sqrt{\sin(x) + 1}}dx\)
\(2\sqrt{\sin(x) + 1} + C\)ans
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\(\int \frac{1}{x (2 \ln(x) + 1)^4}dx\).
\(-\frac{1}{6(2 \ln(x) + 1)^3} + C\)ans
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\(\int (e^{5x} + 1)^9 (e^{5x} )dx\).
\(\frac{1}{2} (e^{5x} + 1)^{10} + C\)ans
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\(\int (8x^3) (x^4 + 1)^2dx\)


4.7: Practice Exercises
4.7 Practice Exercises
1.
After listening to each example, specify the meter (e.g., “compound triple”).
- Lin-Manuel Miranda, “First Burn”
- Al Green, Willie Mitchell, Al Jackson Jr., “Let's Stay Together”
- Kenny Kirkland, “Dienda”
- Traditional Scottish Gaelic tune, text by Eleanor Farjeon, “Morning Has Broken”
- Guy Lawrence, Howard Lawrence, James Napier, and Sam Smith, “Disclosure”
Answer.
2.
For each example, specify the implied time signature and the meter (e.g., “simple duple”).
a. Time Signature: Meter:
b. Time Signature: Meter:
c. Time Signature: Meter:
d. Time Signature: Meter:
Answer.
3.
Use one note value (with one or two dots as necessary) to show the sum of all the rhythmic values given.
Answer.
4.
Specify the meter of each time signature.
- The meter of 42 is:
- The meter of 916 is:
- The meter of 34 is:
Answer.
5.
Correct the incorrect rhythmic notation in each example in order to show the beats. The subdivisions grouped into beats are shown in parentheses above the rhythms to be corrected.
Answer.
Click
here to download the homework.
PDF versions of the textbook, homework exercises, and practice exercises can be found at
musictheory.pugetsound.edu


12.1: Grammar Introduction
Learning Objectives
learning objectives
- Understand basic sentence structure and how to arrange modifying phrases and clauses around a basic sentence structure.
- Understand verb usage and subject-verb agreement.
- Understand nouns and many grammar constructions related to nouns.
- Understand the rules behind various types of punctuation, especially commas.
This chapter explain and provides practice in the rules of English grammar, from a basic to an advanced level. Almost every student will need to practice some of the grammar in this chapter, but what needs practice will vary by person. Some people will need to practice more of the grammar skills. Consult with your instructor about which grammar skills they think will benefit you most. The skills are generally introduced in the order you would learn them in school. Don't be embarrassed if you feel you should already know some of these rules and skills. Some people miss some of them for a number of reasons in school, or it just didn't "click" the first time around. Other people were never taught them in the first place for a variety of reasons. In terms of grammar, learn what you need, and then apply it to your own writing. We have tried to keep the exercise examples at a level of what you are more likely to encounter in your college reading and writing. This prevents them from being overly simplistic and, thus, inapplicable to your own, more sophisticated writing.
Contributors and Attributions
This page was developed by Athena Kashyap and Erika Dyquisto under a Creative Commons 4.0 CC BY-SA license and should be attributed as such.


11.1.1E: Racism
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Racism is the belief that different traits of racial groups are inherent and justify discrimination.
Learning Objectives
- Identify four scenarios which separately illustrate individual-level racism, structural racism, cultural racism, and historical racism
Key Points
- Racism may be expressed individually and consciously, through explicit thoughts, feelings, or acts, or socially and unconsciously, through institutions that promote inequality between races.
- Individual-level racism is seen in prejudice, bias, or discrimination displayed between two or more people.
- Structural racism refers to inequalities built into an organization or system.
- Historical economic or social disparity caused by racism can adversely affect present generations.
- Historical economic or social disparity caused by racism can adversely affect present generations.
- Racial profiling involves the singling out of individuals based upon their race for differential treatment, often in the form of increased surveillance or enforcement by the criminal justice system.
- Color-blind racism involves the avoidance of racial language as a means to suggest that racism is no longer an issue when it actually is.
- Genocide refers to the attempt to completely destroy a racial or ethnic group, usual perpetrated by a dominant racial group against a persecuted one.
Key Terms
- Structural Racism : Unequal practices built into organizations or institutional systems that disproportionately benefit or disadvantage particular racial groups.
- racism : prejudice or discrimination based upon race; the belief that one race is superior to all others
- Cultural Racism : A variation of structural racism that occurs when the assumption of the inferiority of one or more races is embraced by the culture of a given society.
Racism is the belief that different inherent traits in racial groups justify discrimination. In the modern English language, the term “racism” is used predominantly as a pejorative epithet. It is applied especially to the practice or advocacy of racial discrimination of a pernicious nature, which is often justified by recourse to racial stereotyping or pseudo-science.
Racism can refer to any or all of the following beliefs and behaviors:
- Race is the primary determinant of human capacities (prejudice or bias).
- A certain race is inherently superior or inferior to others (prejudice or bias).
- Individuals should be treated differently according to their racial classification (prejudice or bias).
Racism is broadly recognized as an affront to basic human dignity and a violation of human rights. Racism is opposed by almost all mainstream voices in the United States. A number of international treaties have sought to end racism. The United Nations uses a definition of racist discrimination laid out in the International Convention on the Elimination of All Forms of Racial Discrimination adopted in 1965:
“…any distinction, exclusion, restriction or preference based on race, colour, descent, or national or ethnic origin which has the purpose or effect of nullifying or impairing the recognition, enjoyment or exercise, on an equal footing, of human rights and fundamental freedoms in the political, economic, social, cultural or any other field of public life. ”
Expressions of Racism
Racism may be expressed individually and consciously, through explicit thoughts, feelings, or acts, or socially and unconsciously, through institutions that promote inequalities among races.
Individual-Level Racism
Individual-level racism is prejudice, bias, or discrimination displayed in an interaction between two or more people. An example of individual-level racism would be an employer firing someone because of his/her race, without regard to their skills or qualifications as an employee.
Children develop an awareness of race and racial stereotypes quite young (between the ages of 5 and 11), and these racial stereotypes affect behavior. For example, children who identify with a racial minority that is stereotyped as not doing well in school tend to perform worse academically when they learn about the stereotype associated with their race.
Structural Racism
Structural racism refers to inequalities built into an organization or system. An example of structural racism can be seen in recent research on workplace discrimination. There is widespread discrimination against job applicants whose names were merely perceived as “sounding black. ” These applicants were 50 percent less likely than candidates perceived as having “white-sounding names” to receive callbacks for interviews, no matter what their level of previous experience was. The researchers view these results as strong evidence of unconscious biases rooted in the country’s long history of discrimination. This is an example of structural racism as it shows a widespread established belief system that treats people differently based upon their race.
Additional examples of structural racism include apartheid in South Africa, the system of Jim Crow laws in the U.S., and the inequitable lending practices of banks (i.e., redlining).
Cultural Racism
Cultural racial discrimination, a variation of structural racism, occurs when the assumption of inferiority of one or more races is built into the a society ‘s cultural norms. Racism is an expression of culture, passed on through enculturation and socialization.
Historical Racism
Historical economic or social disparity caused by past racism affects the present generation of the victimized group. For example, racial deficits still exist in formal education, and unconscious racist attitudes still exist against certain members of the general population. This perspective seeks to explain how, for example, African Americans in the U.S. have had their opportunities in life adversely affected due to the enslavement of their ancestors. Disparities in wealth, net worth, and education lend credence to this idea.
One response to racial disparity in the U.S. has been affirmative action. Affirmative action is the practice of favoring or benefiting members of a particular race in areas such as college admissions and workplace advancement, in an attempt to create atmospheres of racial diversity and racial equality. Though lauded by many as a boon to society, giving the less privileged a chance at success and working to overcome historical social disparity, the practice is condemned as racially discriminatory by others.
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6.9: The Origin of the Moon
Learning Objectives
By the end of this section, you will be able to:
- Describe the top three early hypotheses of the formation of the Moon
- Summarize the current “giant impact” concept of how the Moon formed
It is characteristic of modern science to ask how things originated. Understanding the origin of the Moon has proven to be challenging for planetary scientists, however. Part of the difficulty is simply that we know so much about the Moon (quite the opposite of our usual problem in astronomy). As we will see, one key problem is that the Moon is both tantalizingly similar to Earth and frustratingly different.
Ideas for the Origin of the Moon
Most of the earlier hypotheses for the Moon’s origin followed one of three general ideas:
- The fission theory—the Moon was once part of Earth, but somehow separated from it early in their history.
- The sister theory—the Moon formed together with (but independent of) Earth, as we believe many moons of the outer planets formed.
- The capture theory—the Moon formed elsewhere in the solar system and was captured by Earth.
Unfortunately, there seem to be fundamental problems with each of these ideas. Perhaps the easiest hypothesis to reject is the capture theory. Its primary drawback is that no one knows of any way that early Earth could have captured such a large moon from elsewhere. One body approaching another cannot go into orbit around it without a substantial loss of energy; this is the reason that spacecraft destined to orbit other planets are equipped with retro-rockets. Furthermore, if such a capture did take place, the captured object would go into a very eccentric orbit rather than the nearly circular orbit our Moon occupies today. Finally, there are too many compositional similarities between Earth and the Moon, particularly an identical fraction of the major isotopes 1 of oxygen, to justify seeking a completely independent origin.
The fission hypothesis, which states that the Moon separated from Earth, was suggested in the late nineteenth century. Modern calculations have shown that this sort of spontaneous fission or splitting is impossible. Furthermore, it is difficult to understand how a Moon made out of terrestrial material in this way could have developed the many distinctive chemical differences now known to characterize our neighbor.
Scientists were therefore left with the sister hypothesis—that the Moon formed alongside Earth—or with some modification of the fission hypothesis that can find a more acceptable way for the lunar material to have separated from Earth. But the more we learned about our Moon, the less these old ideas seem to fit the bill.
The Giant Impact Hypothesis
In an effort to resolve these apparent contradictions, scientists developed a fourth hypothesis for the origin of the Moon, one that involves a giant impact early in Earth’s history. There is increasing evidence that large chunks of material—objects of essentially planetary mass—were orbiting in the inner solar system at the time that the terrestrial planets formed. The giant impact hypothesis envisions Earth being struck obliquely by an object approximately one-tenth Earth’s mass—a “bullet” about the size of Mars. This is very nearly the largest impact Earth could experience without being shattered.
Such an impact would disrupt much of Earth and eject a vast amount of material into space, releasing almost enough energy to break the planet apart. Computer simulations indicate that material totaling several percent of Earth’s mass could be ejected in such an impact. Most of this material would be from the stony mantles of Earth and the impacting body, not from their metal cores. This ejected rock vapor then cooled and formed a ring of material orbiting Earth. It was this ring that ultimately condensed into the Moon.
While we do not have any current way of showing that the giant impact hypothesis is the correct model of the Moon’s origin, it does offer potential solutions to most of the major problems raised by the chemistry of the Moon. First, since the Moon’s raw material is derived from the mantles of Earth and the projectile, the absence of metals is easily understood. Second, most of the volatile elements would have been lost during the high-temperature phase following the impact, explaining the lack of these materials on the Moon. Yet, by making the Moon primarily of terrestrial mantle material, it is also possible to understand similarities such as identical abundances of various oxygen isotopes.
Summary
The three standard hypotheses for the origin of the Moon were the fission hypothesis, the sister hypothesis, and the capture hypothesis. All have problems, and they have been supplanted by the giant impact hypothesis, which ascribes the origin of the Moon to the impact of a Mars-sized projectile with Earth 4.5 billion years ago. The debris from the impact made a ring around Earth which condensed and formed the Moon.
Footnotes
1 Remember from the Radiation and Spectra chapter that the term isotope means a different “version” of an element. Specifically, different isotopes of the same element have equal numbers of protons but different numbers of neutrons (as in carbon-12 versus carbon-14.)
Contributors and Attributions
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Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).


Newton's Third Law states
"For any action, there is an equal and opposite reaction."
By "action" Newton meant a force, so for every force one body exerts on another body, that second body exerts a force of equal magnitude but opposite direction back on the first body. Since all forces are exerted by bodies (either directly or indirectly), all forces come in pairs, one acting on each of the bodies interacting.
Though there may be two equal and opposite forces acting on a single body, it is important to remember that for each of the forces a Third Law pair acts on a separate body. This can sometimes be confusing when there are multiple Third Law pairs at work. Below are some examples of situations where multiple Third Law pairs occur.


6.9: Reading- Studying the Earth’s Interior
The following table summarizes the physical layers of the earth.
| Physical Layers of Earth | ||
|---|---|---|
| Layer | Physical Behavior | Thickness |
| Lithosphere | rigid, brittle at shallow depths | 5–200 km |
| Asthenosphere | ductile | 100–300 km |
| Upper Mesosphere | rigid, not brittle, rapid increase in density with depth | 300–400 km |
| Lower Mesosphere | denser and more rigid than upper mesosphere | 2,300 km |
| Outer Core | liquid | 2,300 km |
| Inner Core | rigid, not brittle | 1,200 km |
Earth’s Magnetic Field Originates in the Core
The liquid outer core is the source of the earth’s magnetic field, as a result of its metallic nature, which means it contains electrons not attached to particular nuclei. Heat is transferred upward to the mantle from the inner core via convective cells, in which the liquid in the outer core flows in looping patterns. The combination of the loose electrons and looping convective flow with the rotation of the earth results in a geodynamo that produces a magnetic field. Because the magnetic field is generated by a dynamically convecting and rotating sphere of liquid, it is unstable. Every now and then, after several hundred thousand to several million years, the earth’s magnetic field becomes unstable to the point that it temporarily shuts down. When it restarts, its north and south magnetic poles must inevitably be reversed, according to the physics of magnetic fields produced spontaneously from geodyamos. (For comparison, the magnetic field of the Sun, which is also produces by convecting electrical charges in a rotating sphere, becomes magnetically unstable and reverses its magnetic field on a more regular basis, every 11 years.)
Given that the inner core is a solid metallic sphere, made mostly of iron and nickel, surrounded entirely by liquid, it can be pictured as a giant ball bearing spinning in a pressurized fluid. Detailed studies of earthquake waves passing though the inner core have found evidence that it is spinning – rotating – just slightly faster than the rest of the earth.
Beyond Simple Layers
The interior of the earth is not simply layered. Some of the layers, particularly the crust and lithosphere, are highly variable in thickness. The boundaries between layers are rough and irregular. Some layers penetrate other layers at certain places. Variations in the thickness of the earth’s layers, irregularities in layer boundaries, and interpenetrations of layers, reflect the dynamic nature of the earth.
For example, the lithosphere penetrates deep into the mesosphere at subduction zones. Although it is still a matter of research and debate, there is some evidence that subducted plates may penetrate all the way into the lower mesosphere. If so, plate tectonics is causing extensive mixing and exchange of matter in the earth, from the bottom of the mantle to the top of the crust.
As another example, hot spots may be places where gases and fluids rise from the core-mantle boundary, along with heat. Studies of helium isotopes in hot spot volcanic rocks find evidence that much of the helium comes from deep in the earth, probably from the lower mesosphere.
How Do We Know?
We humans have no hands-on access to samples of the earth’s interior from deeper than the upper mantle. The earth’s core is so dense and so deep, it is completely inaccessible. Contrary to a popular misconception, lava does not come from the earth’s core. Magma and lava come from only the lithosphere and asthenosphere, the upper 200 km of earth’s 6,400 km thickness. Attempts have been made to drill through the crust to reach the mantle, without success. Given the lack of actual pieces of the earth from deeper than the asthenosphere, how do we know about the internal layers of the earth, what they are made of, and what their properties and processes are?
Igneous Rocks and Fault Blocks
There are two sources of rock samples from the lower lithosphere and asthenosphere, igneous rocks and fault blocks. Some igneous rocks contain xenoliths, pieces of solid rock that were adjacent to the body of magma, became incorporated into the magma, and were carried upward in the magma. From xenoliths in plutonic and volcanic igneous rocks, many samples of the lower crust and upper mantle have been identified and studied.
Another source of pieces of the lower crust and upper mantle is fault zones and exposed orogenic zones (root zones of mountains that have been exposed after much uplift and erosion). Some slabs of thrust-faulted rock contain lithospheric mantle rock. In ophiolites, ultramafic rock from the mantle part of the lithosphere is a defining attribute. Most ophiolites and thrust-faulted slices of rock that contain pieces of the upper mantle are related to either subduction zones or transform plate boundaries.
Seismic Waves
The energy from earthquakes travels in waves. The study of seismic waves is known as seismology. Seismologists use seismic waves to learn about earthquakes and also to learn about the Earth’s interior.
One ingenious way scientists learn about Earth’s interior is by looking at earthquake waves. Seismic waves travel outward in all directions from where the ground breaks and are picked up by seismographs around the world. Two types of seismic waves are most useful for learning about Earth’s interior.
Body Waves
P-waves and S-waves are known as body waves because they move through the solid body of the Earth. P-waves travel through solids, liquids, and gases. S-waves only move through solids (Figure 1). Surface waves only travel along Earth’s surface. In an earthquake, body waves produce sharp jolts. They do not do as much damage as surface waves.
- P-waves (primary waves) are fastest, traveling at about 6 to 7 kilometers (about 4 miles) per second, so they arrive first at the seismometer. P-waves move in a compression/expansion type motion, squeezing and unsqueezing Earth materials as they travel. This produces a change in volume for the material. P-waves bend slightly when they travel from one layer into another. Seismic waves move faster through denser or more rigid material. As P-waves encounter the liquid outer core, which is less rigid than the mantle, they slow down. This makes the P-waves arrive later and further away than would be expected. The result is a P-wave shadow zone. No P-waves are picked up at seismographs 104 o to 140 o from the earthquakes focus.
- S-waves (secondary waves) are about half as fast as P-waves, traveling at about 3.5 km (2 miles) per second, and arrive second at seismographs. S-waves move in an up and down motion perpendicular to the direction of wave travel. This produces a change in shape for the Earth materials they move through. Only solids resist a change in shape, so S-waves are only able to propagate through solids. S-waves cannot travel through liquid.
Where seismic waves speed up or slow down, they refract, changing the direction in which they are traveling. Where seismic waves encounter an abrupt boundary between two very different layers, some of the seismic wave energy is reflected, bouncing back at the same angle it struck. The reflections and refractions of seismic waves allow the layers and boundaries within the earth to be located and studied.
By tracking seismic waves, scientists have learned what makes up the planet’s interior (figure 2).
- P-waves slow down at the mantle core boundary, so we know the outer core is less rigid than the mantle.
- S-waves disappear at the mantle core boundary, so the outer core is liquid.
This animation shows a seismic wave shadow zone .
Here are some examples of what we have been able to distinguish in the earth’s interior from the study of seismic waves and how they travel through the layers of the earth:
- The thickness of the crust. This is a measure of the thickness of the crust based on the abrupt increase in speed of seismic waves that occurs when they enter the mantle. The boundary between the crust and mantle, as inferred from the change in the speed of P- and S-waves, is called the Mohorovicic discontinuity, named after the Croatian seismologist who first discerned it; usually it is referred to simply as the Moho. It is mainly from seismic waves that we know how thin oceanic crust is and how thick continental crust is.
- The thickness of the lithosphere. Where seismic waves pass down from the lithosphere into the asthenosphere, they slow down. This is because of the lower rigidity and compressibility of the rocks in the layer below the lithosphere. The zone below the lithosphere where seismic waves travel more slowly is called the low velocity zone. The low velocity zone is probably coincident with the asthenosphere.
- The boundary between the upper and lower mesosphere (upper and lower mantle). This shows up as an increase in seismic wave speed at a depth of 660 km.
- The boundary between the mantle and the core. This is marked by S-waves coming to an abrupt stop, presumably because the outer core is liquid, and a sudden large reduction in the speed of P-waves, as they enter the liquid core where there is no rigidity to contribute to P-wave speed.
- The inner core. This was first recognized by refraction of P-waves passing through this part of the core, due to an abrupt increase in their speed, which was not shown by P-waves traveling through only the outer part of the core.
- Seismic tomography: imaging slabs and masses at various orientations in the earth, not just in layers. By combining data from many seismometers, three-dimensional images of zones in the earth that have higher or lower seismic wave speeds can be constructed. Seismic tomography shows that in some places there are masses of what may be subducted plates that have penetrated below the asthenosphere into the mesosphere and, in some cases, penetrated into the lower mesosphere, the deepest part of the mantle. In other places, subducted plates appear to have piled up at the base of the upper mesosphere without penetrating into the lower mesosphere.
Gravity
Isaac Newton was the first to calculate the total mass of the earth. This gives us an important constraint on what the earth is made of, because, by dividing the mass of the earth by the volume of the earth, we know the average density of the earth. Whatever the earth is made of, it must add up to the correct amount of mass. Gravity measurements, and the earth’s mass, tell us that the interior of the earth must be denser than the crust, because the average density of earth is much higher than the density of the crust.
Because different parts of the crust, mantle, and core have different thicknesses and densities, the strength of gravity over particular points on earth varies slightly. These variations from the average strength of earth’s gravity are called gravity anomalies. Mapping and analyzing gravity anomalies, in some cases by using satellites, and also be measuring the effect of gravity anomalies on the surface shape of the ocean, has given us much insight into subduction zones, mid-ocean spreading ridges, and mountain ranges, including constraints on the depths of their roots.
Moment of Inertia
The earth’s gravity tells us how much total mass the earth has, but does not tell us how the mass is distributed within the earth. A property known as moment of inertia, which is the resistance (inertia) of an object to changes in its spin (rotation), is determined by exactly how matter is distributed in a spinning object, from its core to its surface. The earth’s moment of inertia is measured by its effect on other objects with which it interacts gravitationally, including the Moon, and satellites. Knowing the earth’s moment of inertia provides a way of checking and refining our understanding of the mass and density of each of the earth’s internal layers.
Meteorites
Studies of meteorites, which are pieces of asteroids that have landed on earth, along with astronomical studies of what the Sun, the other planets, and orbiting asteroids are made of, give us a model for the general chemical composition of objects in the inner solar system, which are made mainly of elements that form rocks and metals, as opposed to the outer planets such as Jupiter, which are made mostly of light, gas-forming elements. The general compositional model of the rocky and metallic part of the solar system has much higher percentages of iron, nickel, and magnesium than is found in the earth’s crust.
If the earth’s mantle is made of ultramafic rock, as is found in actual samples of the upper mantle in xenoliths and ophiolites, that would account for part of the missing iron, nickel, and magnesium. But much more iron and nickel would still be missing. If the core is made mostly of iron, and abundant nickel as well, it would give the earth an overall composition similar to the composition of other objects in the inner solar system, and similar to the proportions of rock and metal-forming elements measured in the Sun.
A mantle with an ultramafic composition, and a core made mostly of iron plus nickel, would make earth’s composition match the composition of the rest of the solar system, and give those layers the right densities to account for the earth’s moment of inertia and total mass.
Experiments
Geology, like other sciences, is based on experiment along with observation and theory. earth scientists and physicists have developed experimental methods to study how materials behave at the pressures and temperatures of the earth’s interior, including core temperatures and pressures. They can measure such properties as the density, the state of matter (liquid or solid), the rigidity, the compressibility, and the speed at which seismic waves pass through these materials at high pressures and temperatures. These studies allow further refinement of our knowledge of what the interior of the earth is made of and how it behaves. These experiments support the theory that the mantle is ultramafic and the core is mostly iron and nickel, because they show that materials with those compositions have the same density and seismic wave speeds as have been observed in the earth.
Contributors and Attributions
- Basics -- Earth's Interior. Authored by : Ralph L. Dawes and Cheryl D. Dawes. Provided by : Wenatchee Valley College. Located at : http://commons.wvc.edu/rdawes/G101OCL/Basics/earthinterior.html#magnetic . Project : Geology 101 - Introduction to Physical Geology. License : CC BY: Attribution
- Seismic Waves. Provided by : CK-12. Located at : http://www.ck12.org/earth-science/Seismic-Waves/lesson/Seismic-Waves-HS-ES/ . License : CC BY-NC: Attribution-NonCommercial


11.2: Elementary Operations
Learning Objectives
- Use elementary operations to find the solution to a linear system of equations.
- Find the and of a matrix.
- Determine whether a system of linear equations has no solution, a unique solution or an infinite number of solutions from its .
- Solve a system of equations using Gaussian Elimination and Gauss-Jordan Elimination.
- Model a physical system with linear equations and then solve.
We have taken an in depth look at graphical representations of systems of equations, as well as how to find possible solutions graphically. Our attention now turns to working with systems algebraically.
Definition \(\PageIndex{1}\) : System of Linear Equations
A system of linear equations is a list of equations, \[\begin{array}{c} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1} \\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2} \\ \vdots \\ a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m} \end{array}\] where \(a_{ij}\) and \(b_{j}\) are real numbers. The above is a system of \(m\) equations in the \(n\) variables, \(x_{1},x_{2}\cdots ,x_{n}\). Written more simply in terms of summation notation, the above can be written in the form \[\sum_{j=1}^{n}a_{ij}x_{j}=b_{i}, \text{ }i=1,2,3,\cdots ,m\]
The relative size of \(m\) and \(n\) is not important here. Notice that we have allowed \(a_{ij}\) and \(b_{j}\) to be any real number. We can also call these numbers scalars . We will use this term throughout the text, so keep in mind that the term scalar just means that we are working with real numbers.
Now, suppose we have a system where \(b_{i} = 0\) for all \(i\). In other words every equation equals \(0\). This is a special type of system.
Definition \(\PageIndex{2}\): Homogeneous System of Equations
A system of equations is called homogeneous if each equation in the system is equal to \(0\). A homogeneous system has the form \[\begin{array}{c} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}= 0 \\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}= 0 \\ \vdots \\ a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}= 0 \end{array}\] where \(a_{ij}\) are scalars and \(x_{i}\) are variables.
Recall from the previous section that our goal when working with systems of linear equations was to find the point of intersection of the equations when graphed. In other words, we looked for the solutions to the system. We now wish to find these solutions algebraically. We want to find values for \(x_{1},\cdots ,x_{n}\) which solve all of the equations. If such a set of values exists, we call \(\left( x_{1},\cdots ,x_{n}\right)\) the solution set .
Recall the above discussions about the types of solutions possible. We will see that systems of linear equations will have one unique solution, infinitely many solutions, or no solution. Consider the following definition.
Definition \(\PageIndex{3}\): Consistent and Inconsistent Systems
A system of linear equations is a list of equations, \[\begin{array}{c} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1} \\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2} \\ \vdots \\ a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m} \end{array}\] where \(a_{ij}\) and \(b_{j}\) are real numbers. The above is a system of \(m\) equations in the \(n\) variables, \(x_{1},x_{2}\cdots ,x_{n}\). Written more simply in terms of summation notation, the above can be written in the form \[\sum_{j=1}^{n}a_{ij}x_{j}=b_{i}, \text{ }i=1,2,3,\cdots ,m\]
A system of linear equations is called consistent if there exists at least one solution. It is called inconsistent if there is no solution.
If you think of each equation as a condition which must be satisfied by the variables, consistent would mean there is some choice of variables which can satisfy all the conditions. Inconsistent would mean there is no choice of the variables which can satisfy all of the conditions.
The following sections provide methods for determining if a system is consistent or inconsistent, and finding solutions if they exist.
Elementary Operations
We begin this section with an example.
Example \(\PageIndex{1}\): Verifying an Ordered Pair is a Solution
Algebraically verify that \( \left( x, y \right) = \left( -1, 4 \right) \) is a solution to the following system of equations. \[\begin{array}{c} x+y=3 \\ y-x=5 \label{eq1.2.1} \end{array}\]
Solution
By graphing these two equations and identifying the point of intersection, we previously found that \(\left(x, y \right) = \left( -1, 4 \right)\) is the unique solution.
We can verify algebraically by substituting these values into the original equations, and ensuring that the equations hold. First, we substitute the values into the first equation and check that it equals \(3\). \[x + y = (-1)+(4) = 3\] This equals \(3\) as needed, so we see that \(\left( -1,4 \right)\) is a solution to the first equation. Substituting the values into the second equation yields \[y -x = (4) - (-1) = 4 + 1 = 5\] which is true. For \(\left( x,y\right) =\left( -1,4\right)\) each equation is true and therefore, this is a solution to the system.
Now, the interesting question is this: If you were not given these numbers to verify, how could you algebraically determine the solution? Linear algebra gives us the tools needed to answer this question. The following basic operations are important tools that we will utilize.
Definition \(\PageIndex{4}\): Elementary Operations
Elementary operations are those operations consisting of the following.
- Interchange the order in which the equations are listed.
- Multiply any equation by a nonzero number.
- Replace any equation with itself added to a multiple of another equation.
It is important to note that none of these operations will change the set of solutions of the system of equations. In fact, elementary operations are the key tool we use in linear algebra to find solutions to systems of equations.
Consider the following example.
Example \(\PageIndex{2}\): Effects of an Elementary Operation
Show that the system \[\begin{array}{c} x+y=7 \\ 2x-y=8 \end{array}\] has the same solution as the system \[\begin{array}{c} x+y=7 \\ -3y=-6 \end{array}\]
Solution
Notice that the second system has been obtained by taking the second equation of the first system and adding -2 times the first equation, as follows: \[2x-y + (-2)(x+y) = 8 + (-2)(7)\] By simplifying, we obtain \[-3y=-6\] which is the second equation in the second system. Now, from here we can solve for \(y\) and see that \(y=2\). Next, we substitute this value into the first equation as follows \[x+y=x+2=7\] Hence \(x=5\) and so \(\left( x,y\right) = \left(5,2 \right)\) is a solution to the second system. We want to check if \(\left(5,2 \right)\) is also a solution to the first system. We check this by substituting \(\left(x, y \right) = \left(5,2 \right)\) into the system and ensuring the equations are true. \[\begin{array}{c} x+y = \left(5 \right)+ \left( 2 \right) = 7 \\ 2x-y= 2 \left(5 \right) - \left( 2 \right) = 8 \end{array}\] Hence, \(\left(5,2 \right)\) is also a solution to the first system.
This example illustrates how an elementary operation applied to a system of two equations in two variables does not affect the solution set. However, a linear system may involve many equations and many variables and there is no reason to limit our study to small systems. For any size of system in any number of variables, the solution set is still the collection of solutions to the equations. In every case, the above operations of Definition \(\PageIndex{4}\) do not change the set of solutions to the system of linear equations.
In the following theorem, we use the notation \(E_i\) to represent an equation, while \(b_i\) denotes a constant.
Theorem \(\PageIndex{1}\): Elementary Operations and Solutions
Suppose you have a system of two linear equations \[\begin{array}{c} E_{1}=b_{1}\\ E_{2}=b_{2} \end{array} \label{system}\] Then the following systems have the same solution set as \ref{system}:
- \[\begin{array}{c} E_{2}=b_{2}\\ E_{1}=b_{1} \end{array} \label{thm1.9.1}\]
- \[\begin{array}{c} E_{1}=b_{1} \\ kE_{2}=kb_{2}\\ \end{array} \label{thm1.9.2}\] for any scalar \(k\), provided \(k\neq0\).
- \[\begin{array}{c} E_{1}=b_{1} \\ E_{2}+kE_{1}=b_{2}+kb_{1} \end{array} \label{thm1.9.3}\] for any scalar \(k\) (including \(k=0\)).
Before we proceed with the proof of Theorem \(\PageIndex{1}\), let us consider this theorem in context of Example \(\PageIndex{2}\). Then, \[\begin{array}{cc} E_{1} = x+y, & b_{1} = 7 \\ E_{2} = 2x-y, & b_{2} = 8 \end{array}\] Recall the elementary operations that we used to modify the system in the solution to the example. First, we added \(\left( -2 \right)\) times the first equation to the second equation. In terms of Theorem Theorem \(\PageIndex{1}\), this action is given by \[E_{2} + \left( -2 \right) E_{1} = b_{2} + \left( -2 \right)b_{1}\] or \[2x-y + \left( -2 \right) \left(x+y \right) = 8 + \left( -2 \right) 7\] This gave us the second system in Example \(\PageIndex{2}\), given by \[\begin{array}{c} E_{1} = b_{1} \\ E_{2} + \left( -2 \right) E_{1} = b_{2} + \left( -2 \right) b_{1} \end{array}\]
From this point, we were able to find the solution to the system. Theorem \(\PageIndex{1}\) tells us that the solution we found is in fact a solution to the original system.
We will now prove Theorem \(\PageIndex{1}\).
- Proof
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The proof that the systems \ref{system} and \ref{thm1.9.1} have the same solution set is as follows. Suppose that \(\left( x_{1},\cdots ,x_{n}\right)\) is a solution to \(E_{1}=b_{1},E_{2}=b_{2}\). We want to show that this is a solution to the system in \ref{thm1.9.1} above. This is clear, because the system in \ref{thm1.9.1} is the original system, but listed in a different order. Changing the order does not effect the solution set, so \(\left( x_{1},\cdots ,x_{n}\right)\) is a solution to \ref{thm1.9.1}.
Next we want to prove that the systems \ref{system}and \ref{thm1.9.2} have the same solution set. That is \(E_{1}=b_{1},E_{2}=b_{2}\) has the same solution set as the system \(E_{1}=b_{1},kE_{2}=kb_{2}\) provided \(k\neq 0\). Let \(\left( x_{1},\cdots ,x_{n}\right)\) be a solution of \(E_{1}=b_{1},E_{2}=b_{2},\). We want to show that it is a solution to \(E_{1}=b_{1},kE_{2}=kb_{2}\). Notice that the only difference between these two systems is that the second involves multiplying the equation, \(E_{2}=b_{2}\) by the scalar \(k\). Recall that when you multiply both sides of an equation by the same number, the sides are still equal to each other. Hence if \(\left( x_{1},\cdots ,x_{n}\right)\) is a solution to \(E_{2}=b_{2}\), then it will also be a solution to \(kE_{2}=kb_{2}\). Hence, \(\left( x_{1},\cdots ,x_{n}\right)\) is also a solution to \ref{thm1.9.2}.
Similarly, let \(\left( x_{1},\cdots ,x_{n}\right)\) be a solution of \(E_{1}=b_{1},kE_{2}=kb_{2}\). Then we can multiply the equation \(kE_{2}=kb_{2}\) by the scalar \(1/k\), which is possible only because we have required that \(k\neq 0\). Just as above, this action preserves equality and we obtain the equation \(E_{2} = b_{2}\). Hence \(\left( x_{1},\cdots ,x_{n}\right)\) is also a solution to \(E_{1}=b_{1},E_{2}=b_{2}.\)
Finally, we will prove that the systems\ref{system} and \ref{thm1.9.3} have the same solution set. We will show that any solution of \(E_{1}=b_{1},E_{2}=b_{2}\) is also a solution of \ref{thm1.9.3}. Then, we will show that any solution of \ref{thm1.9.3} is also a solution of \(E_{1}=b_{1},E_{2}=b_{2}\). Let \(\left( x_{1},\cdots ,x_{n}\right)\) be a solution to \(E_{1}=b_{1},E_{2}=b_{2}\). Then in particular it solves \(E_{1} = b_{1}\). Hence, it solves the first equation in \ref{thm1.9.3}. Similarly, it also solves \(E_{2} = b_{2}\). By our proof of \ref{thm1.9.2}, it also solves \(kE_{1}=kb_{1}\). Notice that if we add \(E_{2}\) and \(kE_{1}\), this is equal to \(b_{2} + kb_{1}\). Therefore, if \(\left( x_{1},\cdots ,x_{n}\right)\) solves \(E_{1}=b_{1},E_{2}=b_{2}\) it must also solve \(E_{2}+kE_{1}=b_{2}+kb_{1}\).
Now suppose \(\left( x_{1},\cdots ,x_{n}\right)\) solves the system \(E_{1}=b_{1}, E_{2}+kE_{1}=b_{2}+kb_{1}\). Then in particular it is a solution of \(E_{1} = b_{1}\). Again by our proof of \ref{thm1.9.2}, it is also a solution to \(kE_{1}=kb_{1}\). Now if we subtract these equal quantities from both sides of \(E_{2}+kE_{1}=b_{2}+kb_{1}\) we obtain \(E_{2}=b_{2}\), which shows that the solution also satisfies \(E_{1}=b_{1},E_{2}=b_{2}.\)
Stated simply, the above theorem shows that the elementary operations do not change the solution set of a system of equations.
We will now look at an example of a system of three equations and three variables. Similarly to the previous examples, the goal is to find values for \(x,y,z\) such that each of the given equations are satisfied when these values are substituted in.
Example \(\PageIndex{3}\): Solving a System of Equations with Elementary Operations
Find the solutions to the system,
\[\begin{array}{c} x+3y+6z=25 \\ 2x+7y+14z=58 \\ 2y+5z=19 \end{array} \label{solvingasystem1}\]
Solution
We can relate this system to Thoerem \(\PageIndex{1}\) above. In this case, we have \[\begin{array}{c c} E_{1} = x + 3y + 6z, & b_{1} = 25\\ E_{2} = 2x+7y+14z, & b_{2} = 58 \\ E_{3} = 2y+5z, & b_{3} = 19 \end{array}\] Thoerem \(\PageIndex{1}\) claims that if we do elementary operations on this system, we will not change the solution set. Therefore, we can solve this system using the elementary operations given in Definition \(\PageIndex{4}\) . First, replace the second equation by \(\left( -2\right)\) times the first equation added to the second. This yields the system \[\begin{array}{c} x+3y+6z=25 \\ y+2z=8 \\ 2y+5z=19 \end{array} \label{solvingasystem2}\] Now, replace the third equation with \(\left( -2\right)\) times the second added to the third. This yields the system \[\begin{array}{c} x+3y+6z=25 \\ y+2z=8 \\ z=3 \end{array} \label{solvingasystem3}\] At this point, we can easily find the solution. Simply take \(z=3\) and substitute this back into the previous equation to solve for \(y\), and similarly to solve for \(x\). \[\begin{array}{c} x + 3y + 6 \left(3 \right) = x + 3y + 18 = 25\\ y + 2 \left(3 \right) = y + 6 = 8 \\ z = 3 \end{array}\] The second equation is now \[y+6=8\] You can see from this equation that \(y = 2\). Therefore, we can substitute this value into the first equation as follows: \[x + 3 \left(2 \right) + 18 = 25\] By simplifying this equation, we find that \(x=1\). Hence, the solution to this system is \(\left( x,y,z \right) = \left( 1,2,3 \right)\). This process is called back substitution .
Alternatively, in \ref{solvingasystem3} you could have continued as follows. Add \(\left( -2\right)\) times the third equation to the second and then add \(\left( -6\right)\) times the second to the first. This yields \[ \ \begin{array}{c} x+3y=7 \\ y=2 \\ z=3 \end{array}\] Now add \(\left( -3\right)\) times the second to the first. This yields \[ \ \begin{array}{c} x=1 \\ y=2 \\ z=3 \end{array}\] a system which has the same solution set as the original system. This avoided back substitution and led to the same solution set. It is your decision which you prefer to use, as both methods lead to the correct solution, \(\left( x,y,z \right) = \left(1,2,3\right)\).