Блог Анкара

In This Chapter

► Strengthening your new, helpful attitudes and beliefs

► Dealing with doubts about a new way of thinking

► Testing out your new ways of thinking in difficult situations Preparing for setbacks

I\ Fter you’ve identified your unhelpful patterns of thinking and developed Ґ \ more helpful attitudes (refer to Chapters 2, 3, 12, and 14), you need to reinforce your new thoughts and beliefs. The process of reinforcing new beliefs is like trying to give up a bad habit and develop a good habit in its place. You need to work at making your new, healthy ways of thinking second nature, at the same time as eroding your old ways of thinking. This chapter describes some simple exercises to help you develop and nurture your new beliefs.

In many ways, Integrating Your new method of thinking with your mind, emotions, and actions is The Critical process in CBT. A parrot can repeat rational philosophies, but the parrot doesn’t understand or Believe What it’s saying. The real work in CBT is turning intellectual understanding into something you that know in your gut to be true.

Defining the Beliefs \lou Want to Strengthen

Many people who work at changing their attitudes and beliefs complain: ‘I know what I Should Think, but I don’t believe it!’ When you begin to adopt a new way of thinking, you may Know That something makes sense but you may not Feel That the new belief is true.

When you’re in a state of Cognitive dissonance You know that your old way of thinking isn’t 100 per cent right, but you aren’t yet convinced of the alternative. Being in a state of cognitive dissonance can be uncomfortable because things don’t feel quite right. However, this feeling is a good sign that things are changing.

In CBT, we often call this disconnection between thinking and truly believing the Head-to-heart problem. Basically, you know that an argument is true in your head, but you don’t feel it in your heart. For example, if you’ve spent many years believing that you’re less worthy than others or that you need the approval of other people in order to approve of yourself, you may have great difficulty Internalising (believing in your gut) an alternative belief. You may find that the idea that you have as much basic human worth as the next person, or that approval from others is a bonus but not a necessity, difficult to buy.

Your alternative beliefs are likely to be about three key areas:

I Yourself

I Other people

I The world

Alternative beliefs may take the following formats:

A Flexible preference, Instead of a rigid demand or rule, such as ‘I’d very much prefer to be loved by my parents, but there’s no reason they absolutely Have To love me.’

An Alternative assumption, Which is basically an if/then statement, such as ‘If I Don’t get an A in my test, Then That won’t be the end of the world. I can still move on in my academic career.’

A global belief, Which expresses a positive healthy general truth, such as ‘I’m basically okay’ rather than ‘I’m worthless’, or ‘The world’s a place with some safe and some dangerous parts’ instead of ‘The world’s a dangerous place’.

When you do experience the head-to-heart problem, we recommend acting As if‘you really do hold the new belief to be true – we explain how to do this in the following section.

One of your main aims in CBT, after you’ve developed a more helpful alternative belief, is to increase how strongly you endorse your new belief or raise your Strength of conviction (SOC). You can rate how much you believe in an alternative healthy philosophy on a 0-100 percentage scale, 0 represents a total lack of conviction and 100 represents an absolute conviction.

Acting As If I/ou Already Belietfe

You don’t need to believe your new philosophy entirely in order to start changing your behaviour. Starting out, it’s enough to Know In your head that your new belief makes sense and then Act According to your new belief or philosophy. If you consistently do the ‘acting as if technique, which we explain here, your conviction in your new way of thinking is likely to grow over time.

You can use the ‘acting as if technique to consolidate any new way of thinking, in pretty much any situation. Ask yourself the following questions:

How would I behave if I truly considered my new belief to be true?

How would I overcome situational challenges to my new belief if I truly considered it to be true and helpful?

What sort of behaviour would I expect to see in other people who truly endorse this new belief?

You can make a list of your answers to the above questions and refer to it before, after, and even during an experience of using the ‘acting as if technique. For example, if you’re dealing with social anxiety and trying to get to grips with self-acceptance beliefs, use the ‘acting as if techniques that follow, and ask yourself similar kinds of questions, such as:

Act consistently with the new belief: If I truly believed that I was as worthy as anyone else, how would I behave in a social situation?

Be specific about how you’d enter a room, the conversation you may initiate, and what your body language would be like.

Troubleshoot for challenges to your new belief: If I truly believed that I was as worthy as anyone else, how would I react to any social hiccups?

Again, be specific about how you may handle lulls in conversation and moments of social awkwardness.

Observe other people. Does anyone else in the social situation seem to be acting as if they truly endorse the belief that I am trying to adopt?

If so, note how the person acts and how they handle awkward silences and normal breaks in conversation. Imitate their behaviour.

When you act in accordance with a new way of thinking or a specific belief, you reinforce the truth of that belief. The more you experience a belief In action, The more you can appreciate its beneficial effects on your emotions. In essence, you are rewiring your brain to think in a more helpful and realistic way. Give this technique a try, even if you think that it’s wishful thinking or seems silly. Actions do speak louder than words. So if a new belief makes sense to you, follow it up with action.

Budding a Portfolio of Arguments

When an old belief rears its ugly head, try to have on hand some strong arguments to support your new belief. Your old beliefs or thinking habits have probably been with you a long time, and they can be tough to shift. You can expect to argue with yourself about the truth and benefit of your new thinking several times, before the new stuff well and truly replaces the old.

Your portfolio of arguments can consist of a collection of several arguments against your old way of thinking and several arguments in support of your new way of thinking. You can refer to your portfolio anytime that you feel conviction in your new belief is beginning to wane. The following sections help to guide you towards developing sound rationales in support of helpful beliefs and in contradiction of unhelpful beliefs.

Generating arguments against an unhelpful belief

To successfully combat unhealthy beliefs, try the following exercise. At the top of a sheet of paper, write down an old, unhelpful belief you want to weaken. For example, you may write: ‘I have to get approval from significant others, such as my boss. Without approval, I’m worthless.’ Then, consider the following questions to highlight the unhelpful nature of your belief:

Is the belief untrue or inconsistent with reality? Try to find evidence that your belief isn’t factually accurate (or at least not 100 per cent accurate for 100 per cent of all of the time). For example, you don’t Have To get approval from your boss: The universe permits otherwise, and you can survive without such approval. Furthermore, you cannot be defined as worthless on the strength of this experience, because you’re much too complex to be defined.

Considering why a certain belief is Understandable Can help you to explain why you hold a particular belief to be true. For example, ‘It’s understandable that I think I’m stupid because my father often told me I was when I was young, but that was really due to his impatience and his own difficult childhood. So, it follows that I believe myself to be stupid because of my childhood experiences, and not because there is any real truth in the idea that I am stupid. Therefore, the belief that I am stupid is consistent with my upbringing but inconsistent with reality.’

Is the belief rigid? Consider whether your belief is flexible enough to allow you to adapt to reality. For example, the idea that you Must Get approval or that you Need Approval in order to think well of yourself, is overly rigid. It is entirely possible that you will fail to get approval from significant others at some stage in your life. Unless you have a flexible belief about getting approval, you are destined to think badly of yourself whenever approval is not forthcoming. Replace the word Must With Prefer In this instance, and turn your demand for approval into a flexible preference for approval.

Is the belief extreme? Consider whether your unhelpful belief is extreme. For example, equating being disliked by one person with worthlessness is an extreme conclusion. It is rather like concluding that being late for one appointment means that you will always be late for every appointment you have for the rest of your life. The conclusion that you draw from one or more experiences is far too extreme to accurately reflect reality.

Is the belief illogical? Consider whether your belief actually makes sense. You may want approval from your boss, but logically she doesn’t Have To approve of you. Not getting approval from someone significant doesn’t logically lead to you being less worthy. Rather, not getting approval shows that you’ve failed to get approval on this occasion, from this specific person.

Is the belief unhelpful? Consider how your belief may or may not be helping you. For example, if you worry about whether your boss is approving of you, you’ll probably be anxious at work much of the time. You may feel depressed if your boss treats you with indifference or visibly disapproves of your work. You’re less likely to say no to unreasonable requests or to put your opinions forward. You may actually be less effective at work because you’re so focused on making a good impression. You may even assume that your boss is disapproving of you when actually this isn’t the case. So, is worrying about your boss’s approval helpful? Clearly not!

Running through the preceding list of questions is definitely an exercise that involves putting pen to paper or fingertips to keyboard. Try to pick out your unhelpful beliefs and to formulate helpful alternatives, then generate as many watertight arguments against your old belief and in support of your new belief as you can. Try to fill up one side of A4 paper for each belief you target.

You can include in your portfolio evidence gathered from other CBT techniques you use to tackle your problems, such as ABC forms (Chapter 3) and behavioural experiments (Chapter 4). You can use any positive results observed from living according to new healthy beliefs as arguments to support the truth and benefits of these new beliefs.

Generating arguments to support your helpful alternative belief

The guidelines for generating sound arguments to support alternative, more helpful ways of thinking about yourself, other people, and the world, are similar to those suggested in the preceding section, ‘Generating arguments against an unhelpful belief.

On a sheet of paper, write down a helpful alternative belief that you want to use to replace a negative, unhealthy view you hold. For example, a helpful alternative belief regarding approval at work may be: ‘I want approval from significant others, such as my boss, but I don’t Need It. If I don’t get approval, I still have worth as a person.’

Next, develop arguments to support your alternative belief. Ask yourself the following questions to ensure that your helpful alternative belief is strong and effective:

Is the belief true and consistent with reality? For example, you really can want approval and fail to get it sometimes. Just because you want something very much doesn’t mean to say you’ll get it. Lots of people don’t get approval from their bosses, but it doesn’t mean they’re lesser people.

Is the belief flexible? Consider whether your belief allows you to adapt to reality. For example, the idea that you Prefer To get approval but that it isn’t a dire necessity for either survival or self-esteem, allows for the possibility of not getting approval from time to time. You don’t have to form any extreme conclusions about your overall worth in the face of occasions of disapproval.

Is the belief balanced? Consider whether your helpful belief is balanced and non-extreme. For example, ‘Not being liked by my boss is unfortunate but it’s not proof of whether I’m worthwhile as a person.’ This balanced and flexible belief recognises that disapproval from your boss is undesirable and may mean that you need to reassess your work performance. However, this recognition does not hurl you into depression based on the unbalanced belief that you’re unworthy for failing to please your boss on this occasion.

Is the belief logical and sensible? Show how your alternative belief follows logically from the facts, or from your preferences. It follows logically that your boss’s disapproval about one aspect of your work is undesirable and may mean that you need to work harder or differently. It does not follow logically that because of his disapproval you are an overall bad or worthless person.

Is the belief helpful? When you accept that you want approval from your boss but that you don’t Have To get it, you can be less anxious about the possibility of incurring your boss’s disapproval or failing to make a particular impression. You also stand a better chance of making a good impression at work when you prefer, but are not desperate for, approval. You can be more focused on the job that you’re doing and less preoccupied by what your boss may be thinking about you.

Imagine you’re about to go into court to present to the jury arguments in defence of your new belief. Develop as many good arguments that support your new belief as you can. Most people find that listing lots of ways in which the new belief is helpful makes the most impact. Try to generate enough arguments to fill one side of A4 paper for each individual belief.

Review your rational portfolio regularly, not just when your unhealthy belief is triggered. Doing so helps you reaffirm your commitment to thinking in healthy ways.

Understanding That Practice Makes Imperfect

Despite your best efforts, you may continue to think in rigid and extreme ways and experience unhealthy emotions from time to time. Why? Well, – oh yes, we say it again – you’re only human.

Practicing your new, healthy ways of thinking and putting them to regular use minimises your chances of relapse. However, you’re never going to become a perfectly healthy thinker – human beings seem to have a tendency to develop thinking errors and you need a high degree of diligence to resist unhelpful and unhealthy thinking (refer to Chapter 2).

Be wary of having a perfectionist attitude about your thinking. You’re setting yourself up to fail if you expect that you can always be healthy in thought, emotion, and behaviour. Give yourself permission to make mistakes with your new thinking, and use any setbacks as opportunities to discover more about your beliefs.

Beating With your doubts and reservations

You must give full range to your scepticism when you’re changing your beliefs. If you try to sweep your doubts under the carpet, those doubts can

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Re-emerge when you least expect it – usually when you’re in a stressful situation. Consider Sylvester’s experience:

Sylvester, or Sly for short, believes that other people must like him and goes out of his way to put people at ease in social situations. Sly takes great care to never hurt anyone’s feelings and puts pressure on himself to be a good host. Not surprisingly, Sly’s often worn out by his efforts. Because Sly’s work involves managing other staff, he also feels anxious much of the time. Sly also worries about confrontation and what his staff members think of him when he disciplines them.

After having some CBT, Sly concludes that his beliefs need to change if he’s ever going to overcome his anxiety and feelings of panic at work. Sly formulates a healthy alternative belief: ‘I want to be liked by others, but I don’t always Have To be liked. Being disliked is tolerable and doesn’t mean I’m an unlikeable person.’

Sly can see how this new belief makes good sense and can help him feel less anxious about confronting staff members or being not-so-super-entertaining in social situations. But deep inside, Sly feels stirrings of doubt. Still, Sly denies his reservations about the new belief and ignores niggling uncertainty. One day, when Sly’s confronting a staff member about persistent lateness, his underlying doubts rear up. Sly resorts to his old belief because he hasn’t dealt with his doubts effectively. Sly ends up letting his worker off the hook and feeling angry with himself for not dealing with the matter properly.

Had Sly faced up to his misgivings about allowing himself to be disliked, he may have given himself a chance to resolve his feeling. Sly may then have been more prepared to deal with the stressful situation without resorting to his old belief and avoidant behaviour.

Zigging and zagging through the zigzag technique

Use the zigzag technique to strengthen your belief in a new healthy alternative belief or attitude. The zigzag technique involves playing devil’s advocate with yourself. The more you argue the case in favour of a healthy belief and challenge your own attacks on it, the more deeply you can come to believe in it. Figure 15-1 shows a completed zigzag form based on Sly’s example.

HEALTHY BELIEF

I want to be liked by other people but I don’t Always haveXo Be liked. It’s tolerable to be disliked and it doesn’t mean that I’m an unlikable person.

Rate conviction in Healthy Belief 40 %

THE

ZIG-ZAG

FORM

ATTACK

Yeah but, if LOTS of People don’t like me it’s awful! I can’t stand that.

DEFENCE

Lots of people not liking me would be Unfortunate But not the worst thing in the world. Trying to get everyone to like me makes me really clumsy and anxious socially.

ATTACK

But lots of people not liking me Must Mean there’s something wrong with me. It proves I’m unlikable.

DEFENCE

Figure 15-1:

Sly’s completed zigzag form.

First of all, I’m more likely to Assume Lots of people don’t like me and I don’t actually know that it’s true. I simply can’t be everyone’s cup of tea. /like some people more than others and it doesn’t mean there’s something wrong with them.

Rate conviction in Healthy Belief 75 %

You can find a blank zigzag form in Appendix B. To go through the zigzag technique, do the following steps:

1. Write down in the top left-hand box of the zigzag form a belief that you want to strengthen.

On the form, rate how strongly you endorse this belief, from 0 to 100 per cent conviction.

Be sure that the belief’s consistent with reality or true, logical, and helpful to you. Refer to the section that covers generating arguments to support your helpful alternative beliefs, for more on testing your healthy belief.

2. In the next box down, write your doubts, reservations, or challenges about the healthy belief.

Really let yourself attack the belief, using all the unhealthy arguments that come to mind.

3. In the next box, dispute your attack and redefend the healthy belief.

Focus on defending the healthy belief. Don’t become sidetracked by any points raised in your attack from Step 2.

4. Repeat Steps 2 and 3 until you exhaust all your attacks on the healthy belief.

Be sure to use up all your doubts and reservations about choosing to really go for the new, healthy alternative way of thinking. Use as many forms as you need and be sure to stop on a defence of the belief you want to establish rather than on an attack.

5. Re-rate, from 0 To 100 Per cent, how strongly you endorse the healthy belief after going through all your doubts.

If your conviction in the healthy belief hasn’t increased or has increased only slightly, revisit the previous instructions on how to use the zigzag form. Or, if you have a CBT therapist, discuss the form with her and see whether she can spot where you zigged when you should have zagged.

Putting your new beliefs to the test

Doing pen-and-paper exercises is great – they really can help you to move your new beliefs from your head to your heart.

However, the best way to make your new ways of thinking more automatic is to put them to the test. Putting them to the test means going into familiar situations where your old attitudes are typically triggered, and acting according to your new way of thinking.

So, our friend Sly from earlier in the chapter may choose to do the following to test his new beliefs:

Sly confronts his member of staff about her lateness in a forthright manner. Sly bears the discomfort of upsetting her and remembers that being disliked by one worker doesn’t prove that he’s an unlikeable person.

Sly throws a party and resists the urge to make himself busy entertaining everyone and playing the host.

Sly works less hard in work and social situations at putting everyone at ease and trying to be super-likeable mister nice guy.

If you’re really, really serious about making your new beliefs stick, you can Seek out Situations in which to test them. On top of using your new beliefs and their knock-on new behaviours in everyday situations, try setting difficult tests for yourself. Sit down and think about it: If you were still operating under your old beliefs, what situations would really freak you out? Go there. Doing this will ‘up the ante’ with regard to endorsing your new beliefs.

Coping with everyday situations, such as Sly’s previous example, is very useful, and they’re often enough to move your new belief from your head to your heart. But if you really want to put your new beliefs under strain, with a view to making them even stronger, put yourself into out-of-the-ordinary situations. For example, try deliberately doing something ridiculous in public or being purposefully rude and aloof. See if you can remain resolute in your new belief such as ‘disapproval does not mean unworthiness’ in the face of your most feared outcomes. We think you can! This is a tried and tested CBT tool for overcoming all sorts of problems, such as social anxiety. (Refer to Chapter 12 for more guidance on developing Self-acceptance And Chapter 22 for more on devising Shame attacking Exercises.)

Here are some tests that Sly (or we could now call him ‘Braveheart’) may set up for himself:

Go into shops and deliberately be impolite by not saying ‘thank you’ and not smiling at the shop assistant. This test requires Sly to bear the discomfort of possibly leaving the shop assistant unhappy after making a poor impression.

Say good morning to staff without smiling and allow them to form the impression that he was ‘in a bad mood’.

Mooch about, deliberately trying to look moody and aloof in a social setting.

I Make a complaint about faulty goods he’s purchased from a local shop where the staff know him.

I Bump into someone on public transport and do not apologise.

You may think that Sly’s setting himself up to be utterly friendless as a result of this wretched belief change lark. Au contraire, nos chere! Sly has friends. Sly still has a reputation of being a generally kind and affable bloke. What Sly doesn’t have now is a debilitating belief that he has to please all the people all the time. Rather, Sly can come to truly believe that he can tolerate the discomfort of upsetting people occasionally and that being disliked by one or more people is part of being human. That’s life. That’s the way it goes sometimes. Sly can believe in his heart that he’s a fallible human being, just like everyone else, that he’s capable of being liked and disliked but basically he’s okay.

Nurturing \lour NeuJ Beliefs

As you continue to live with your alternative helpful beliefs, gather evidence that supports your new beliefs. Becoming more aware of evidence from yourself, other people, and the world around you that supports your new, more helpful way of thinking, is one of the keys to strengthening your beliefs and keeping them strong.

A Positive data log Helps you overcome the biased, prejudiced way in which you keep unhelpful beliefs well-fed, by soaking up evidence that fits with them and discounting or distorting evidence that doesn’t fit. Using a positive data log boosts the available data that fit your new belief and helps you to retrain yourself to take in the positive.

Your positive data log is simply a record of positive results arising from acting in accordance with a healthy new belief and evidence that contradicts your old unhealthy belief. You can use any type of notebook to record your evidence. Follow these steps:

1. Write your new belief at the top of a page.

2. Record any experiences that support your new belief.

Be specific and include even the smallest details that encourage you to doubt your old way of thinking. For example, even a newspaper vendor making small talk when you buy your paper can be used as evidence to support a belief that you are likeable.

3. Record positive reactions that you get from others when you act in accordance with new beliefs.

4. Record evidence that your new belief is helpful to you; include changes in your emotions and behaviour.

Fill up the whole notebook if you can.

If you still have trouble believing that an old, unhelpful belief is true, start by collecting evidence on a daily basis that your old belief isn’t 100 per cent true, 100 per cent all of the time. Collecting this sort of evidence can help you steadily erode how true the belief seems.

In your positive data log, you can list the benefits of operating under your new belief, including all the ways in which your fears about doing so have been disproved.

For example, Sly might record the following observations:

His staff members still seem to generally like being managed by him, despite the fact that he disciplines them when needed.

Being less gregarious at parties doesn’t stop others from having a good time or from engaging with him.

His anxiety and panic about the possibility of being disliked have reduced in response to his belief change.

Your positive data log can not only remind you of the good results you have reaped from changing your unhealthy beliefs to healthy ones, but also help you be Compassionate With yourself when you relapse to your unhealthy beliefs and corresponding behaviours. Use your positive data log to chart your progress, so when you Do Fall back you can assure yourself that your setback need be only temporary. After all, practice makes imperfect.

Many people add to their positive data log for months or even years. Keeping the log provides them with a useful antidote to the natural tendency to be overly self-critical.

Be sure to refer to your positive data log often, even daily, or several times each day when you are bedding down new beliefs. Keep it in your desk or handbag or wherever you are most likely to be able to access it during the day. As a general rule, you can’t look at your positive data log too often!

In This Chapter

^ Finding out what relaxation therapies are all about ^ Exploring different ways to relax

^ Discovering what relaxation therapies can be good for

^ Examining the evidence

^ Knowing what to expect in a typical session

^ Knowing how to find safe and effective practitioners

M LO you suffer from stress? Then you’re not alone. A moderate amount of stress can be a useful stimulant but too much wears you down and can contribute to a host of common ailments such as fatigue, lethargy, depression, insomnia, and headaches, as well as more serious conditions such as heart disease.

Relaxation of mind and body, healthy breathing, and meditation can all be powerful antidotes to stress as well as great health promoters. Many complementary therapies provide relaxation and stress relief as an indirect part of their healing approach but some therapies are also aimed specifically at promoting physical and mental relaxation. These therapies can be used in their own right or successfully combined with other forms of complementary medicine.

In this chapter, I outline some ancient and new ways of relaxing the mind and body. You’ll find out about the importance of correct breathing and investigate techniques for releasing muscle tension and calming and expanding the

Mind. You’ll also discover the evidence for the different therapies to determine what each may be helpful for.

At the end of the chapter, I pass on to you a simple self-help relaxation technique that you can use in your everyday life.

Finding Out about Relaxation Therapies

In ancient times people may not have been as stressed as we are now but they still valued relaxation and saw its health benefits! All the ancient healing traditions of China, Japan, India, Tibet, and so on (take a look at the chapters in Part II to find out more about these traditional healing systems) emphasise the importance of good breathing, of having a relaxed and healthy body, and of maintaining a calm, clear mind.

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In the ancient medical systems of Asia and the Far East good health and relaxation almost always start with the breath.

In Traditional Chinese Medicine (TCM), for example, good breathing is said to facilitate the flow of Qi (vital energy) in the Meridians (vital energy channels), and to promote healthy internal organ function.

On a physical level, each of the great ancient medical traditions also employed techniques for relaxing the muscles and mobilising the joints. For example, Ayurvedic medicine involves yoga exercises and special Panchakarma Hot oil massage techniques; TCM uses Qi gong Exercises and tui na massage; Japanese medicine utilises Do-in Exercises, Anma Massage, and Shiatsu; And Tibetan medicine involves Kum Nye Exercises and various massage techniques (check out the chapters in Part II and Chapter 16 on bodywork therapies for more about these).

These traditions also emphasised mental relaxation, so breathing and physical exercises were combined with training in stillness and meditation. Meditation techniques range from observing a flame or controlling the breath (Ayurveda and yoga), to visualisation of deities and reciting mantras (Tibet) and visualising the circulation of light, or observing the breath and practising ‘mindful awareness’ (Japan and China).

Some of these ancient therapies have become popular forms of relaxation today – such as yoga and yoga breathing techniques. Other techniques – such as Jacobson’s progressive relaxation and Buteyko’s breathing technique – have been developed more recently. In the next section, I take a look at these therapies in a bit more detail.

Looking at Popular Relaxation Therapies Today

First, I introduce you to some breathing therapies and then I tell you about some relaxation and meditation techniques. In this section, I cover the following:

Pranayama (yoga breathing)

Buteyko

Yoga Asanas

Progressive relaxation

Flotation

Meditation

Breathing therapies

Breathing therapies have been an important part of healing systems across the ages. The underlying belief seems to be that abnormal breathing can contribute to disease and that by bringing it under conscious control and regulating it, health can be promoted and healing facilitated.

Not enough space exists within the scope of this book to go through the breathing techniques from each of the ancient traditions. So I concentrate here on one of those most commonly practised and researched types of breathing therapy – Pranayama. Then I tell you about what could be the most important modern breathing therapy – the Buteyko breathing method.

Pranayama

The ancient Sanskrit word Pranayama Means bringing your Prana, Or life force, under control (yama). Pranayama Breathing exercises were first described by the legendary Indian Yogi, Patanjali, said to have lived around the third century BC.

How does it work?

The exercises involve controlling your in – and out-breaths and holding your breath. Their role is to help cleanse the body, improve powers of concentration and to aid personal and spiritual self-development.

Practising pranayama

You can learn many different types of Pranayama, From simple beginners’ techniques through to quite advanced practices that should only be practised under the guidance of a qualified yoga teacher.

Introductory Pranayama Exercises include the following:

Diaphragmatic breathing: Locating your Diaphragm (the big bell-shaped muscle across your abdomen) and using it more effectively when breathing.

Full yogic breath: Breathing in three stages, filling the lower part of the lungs, then the mid-section, and then the upper part, and then exhaling air from the lungs in the reverse order.

Anula viloma (alternate nostril breathing): This technique involves using the thumb and forefinger to close each nostril alternately and enable in – and out-breaths through one nostril at a time. It may also be combined with breath retention. Take a look at the ‘Helping Yourself with Relaxation Therapies’ section at the end of this chapter for details on how to practise this technique.

Ujjayi pranayama (victorious breath): This technique involves breathing in through the nose while tightening the throat to produce a throaty sound.

Kapalabhati pranayama (skull cleansing Or Skull shining): This technique is a type of bellows breathing with a soft inhalation and then forceful exhalation achieved by contracting the abdominal muscles.

Whom or what is it good for?

Anyone can benefit from Pranayama, But you need to practise the techniques properly and safely, ideally from an experienced yoga teacher. Only attempt the more advanced techniques under careful supervision, because an excess of deep breathing can cause dizziness and discomfort, and you need to avoid excessive strain on the lungs. Anyone with lung disease or a serious lung condition should not practise Pranayama Without medical supervision.

What’s the evidence?

Research suggests that Pranayama Exercises can gradually increase lung capacity and improve the balance of oxygen and carbon dioxide in the body and may also slow brain waves, heart rate, and blood pressure.

The first-ever world survey of yoga, including a study of Pranayama, Is being carried out online in 2007. Visit Www. yogainaustralia. com to find out more and register.

Where can I learn pranayama?

In the UK, the Yoga Biomedical Trust, founded by Robin Munro in the 1980s, has pioneered the therapeutic applications of yoga (Tel: 020 7689 3040; Www. yogatherapy. org).

Worldwide, excellent Pranayama Training is offered by the Sivananda Yoga Vedanta Centres. (I did their yoga teacher training course myself in Grass Valley, California many years ago.) For details, see Www. sivananda. org or, in the UK, contact London’s Sivananda Yoga Vedanta Centre (Tel: 020 8780 0160).

Buteyko breathing training

Buteyko (pronounced Bu-tay-ko) Is a method for re-training the way that you breathe devised by Russian medical researcher Konstantin Buteyko in the 1950s. Buteyko was assigned to monitor the breathing patterns of dying patients. He noticed that the sicker people became – especially shortly before death – the more they seemed to need larger volumes of air and took deeper breaths. He developed a theory that improper over-breathing (hyperventilation) contributed to disease and that the less you breathed the longer you’d live!

Buteyko then spent several decades devising a correct breathing technique and thoroughly researching its effects. Despite initial medical opposition his method was approved by the Russian authorities in 1981, for the treatment of asthma. Since then Buteyko breathing training has spread to Australia, New Zealand, Europe, and elsewhere.

How does it Work?

Buteyko’s method is based on the idea that incorrect over-breathing (hyper-ventilation) lowers carbon dioxide levels in the body – carbon dioxide is as necessary as oxygen for certain body processes and the two need to be in a correct ratio. He argued that over-breathing, and breathing through the mouth, means that the body cannot utilise oxygen efficiently and so people may become breathless, nervous, tired, or unwell.

His method emphasises breathing through the nose, using the diaphragm to breathe fully, and slowing down and holding the breath to correct this imbalance. The idea is that you train yourself to use a lesser volume of air more effectively.

Learning Buteyko

The Buteyko method is usually taught by a trained Buteyko teacher in a series of five 90-minute lectures and practical sessions taken over five days. However, some shorter training courses are also now available and follow-up, regular self-practice is essential. The aim is to reduce your number of breaths

Per minute from 12 or more down to 4. The exercises are easy to learn, and you can do them sitting in a chair. Simple stress-management and dietary advice may also be recommended alongside the breathing training.

Whom or what is it good for?

Children (from the age of four) and adults of any age can learn the Buteyko method. It has been reported to help relieve a range of respiratory problems including asthma, hyperventilation, panic attacks, sinusitis, bronchitis, hay fever, emphysema, and snoring, as well as arthritis, high blood pressure, and digestive problems.

What’s the evidence?

A 1980 Russian trial by the Government’s Committee for Science and Technology and many trials by Dr Buteyko himself, found significant improvements in people with respiratory and other symptoms after learning the Buteyko method. A 1994 Australian trial with asthma sufferers also showed that training in the technique led to a reduction of symptoms and inhaler use. However more controlled trials are needed and many Western medical doctors remain ignorant or sceptical about this technique.

How can I find a Buteyko teacher?

For lots more about Buteyko, see Www. buteyko. com. In the UK, you can find a Buteyko teacher via The Buteyko Breathing Association (Tel: 01277 366 906; Www. buteykobreathing. org).

Relaxation therapies

In this section, I introduce you to three relaxation therapies that are popular today: an ancient one – yoga, and two more modern ones; progressive relaxation and flotation therapy.

Yoga

The Sanskrit word Yoga Means ‘union’, and refers to union of the body, mind, and spirit. Yoga is believed to have originated several thousand years ago in India. Of the many different types of yoga, the type mainly used in the West for relaxation is known as Hatha yoga And was first formalised in the 15th century by the Indian yogi Swatmarama.

Hatha yoga is based on a series of Asanas, Or ‘body postures’, that are primarily intended to improve physical health. However, their ultimate aim goes further and, combined with Pranayama Breathing exercises and meditation, the Asanas Are designed to help you on the path to mental peace, personal development, and spiritual upliftment.

Ft

—

Practising yoga asanas

The yoga Asanas Are a series of body poses, from simple ones to more advanced ones, either held statically or performed in a dynamic sequence. Different Asanas Are believed to stimulate different internal organs and body systems.

The Shavasana, Or ‘corpse’ pose, which is a resting pose performed lying down, is often used as the final pose of a yoga session and is especially useful for relaxation.

How do they work?

The Asanas Facilitate muscle stretching, toning and endurance, flexibility, relaxation, and correct breathing. On a more subtle level, the Asanas Are also believed to facilitate the flow of vital energy, or Prana, In the energy channels known as the Nadis - these are believed to correspond closely to the meridian system of acupuncture – and to promote healing.

Whom or what are yoga asanas good for?

Anyone, from young children to adults of any age, can practise the yoga Asanas. Yoga has been found to be beneficial for many conditions, including back pain, digestive problems, and anxiety.

Only perform advanced poses under the guidance of an experienced yoga instructor. If you have neck problems or high blood pressure, don’t perform the inverted poses (such as headstands).

What’s the evidence?

Research suggests that practise of yoga Asanas Have a range of physiological effects including slowing brain waves, heart rate, and respiration rate; decreasing muscle tension and stress; and increasing the sense of well-being.

How can I learn yoga asanas?

Check out the resources listed earlier in this chapter in the section on Pranayama, Or, in the UK, contact The British Wheel of Yoga (Tel: 01529 306 851; Www. bwy. org. uk) or, in the US, the Yoga Alliance (Tel: 00 1877-964-2255 (toll free) or 00 1301-868-4700; Www. yogaalliance. org).

Progressive relaxation (PR)

Progressive muscle relaxation is a technique devised by an American doctor and researcher, Edmund Jacobson, in the 1930s. He noticed that anxiety was accompanied by muscle tension and reasoned that if you can relax the muscles you may be able to reduce anxiety. He also found that if you tense a muscle for a short period of time, when you release it, it will relax. He therefore devised a systematic sequence for going through each of the major muscle groups in the body and tensing and then relaxing them.

How does it work?

By isolating individual muscles and practising tensing and then relaxing them, you increase your awareness of muscular tension and become able to voluntarily release muscular tension.

Learning PR

To learn PR you need to sit or lie comfortably and then go through a sequence of tensing and relaxing each major muscle, or muscle group, from the hands, arms, and neck, down to the toes. Each muscle is contracted tightly and then held for about 10 seconds. At the end of the sequence you just rest for a while breathing normally. The whole sequence takes 15 to 20 minutes. You need to practise daily until you’ve mastered the technique.

Whom or what is it good for?

Anyone – from children to adults – can learn and use the PR technique. What’s the evidence?

Research has shown that PR is effective in reducing muscular tension and in promoting the relaxation response in the body.

How can I learn PR?

No association for PR teachers exists at present. You can train yourself in the technique from books or audio tapes.

Flotation therapy

Flotation therapy developed out of work in the 1950s, by American physiologist and psychoanalyst John C. Lilly on sensory deprivation, investigating the effects of being deprived of normal sensory experiences such as sight, hearing, and so on. He built Sensory deprivation chambers That excluded light and reduced sounds, and contained several inches of a highly concentrated Epsom salt solution to make trial participants float and feel weightless. He found that when people were put in these chambers for short periods, they found the experience very relaxing.

In the 1970s, two other researchers, Peter Suedfeld and Roderick Borrie, coined the term ‘restricted environmental stimulation technique’, otherwise known as REST, to describe tanks that re-created these conditions for a specifically therapeutic purpose. Wet REST systems involve floating in a concentrated warm-water salt solution, while dry REST systems enable the person to lie on a piece of foam suspended on the water and stay dry.

Having flotation therapy

You float inside an enclosed tank, usually about 8 feet (2.5 metres) by 4 feet (1.25 metres) wide and with 10 inches (25 cm) or so of warm water that is highly salted so that you float easily. The tanks are often egg-shaped and have one whole side that lifts up or have a door by which you can enter. Once

Inside there may be a control panel to regulate light and sound. Earplugs can be worn to block out sound or some tanks have speakers and/or screens so that music or educational audios, or even videos, can be played. (It has been shown that learning is facilitated in this relaxed state!) Some tanks also have microphones so that you can communicate with a therapist or other person outside the tank. Sessions generally last for 60 to 90 minutes, but you lose all sense of time when floating and it feels like five minutes!

How does it work?

During flotation, the body and brain enter a deeply relaxed state. Your brain produces feel-good, relaxing chemicals known as Endorphins, And your pulse rate, brain waves, and so on all slow down. Being freed from gravity appears to have a therapeutic effect that deepens the relaxation state, and the body’s natural homeostatic mechanism for healing may be able to kick in as a result.

Whom or what is it good for?

Flotation is recommended for stress and pain relief, and for conditions such as arthritis, headaches, and fatigue. People also report that it helps their concentration and mental sharpness.

Flotation therapy isn’t advisable for people who suffer from claustrophobia or serious mental disturbance.

What’s the evidence?

Research shows decreases in blood pressure, muscle tension, and heart rate after flotation therapy, as well as a decrease in the production of stress hormones. Clinical studies have shown that flotation can be helpful in reducing pain and inflammation and may help ease a range of disorders, including high blood pressure, insomnia, premenstrual syndrome, and rheumatoid arthritis. More good studies are needed.

Where can I try flotation therapy?

You need a flotation tank! These tanks can be found in various complementary medicine clinics or you can buy your own for home use – but they’re expensive.

In the UK and Eire, clinics and centres with flotation tanks can be located

Via Www. floatationtankassociation. net and in the US via www. Floatation. com (Tel: 00 1530 432 4502).

Meditation

Many different types of meditation exist and going into them all is beyond the scope of this book. I therefore concentrate on the therapeutic aspects of meditation, outline some of the main types for you, and then point you in the direction for further information.

What is it?

Meditation is a mental technique that, on a physical level, may be used to calm the mind and induce relaxation. It involves concentration on an object or image (such as a candle flame or religious picture), or a sound (such as the breath or chanting) in order to eliminate other external distractions or inner thoughts or feelings.

However, meditation also has a higher purpose. All the religions of the world involve meditation in one form or another, either in terms of prayer and contemplation, the recitation of mantras, or specific meditation techniques, and meditation is seen as a method of personal development and a way of transcending the material world in order to draw closer to the spiritual. Meditation may be devoid of religious trappings and used as a way of evolving consciousness and of entering a state of increased awareness and even bliss.

Types of meditation

Here are some of the main types of meditation (in alphabetical order):

Raja yoga meditation: Taught free by the Brahma Kumaris World Spiritual University all over the world, this technique involves an exploration of universal values through meditation and positive thinking. For worldwide locations, see Www. bkwsu. org.

Sufi meditation: As part of the Sufi tradition (a mystic tradition derived from the Islamic faith), Sufi meditation uses chants, sacred dance, and philosophical lectures, taught in small groups under instruction from a master, to guide personal growth. For more details in the UK, see Www. sufimeditation. org (Tel: 0794 44 89527), or for worldwide information, check out Www. sufimovement. org.

Transcendental meditation: Brought to the West by the Maharishi Mahesh Yogi, whose most famous students were the Beatles in the 1960s, this technique involves 20 minutes daily repetition of a simple mantra. For details of UK courses, check out Www. t-m. org. uk (Tel: 08705 143 733), or contact the Association of Independent TM Teachers, who charge more affordable fees, at Www. tm-meditation. co. uk (Tel: 01843 841 010 or 0191 213 2179). For worldwide and US information, see Www. tm. org or, in the US, call 00 1 888 LEARN TM to find your nearest TM teacher.

Vipassana meditation: Vipassana Means ‘to see things as they really are’. This form of mindfulness meditation is based on observing the breath. It was brought to the West by S. N. Goenka and is taught over ten days of silent practice with lectures and individual guidance from a trained teacher. The technique is derived from Buddhist teachings in India but can be practised by anyone regardless of religious background.

The teaching is free and offered all over the world. For more information, see Www. dhamma. org or, in the UK, call 01989 730 234.

Zen meditation: This is also called the ‘practice of no mind’ and is based on emptying the mind of all thoughts and following the breathing. This technique is a Buddhist practice that travelled from India to the Orient and then to the West. It is practised sitting upright in either a cross-legged or kneeling position or seated in a chair. For a funny, visual, online introduction to Zen from the famous Kodaiji Temple in Kyoto, Japan, see Www. do-not-zzz. com. Search online or in local phone directories for your local Zen centre.

How does it work?

During meditation you produce brain waves associated with relaxation and heart and respiration rate decrease.

Whom or what is it good for?

Anyone can benefit from meditation – both children and adults. It appears to be effective for stress relief and relaxation.

What’s the evidence?

Meditation has been well researched for many decades, especially since the pioneering work of Professor Herbert Benson at Harvard University in the US from the 1970s onwards. Benson was the first to identify the ‘relaxation response’ showing the physical effects of meditation on the body. For more on this work, see the Mind/Body Medical Institute site at Www. mbmi. org.

Helping Yourself with Relaxation Therapies

To relax yourself try the following alternate nostril breathing exercise from L/tjYl The yoga tradition:

^"^^ Sit in a comfortable position.

Raise your right hand and place the thumb against the right nostril to close it.

I Inhale through your left nostril to the count of eight.

Close your left nostril by pressing the ring finger against it.

I Release the thumb and exhale through the right nostril also to the count of eight.

IU Repeat the process, this time inhaling through the open right nostril to a count of eight, then close this nostril with the thumb and release the ring finger, exhaling though your left nostril to a count of eight again.

I Repeat this sequence seven times slowly and gently.

This exercise helps to relieve stress and aid relaxation. In more advanced versions of this exercise, the sequence of counts varies and the breath may be held at the end of the out-breath.

In this part. . .

I\ Toms are built with three kinds of particles, two of Ґ \ which are charged. This means that charge is important in chemistry. Swapping or transferring charge between reactants alters their properties. Two critical classes of reactions revolve around such movements of charge: acid-base reactions and oxidation-reduction reactions. In this part, we give you the tools to deal with these charge-centric reactions. In addition, we describe nuclear chemistry, which deals with transformations in atomic nuclei, the positively charged hearts of atoms.

In This Chapter

► Receiving massage around the world

► Getting massaged while on the road

► Using massage to relax while flying

MMyherever Your hands can go, massage can go, too. And there’s no limit ▼ ▼ to the strange and wonderful environments you can find yourself in when seeking out professional massage or exchanging one with a traveling partner.

The only problem is, you have to get to your destination in order to enjoy the massage offered there, and the getting-there part often causes quite a bit of tension. That’s why I share a coach-class massage with you at the end of this chapter, to help make getting there a little less of a pain.

One World, Many Massages

If you travel around much and receive massages in different parts of the world, one thing you’ll notice pretty quickly is that each culture has its own distinct attitudes about massage and its own unique ideas about what a massage should be. In Turkey, for example, the massage you receive in a traditional Hamam, Or bathhouse, may include a tremendously vigorous rub-down in a big steamy chamber by a silent giant who is apparently indifferent to your discomfort, or your bliss, for that matter. On the other extreme, a massage on a cruise ship sailing the warm waters of the Caribbean may be an airy-light, soothing experience given to you by a sensitive and delicate Englishwoman.

Just say the word

Wherever you are in the world, people often appreciate it if you attempt a few words of their language. And what better opportunity to practice your language skills than when seeking a massage in Madagascar, a backrub in Bangalore, or some reflexology in Rotterdam? With that in mind, I include a list of phrases in several different languages in the following table, all of which will get you the same result no matter where you are in the world — a massage!

Language Phrase

German Ich mochte eine Massage. Spanish Me gustaria un masaje.

M

In Thailand, you can enter one of the public pavilions at a sacred temple to receive your massage on a low, wide bed, while dressed in loose-fitting pajamas.

The Japanese take their bathing and their massages seriously, and they’ve developed a very elaborate system of hot spring resorts called Onsen. If you visit one, you’ll get to immerse yourself in a series of ever-hotter baths and receive a massage directly afterward.

And, in Mexico, you may find yourself in an adobe enclosure up in the mountains where white-robed massage therapists try to attune you to the inner rhythms of the surrounding environment.

You get the idea — massage can be found almost anywhere you go these days, even in the places you least expect it. I have a client who was on a trip through the Yucatan, visiting ancient Mayan ruins, when she received a message about massage from a very unexpected source.

She had heard of a spiritual healer who lived in a remote Mayan village, and she wanted to meet this woman. After traveling for hours on a dirt road, she was escorted into the healer’s hut. This ancient woman, way out in the jungle, took a close look at her and said (through a translator), "You need to get a massage. And not just any massage. Special massage! Many many special massages." Sure enough, my client has been on a quest to receive as many massages as possible ever since, and the massages have helped her recover from a number of injuries.

Massage on the Road

You don’t have to travel by Jeep to a remote Mayan village to have someone tell You To get lots of massages, right? No, you’ve probably already figured that one out for yourself. And when you travel, chances are you’re much more likely to receive your massage-on-the-road in a hotel room rather than in the Yucatan outback.

Hotel rooms, even though they’re often expensive and advertised as luxurious getaways, offer their own distinctive brand of discomfort. They are, after all, not home. Massage is the perfect antidote for hotel discomfort. Chapter 8 shows you what to do to get a professional massage while staying in a hotel and how to deal with the hotel concierge. In this section, I offer a few words of advice for exchanging massages with each other in hotel rooms.

The most important thing about on-the-road massage is to bring along your own traveling Inner chamber Like the one described in Chapter 9. That way you can transform almost any blah hotel space into your own personal massage sanctuary. Then simply follow the instructions from Chapter 11 as you trade massages with each other.

To create your traveling inner chamber, remember to pack: Massage oil

A little massage gizmo (see Chapter 10) A portable CD player with mini-speakers Candles and matches A familiar photograph Incense ^ A bathrobe

In a pinch, you can always try Magic fingers, If it’s available in your room. Magic fingers is a rather hokey-looking device installed in the beds at many hotels and motels. When fed with coins, it vibrates the entire bed, and can be actually quite pleasurable. As singer/songwriter Jimmy Buffet says, "Put in a quarter, turn out the lights, magic fingers makes ya feel all right."

18-wheeler massage

All around the world, the people who probably put in the most hours traveling are truckers. Day after day, for thousands of miles, with their butts glued into high-bucket seats, they roll across the countryside, their necks, shoulders, and backs getting more and more tense as they go.

You can’t get much tougher than truckers. They’re not the type to complain, but recently even they, too, have seen the light about massage. The Triple T Truck Stop in Arizona now

Offers therapeutic massage to the guys and gals hauling goods in their big rigs across the U. S.

If you’ve ever had any thoughts along the lines of, "I’m too tough for massage. That’s for wusses who can’t stand pain," just consider the example of the truckers.

If they see the value of massage therapy, it’s good enough for me — and for you!

Massage in Coach Class

The toughest thing about travel is that it involves an awful lot of moving around, often in air-tight steel tubes hurtling through the upper atmosphere. Air travel is one sure way to get stressed out. Whoever uttered that famous phrase, "Life is a journey, not a destination," probably wasn’t sitting in coach class on a transoceanic flight at the time.

Although there’s just no way to get what you need the most on flights (namely, a hot shower and lots of fresh oxygen), you can still offer yourself or your traveling companions a little relief with a coach-class massage. By the way, this technique works equally well in first or business class, although it may not be quite as necessary.

The coach-class self-massage

Imagine yourself sitting in your coach-class seat like a good little passenger, all packed in like fruitcake in a tin, when suddenly the thought strikes you that you are, indeed, exceedingly uncomfortable, and you need to do something about it. You’ve already stood up twice to stretch, climbing all over other passengers to do so. Nothing seems to help.

The time to try massage has arrived. The five-step routine shown here is especially good for relieving tension in your sinus area, which can become sore due to cabin air pressure changes and dehydration. Just follow these steps and take a look at Figure 17-1:

1. Lean forward slightly (not too far, or else you’ll bang your head into the tray table of the seat in front of you), and hook your thumbs into the tender neck muscles just below the bony ridge at the base of your skull. Press in here, making little circles with your thumbs as you apply firm pressure, as shown in Figure 17-la.

2. Use your thumbs to press in to the upper inner corner of your eyes, right next to the nose, as shown in Figure 17-lb.

Pressure against the nose bones can help relieve sinus soreness. If you’re wearing glasses, take them off first. And if you’re wearing contacts, don’t press directly against them through your closed eyelids.

This move can be done very discreetly, but if you end up sitting next to someone who looks at you strangely, just smile, point at your own head, and say, "Sinus trouble. If I don’t do this my eyes will fall out."

3. With firm pressure, make fingertip circles on your temples, moving the skin and muscle over the bone below while keeping your fingers firmly anchored to one spot on the skin (see Figure 17-lc).

4. Press straight in on the temples and hold for 5 to 10 seconds (as shown in Figure 17-Id).

You may also try moving your jaw around a bit at the same time, which will allow you to press at slightly different depths into the muscles.

5. Press the heels of your hands into both sides of your head, above the ears, compressing the junction between your Parietal bones And your Temporal bones (as shown in Figure 17-le).

Hold this for up to 30 seconds. It often helps reduce headaches and take pressure off your poor skull in those pressurized airplane cabins.

Be careful not to wallop the passengers on either side of you with your elbows when performing this maneuver.

A couple extra tips

Try stretching your legs out a little and pushing several points along the outside of your thighs (being careful not to kick the person in the seat in front of you and cause a mid-air scene!). Then bend forward a bit and reach underneath your knees to massage the upper calves.

If you have stuffy sinuses, and nothing else has helped, try doing what the scuba divers do: Pinch your nose shut and blow gently into it (be careful not to blow too hard). This simple technique often equalizes pressure inside and outside your head.

Ah… relief.

Chapter 18

In This Chapter

^ Writing more than one equation for the problem ^ Solving systems of equations using substitution ^ Finding the break-even point in profits ^ Solving systems with multiple equations

Riting an equation to use for solving a story problem is more than half the battle. Once you have a decent equation involving a variable that

Represents some number or amount, then the actual algebra needed to solve

The equation is typically pretty easy.

Many word problems lend themselves to more than one equation with more than one variable. It’s easier to write two separate equations, but it takes more work to solve them for the unknowns. And, in order for there to be a solution at all, you have to have at least as many equations as variables.

Most of the problems in this chapter deal with the more typical two-equation solutions, but I include a section on dealing with three or more equations, too.

Writing Two Equations and Substituting

Word problems often deal with how many of two or more coins, how many ducks and elephants, how much to invest in this or that, how many red and green jelly beans, and so on. You let variables represent the numbers of coins or ducks or dollars or jelly beans. When working with two different equations written about the same situation, then you have two different variables and need to do some algebra to knock that down to one equation. That’s where substitution comes in.

Solving systems by substitution

A system of two linear equations, such as 2x + 3y = 31 and 5x – Y = 1 is usually solved by Elimination Or Substitution. (Refer to Algebra For Dummies If you want a full explanation of each type of solution method.) For the problems in this chapter, I use the Substitution Method, to solve for a variable. This means that you change the format of one of the equations so that it expresses what one of the variables is equal to in terms of the other, and then you Substitute Into the other equation. For example, you solve for Y In terms of X In the equation 3X + Y = 11 if you subtract 3X From each side and write the equation as Y = 11 – 3X.

Consider solving the system 2x + 3y = 31 and 5x – Y = 1. First go to the second equation and rewrite it with Y On one side and everything else on the other side. You choose the second equation, because the Y Variable has a coefficient of -1. Having a coefficient of 1 or -1 is desirable, because you can avoid working with fractions.

A Coefficient Is a factor or multiplier of a variable. The term 3x has a coefficient of 3, and the term Kx Has a coefficient of k.

To solve for Y In terms of X In 5x – Y = 1, you first add Y To each side of the equation and then subtract 1 from each side.

5x — Y = 1

5x = Y + 1 5x — 1 = Y

Now substitute into the first equation. Because Y = 5x – 1, replace the Y In the first equation with 5x – 1 and solve for X.

2x + 3 (5x — 1) = 31

2x + 15x — 3 = 31

17x — 3 = 31

17x = 34

34 0 X = T7 = 2

You determine that X = 2. Now determine what Y Is equal to by putting the 2 in for X In the equation Y = 5x – 1. You get that Y = 5(2) – 1 = 10 – 1 = 9. So the solution of the system of equations is that X = 2 and Y = 9.

The rest of this chapter deals with how to use substitution in systems of equations to solve word problems.

Working with numbers and amounts of coins

Some problems involving coins are more easily solved using two equations rather than just one, as you find in Chapter 8. If you’re nickeling and diming, then you let N Represent the number of nickels and D Represent the number of dimes. And if you have a total of ten coins, then N + D = 10. The rest of the information in the problem helps you write the other equation.

The Problem: You have ten coins in nickels and dimes. You have two less than five times as many dimes as nickels. How much money do you have?

^VLA/y First, you solve for the number of each type of coin. Then you determine

What the total worth is, multiplying the number of nickels by 5<t and dimes by 10<t. Let the number of nickels be represented by N And the number of dimes be represented by D. You write that the total number of coins is ten with the equation N + D = 10. You write that the number of dimes is two less than five times the number of nickels with the equation D = 5N - 2. As you see, the second equation already has D In terms of N And -2, so substitute 5N - 2 in for D In the first equation and solve for N.

N + (5n — 2) = 10 6n — 2 = 10 6n = 12 N = 2

You have two nickels, so there are 5(2) – 2 = 10 – 2 = 8 dimes. Two nickels are 10<t, and eight dimes are 80<t, so you have a total of 90<t.

The Problem: You have a total of 40 coins in nickels and quarters. If you double the number of quarters and add 12, you’ll get the same number of coins as you’d have if you multiply the number of nickels by 4 and subtract 4. How much money do you have in quarters?

^VLA/y First, determine how many nickels and quarters by writing one equation

Where the sum of the number of each type of coin is 40. Then write another equation involving the relationship between the number of each coin. You can determine how much money you have in quarters when you know how many quarters you have.

Let N Represent the number of nickels and Q Represent the number of quarters. Your first equation is N + Q = 40. The second equation sets the two relationships between the number of coins equal to one another. Let Double the number of quarters and add 12 Be represented by 2q + 12, and

Let Multiply the number of nickels by 4 and subtract 4 Be represented by 4n – 4. The second equation reads: 2q + 12 = 4n – 4. You can do some simplifying and rearranging with the second equation, but I’m going to just solve for Q In the first equation and substitute into the second equation. If N + Q = 40, then Q = 40 – N. Substituting and solving for Q,

You have 24 quarters, giving you 24(0.25) = $6. Does the number of coins check out? If you have 40 coins in all, and 24 are quarters, then you have 16 nickels. Doubling the number of quarters and adding 12 gives you 2(24) + 12 = 48 + 12 = 60. Multiplying the number of nickels by 4 and subtracting 4, you get 4(16) – 4 = 64 – 4 = 60. It checks.

Figuring out the purchases of fast food

You’re sent out to pick up some refreshments for the guys working on a project. You’ve gotten their orders and collected the money, but you’ve lost the piece of paper with the exact listing of what everyone wants. Good thing you know how to solve word problems with numbers of items and cost per item. Math saves the day, yet again.

The Problem: You collected a total of $25 for hamburgers and soft drinks. The hamburgers cost $2.50 each, and the soft drinks cost $1.50 each. Also, the number of hamburgers ordered is three less than twice as many soft drinks. How many hamburgers and how many soft drinks were ordered?

,A? LA/V Write two equations — one involving the number of each item times its

Respective cost, and the other involving the total number of items ordered. Let the number of hamburgers be represented by H And the number of soft drinks be represented by D. Multiplying H By 2.50 and D By 1.50, the total for the whole order is 2.50h + 1.50d = 25. The number of hamburgers, H, Is three less than twice the number of soft drinks, 2d – 3. Set H = 2d – 3. Now replace the H In the first equation with 2d – 3 and solve for H.

2.50 (2d – 3) + 1.50d = 25.00 5d – 7.50 + 1.50d = 25.00 6.50d – 7.50 = 25.00

6.50D = 32.50 32.50

6.50

So 5 soft drinks were ordered. The number of hamburgers is 2(5) – 3 = 10 – 3 = 7 hamburgers. Five soft drinks at $1.50 each is $7.50. Seven hamburgers at $2.50 each is $17.50, for an order that totals $25.

The Problem: Last night, the Fish House Diner sold some crab-cake dinners and shrimp baskets for a total of $705. The crab-cake dinners cost $9, and the shrimp baskets cost $11.50. If you multiply the number of crab-cake dinners by three and add it to twice the number of shrimp baskets, you get 180 dinners. How many of each were sold?

^VLA* Let the number of crab-cake dinners be represented with C And the number of shrimp baskets be represented with B. Multiply the number of each type item by its price and set the sum equal to $705. The equation is 9C + 11.50B = 705. Now write an equation involving the relationship between the numbers of orders. Three times the number of crab-cake dinners is 3C, And twice the number of shrimp baskets is 2B. So 3C + 2B = 180. This system of equations doesn’t have a variable with a coefficient of 1 or -1.

9.00c + 11.50b = 705 3c + 2b = 180

You could solve for C Or B In one of the equations by dividing each term in the

Equation by the coefficient of the respective variables. For instance, solving

3

For B In the second equation, you’d get B = 90 – ^ C. A better move would be

To take advantage of the fact that the C Variable has coefficients in one equation that is a multiple of the coefficient in the other equation. Just multiply each of the terms in the second equation by 3 and solve for 9C.

9c + 6b = 540

9C=540 -6B

Now replace the 9C In the first equation with 540 – 6B, And solve for B.

(540 – 6b) + 11.50b = 705.00 540 + 5.50b = 705.00 5.50b = 165.00

B 165.00 30

Weighing in on a million dollars

You’re on one of those game shows where you lie on the floor with $1 million in $1 bills on your get to pick a box or a curtain or some container chest for one minute, then you get to take all so you can win lots of money. You make your that money home. Your other choice is to take choice and see the words One Million Flashing $50,000 in cash. Which do you choose? in bright neon. You’re now told that, if you can

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They sold 30 shrimp baskets. The number of crab-cake dinners is found by solving 3C + 2B = 180 for C After replacing the B With 30.

3c + 2b = 180 3c + 2 (30) = 180 3c + 60 = 180 3C = 120 C = 40

So 40 crab-cake dinners and 30 shrimp baskets were sold.

Breaking Even and Making a Profit

Anyone in business will tell you that she’s interested in making a profit. Oh, well, the movie and musical The Producers Violates that premise, but you don’t hold with make-believe. In general, the profit from a venture is computed by taking the revenue earned and subtracting the cost that it takes to earn the revenue. Different factors keep pulling the costs up and the prices down, so it’s a real balancing act to be in business nowadays and make that profit.

Finding the break-even point

The Break-even point In business is when the Revenue (the money brought in) is equal to the Cost (the money spent to earn the revenue). When a business gets past the break-even point, it shows a profit. To find the break-even point,

You determine when the revenue and the costs are equal. You use a system of equations to solve the problem.

The Problem: You decide to go into the sandal-making business. Your startup costs are $1,600, and it costs you $40 per pair of sandals to produce them. You write your total cost function as C = 1,600 + 40x, Where X Is the number of pairs of sandals that you produce. The price at which you sell the sandals is dependent on the number of pairs you sell, so there isn’t a fixed price (you lower the price to be able to sell more). In this case, the amount of revenue you get from selling X Pairs of sandals is found with R = 100x – 0.5X2. What is the break-even point? How many pairs of sandals do you have to produce and sell to start making a profit?

Consider some cost and revenue values. Table 17-1 shows the revenue, total cost, and net result for X = 10, 20, 30, and 40 pairs of sandals.

Table 17-1 Revenue, Cost, and Net Result

When 40 pairs of sandals are produced, the revenue and the cost are the same, so X = 40 is the break-even point.

Making this table is helpful, but it isn’t very practical. A better method is to solve the system of equations. The two equations involved are C = 1,600 + 40X And R = 100x – 0.5×2. Replace the C And the R With Y, Letting the Y Represent the amount of money spent in the cost function and the amount of money earned in the revenue function. You want to find out when those two amounts are the same to find the break-even point. Then replace the Y In Y = 1,600 + 40x with 100x – 0.5×2 from the equation Y = 100x – 0.5×2. Solve for the value of X.

1,600 + 40x = 100x – 0.5×2 0.5×2 – 60x + 1,600 = 0

You end up with a quadratic equation. You can factor the equation or you can use the quadratic formula. (See the Cheat Sheet for the formula.) I show you factoring, in the following equation, after multiplying every term by 2 so that the coefficient of the X2 Term becomes a 1.

1×2 – 120x + 3,200 = 0 (x – 40)(X – 80) = 0

X = 40 or X = 80

You end up with two different solutions. The first solution is the one shown in Table 17-1. After you hit sales of 40 pairs of sandals, you’ve broken even and can start to make money. Unfortunately, if you make and sell more than 80 pairs, you start to lose money. Your expenses get too high, and it becomes too costly to produce and sell at that level.

Determining the profit

The break-even point can be where you start seeing some profit to your venture. After you’ve reached the point where increased sales will start making you some money, you earn profit. The amount of the profit is determined by subtracting the cost from the revenue. P = R - C. What level of sales will earn a particular amount of profit?

The Problem: A businessman wants to know when the sale of a particular item reaches a profit level of $2,000. The revenue equation is R = 200x – 0.4×2, and the cost to produce X Items is determined with C = 4,000 + 100x. How many items have to be produced and sold to net a profit of $2,000?

Use the equation P = R - C And replace the P With 2,000. Put the revenue equation and cost equation in their respective positions and solve the equation for X.

2,000 = (200x – 0.4×2) – (4,000 + 100x) 2,000 = 200x – 0.4×2 – 4,000 – 100x 2,000 = 100x – 0.4×2 – 4,000 0.4×2 – 100x + 6,000 = 0

The quadratic equation is solved by either factoring or using the quadratic formula. I choose to factor, but I’ll multiply each term by 2.5, first, to make the coefficient of the X2 Term equal to 1.

(2.5) 0.4×2 – (2.5) 100x + (2.5) 6,000 = 0 1×2 – 250x + 15,000 = 0 (X – 100)( X – 150) = 0

When X = 100 or X = 150, the profit is $2,000. At either level, the difference between the revenue and cost is $2,000.

Mixing It Up with Mixture Problems

Mixture problems are covered in great detail in Chapter 14. You mix so many quarts of one substance with quarts of another substance. Some rather interesting solutions or mixture problems are made possible by introducing the second equation and solving the system.

Gassing up at the station

It’s no easy choice when you pull up at the gas station to fill up your vehicle. First, you have to take a deep breath about the price, and then you have to choose between regular gas, premium gas, or even gas that contains ethanol.

The Problem: Stefanie finds that she gets 19 miles per gallon with regular gas that costs $2.70 per gallon and 23 miles per gallon with the premium gas that costs $3.15 per gallon. She paid a total of $104.40 for gas on a trip of 748 miles. How many gallons of each type gas did she buy?

^VLA* One of the equations you need deals with cost, and the other deals with the

Number of miles. The common element in both equations is the number of gallons of each type of gas — and the number of gallons answers the question, too. Let R Represent the number of gallons of regular gas and P Represent the number of gallons of premium gas. The total cost, $104.40 = $2.70r + $3.15p. The total number of miles, 748 = 19r + 23p. None of the coefficients of the variables is equal to 1, so you have to make a choice as to which variable to solve for. Because the coefficient 19 is the smallest number, I opt to solve for R In the second equation and replace the R In the first equation with that equivalence in terms of P.

19r + 23p = 748

19r = 748 – 23p

R = 748 – 23 P r 19 19 P 2.70r + 3.15p = 104.40

2.70 (- f| P) + 3.15p = 104.40

Distributing the 2.70 and simplifying don’t make for very pretty computations, but a calculator makes short work of all the operations. I choose to find a common denominator to combine the fractions and decimals, because the decimal you get with a denominator of 19 just keeps repeating. Then you solve for P By multiplying each side of the equation by the reciprocal of its coefficient.

2,019.6 62.1 ±,1C ,r, AA(,

1A—TjT – p + 3.15p = 104.40

19 19

62.1 Pp 2,019.6

—1g – P + 3.15p = 104.40–—

- 621 + 315 = 10,440 – 2,019.6 19 P + 100 P = 100 19 - 225 P = - 3,600 1,900 P 1,900

- 3,600 1,900

1,900 – 225

Stefanie used 16 gallons of premium gas. Substitute the 16 for P In the equation 19r + 23p = 748, and you get 19r + 23(16) = 748. This simplifies to 19r + 368 = 748. Subtracting 368 from each side, the equation becomes 19r = 380. Dividing each side by 19, you get that R = 20. She bought 20 gallons of regular gas.

Backtracking for all the answers

Problems involving mixtures have the upfront answers of how many gallons of this or how many items of that. Applications of these problems often involve more than just the numbers. In a gallons-of-gasoline problem, you may want to know the average cost per gallon for the trip. In a fund-raising project, you probably want to know something about the total profit.

The Problem: A service group is selling candy bars and bags of almonds for a fund-raiser. They’re selling the candy bars (which cost them 40 cents each) for $1 and the bags of almonds (which cost them 50<t each) for $1.25. Their total receipts (revenue) for the sale of 1,350 items was $1,500. How many of each item did they sell, and what was their profit?

A^iAfl Write an equation about the total receipts by multiplying the amount charged R By the number of each type of item. Write an equation about the total number

S^>yf ) Of items by adding the number of each type together and setting it equal to ^Mj^’ 1,350. Letting the number of candy bars be represented by C And the number of bags of almonds be represented by A, The two equations are: $1c + $1.25a = $1,500 and C + A = 1,350. Solve for C In the second equation and substitute the equivalent into the first equation.

C + A = 1,350 C = 1,350 – A 1.00 (1350 – A) + 1.25a = 1,500

Now simplify the equation, subtract 1,350 from each side, and solve for A By dividing each side by the coefficient.

Cows and hens

In a recent Ask Marilyn Column in the Sunday hens — had a total of 74 feet, then how many Parade magazine, the following question was cows and how many hens were in the barnyard? posed: If a barnyard of 30 animals — cows and

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1,350 – a + 1.25a = 1,500 1,350 + 0.25a = 1,500 0.25A = 150 A = 600

The group sold 600 bags of almonds. That means that 1,350 – 600 = 750 candy bars were sold. To figure out the cost for these items, multiply 750 times 40<t and 600 times 50<t. Their cost was 750($0.40) + 600($0.50) = $300 + $300 = $600. If the total receipts were $1,500, then subtract the $600 from $1,500 to get $900 profit.

Making Several Comparisons with More Than Two Equations

Some situations have more than two or three things acting on an outcome. The type of gasoline used may involve not only the cost of the fuel and the mileage the car gets on the fuel, but it also may involve the availability of the fuel or the speed at which the pump flows. When a problem involves three or more different influences, a system of equations involving three or more equations is used.

Picking flowers for a bouquet

Florists make up bouquets by mixing complementary colors and types of flowers and greenery. The cost of the bouquet depends on the different types of flowers and materials used.

The Problem: A bouquet is to contain 24 flowers, made up of roses, carnations, and daisies. The person buying the bouquet wants twice as many carnations as roses and doesn’t want to spend more than $8.20. If roses cost 50<t each, carnations 25<t each, and daisies 40<t each, then what composition of flowers should be used to match that maximum price?

Write three different equations: one involving the total number of flowers, one involving the total cost for the bouquet, and one involving the relationship between the number of carnations and roses. Letting the number of roses be represented by r, the number of carnations represented by C, And the number of daisies by D, The three equations are: R + C + D = 24, 0.50r + 0.25c + 0.40 D = 8.20, and C = 2r. You can hone this system of three equations down to two equations by substituting 2R For C In each of the first two equations. Then you have the two equations:

R + 2r + D = 24

0.50r + 0.25 (2r) + 0.40d = 8.20

Simplifying, the two equations become:

3r + D = 24

1r + 0.40d = 8.20

Solve for D In the first equation and substitute the equivalence into the second equation.

D=24 -3R

1r + 0.40 (24 – 3r) = 8.20 R + 9.6 – 1.20r = 8.20 9.6 – 0.20r = 8.20

-0.20R=-1.40 – 1.40

0.20

7

The number of roses is 7, so twice as many carnations is 14. That leaves room for 3 daisies.

Coming up with a game plan for solving systems of equations

In the example in the preceding section, one of the equations in the system of equations had a convenient relationship. The equation C = 2R Allowed for a quick substitution back into the other two equations, creating a nice system

Of equations that solved by substitution. When you aren’t quite so lucky to have a problem with such a nice feature to it, you need a more general game plan for solving.

When solving a system of three or more linear equations, the following guidelines will help with the solution:

Select variables to represent the different amounts or units.

Create a table or chart with the variables and their descriptions down one side and the different relationships (prices, numbers, and so on) across the top.

Write equations using the variables and relationships from the table.

Solve for one of the variables in one of the equations and substitute back into the other equations. Repeat the process until you have one equation with just one variable.

Graphing calculators and spreadsheets are very useful when doing problems involving a large number of variables. The following problem shows you something quite manageable using the substitution method.

The Problem: Sherrill bought 40 dozen cookies for $350. The chocolate-chip cookies cost $10 per dozen; the oatmeal-raisin cookies cost $9 per dozen; and the peanut-butter cookies and snickerdoodles each cost $8 per dozen. Sherrill got two more dozen chocolate-chip cookies than peanut-butter cookies, and she bought twice as many snickerdoodles as oatmeal-raisin cookies. How many dozen of each type did she buy?

This problem is solved by writing four different equations. Let C Represent how many dozen chocolate-chip cookies, R Represent the number of dozen of oatmeal-raisin cookies, P The number of dozen of peanut-butter cookies, and D The number of dozen snickerdoodles. Make a table representing the different relationships (see Table 17-2).

Now write equations using the columns of the table. In the "Dozen Of" column, you add up the dozens and let the sum be 40, giving you:

C + R + P + D = 40

Next, write an equation multiplying the prices times their respective types and having the total be $350:

10C + 9R + 8P + 8D = 350

Next, write the relationship between the dozens of chocolate-chip and peanut-butter cookies: C = P + 2. And finally, write the relationship between the dozens of oatmeal-raisin and snickerdoodles: D = 2r. The following shows the system of equations written in Rectangular Form, with all the same variables lined up under one another. This is the form you’d use in a calculator, replacing all the blank spots with zeros.

C + r + p + d = 40 10c + 9r + 8p + 8d = 350 C — p = 2 2r — D = 0

Using substitution, you replace the ds in the first two equations with 2r, and you replace C In the first two equations with P + 2.

_ P + 2) + R + p + 2R = 40 10 _ P + 2) + 9r + 8p + 8 (2r) = 350

Then simplify the two equations that now have only the variables P And R In them.

2p + 3r = 38 18p + 25r = 330

Solve for P In the first equation and substitute the equivalence into the second equation. Then solve the equation for the value of P.

2P=38—3R

P = 19 — 2 R

18 (19 — 2 R J + 25r = 330 342 — 27r + 25r = 330

—2R =—12

R = 6

Drawing money out of a bag

A bag contains ten $5 bills, ten $10 bills and ten He can keep all the money drawn from the bag $20 bills. Jack is blindfolded and told that he until he has to stop. What is the greatest amount may draw bills out of the bag, one at a time, until of money Jack may draw from the bag? he has three of the same denomination of bill.

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Sherrill bought 6 dozen oatmeal-raisin cookies. She bought twice as many (or 12 dozen) snickerdoodles. Replacing R With 6 in the equation 2p + 3r = 38, you get 2P + 18 = 38, which becomes 2P = 20, or P = 10. She bought 10 dozen peanut-butter cookies. And, if she bought two more dozen chocolate chip than peanut butter, she bought 12 dozen chocolate-chip cookies.

Solving Systems of Quadratic Equations

A quadratic equation occurs when you have one or more terms containing a variable raised to the second power. For example, the equation Y = 3k1 + 2x is a quadratic equation. Solving systems of equations involving quadratic equations leads to some interesting results. You usually end up with more than one solution to the system. Sometimes more than one solution means that there’s more than one answer. Often, though, it means that one of the answers fits the situation and the other is just Extraneous — it satisfies the system of equations, but it doesn’t really have any meaning in the practical problem.

Counting on number problems

A common type of problem involving systems of equations is a Number problem, Where you solve for a certain number that has a particular relationship to one or more other numbers. You incorporate operations such as addition and multiplication, and you often use squares of numbers or their reciprocals.

The Problem: The sum of two numbers is 40, and the product of those same two numbers is 204. What are the numbers?

^VLA* Let the two numbers be represented by the variables X And Y. Writing that

Their sum is 40, the equation is X + Y = 40. You write that their product is 204 with Xy = 204. Solve for Y In the first equation to get Y = 40 – X. Now replace the Y In the second equation with 40 – X, Simplify the equation by setting it equal to 0, and solve the quadratic equation by factoring.

X (40 – X) = 204 40x – X2 = 204

0 = X2 — 40x + 204 0 = (X — 34)( X — 6)

You get that X Is either 34 or 6. These are the two numbers whose sum is 40 and whose product is 204.

The Problem: The difference between two positive numbers is 20. If you square the larger number and subtract ten times the smaller number from the square, you get 575. What are the two numbers?

^VLA* Let the two numbers be represented by X And Y. Write their difference as

X - Y = 20. Squaring the larger number and subtracting ten times the smaller, you get X - 10y = 575. Why does X Have to be the larger number? It’s because the two numbers are positive, and the result of subtracting Y From X Has to be a positive 20.

Now, solve for X In the first equation to get X = 20 + Y. Substitute this into the second equation and solve for Y.

(20 + Y ) — 10y = 575 400 + 40y + Y2 — 10y = 575 Y2 + 30y — 175 = 0 (y — 5)( Y + 35) = 0

The two solutions of the equation are Y = 5 and Y = -35. You discard the Y = -35, because the problem asks for positive numbers. If Y = 5, then X = 20 + 5 = 25. The two numbers are 25 and 5.

The Problem: One integer is four smaller than another. The sum of their

Reciprocals is J5,-. What are the numbers? 24

AVLA/V Let the two numbers be represented by X And X - 4. Writing that the sum of the

Reciprocals of these two numbers is equal to 24, you have: X +–1~n = t^-.

X x— 4 24

Technically, this is a system-of-equations problem, because you could write that the second number, Y, Has the relationship Y = X - 4. The substitution is done here almost automatically.

Now multiply each term by the common denominator, 24x(x – 4), simplify, and solve the quadratic equation.

I X 24x(X — 4) + —^ x 24x (x—4) = 4 # 24x (X — 4)

24x — 96 + 24x = 5×2 — 20x

0 = 5×2 — 68x + 96 0 = (5x — 8)( X — 12)

The solutions of the quadratic equation are 5 and 12. The first solution isn’t

An integer, so discard it as being extraneous. However, when X = 12, you have an integer and another that’s four smaller, which is 8. And, when you add their reciprocals, you get

1 + J_ = _2_ + J3_ = _5_ 12 8 24 24 24

Picking points on circles

The equation of a circle with its center at the origin is x2 + Y2 = R2, Where R Is the radius of the circle. It isn’t always easy to pick nice numbers — integers or fractions — that are coordinates of the points on a circle. For example, if a circle has a radius of 5, you can use 3s and 4s for the XS and YS. The points (3,4), (-3,4), (-4,-3) and so on work. So do combinations of 0s and 5s. The number of Nice Coordinates is limited.

You can find an infinite number of nice Fractions That work for the coordinates of the points on a Unit circle (a circle with a radius of 1).

• /1 —M2 2m

Let M Be any rational number, and the ordered pair I -z-2, + -z-2

*u • 1 U + M2 1 + m2

Point on the unit circle.

Using the formula to find a few points, let M = 2, for example. Then

1 — 22 2 • 2 \ /-3 4\ . . . r.

1 + 22, + 1 + 22 J = \ ~5T, + 5) are on the unit circle and satisfy the equation

X + Y2 = 1. The ± symbol just means that you can let the y-coordinate be either positive or negative — you get two points for the input value.

The formula for these points on the unit circle comes from solving a system of equations.

The Problem: Find the coordinates of the points on the unit circle with its center at the origin that are shared with any line going through the point (-1,0) on that circle. Distinguish between the different lines that go through (-1,0) by letting M Represent the slopes of the different lines.

Find the common solution of the circle, X2 + Y2 = 1 and the line Y = Mx + B. The particular line that goes through the point (-1,0), has an equation that can be written a little more specially. Finding the slope of this line, M, Using the slope

Y— 0 Y

Formula and the points (x, y) and (-1,0), you get M = ——-,—rv = —tt. Solving

F V Tjj V & X — (—1) X + 1 &

This equation for Y, You get that the equation of the line is Y = M(x + 1).

Y

M = ———r X + 1

M (x + 1) = Y

Using this new form for the equation, substitute the M(x + 1) for Y In the equation of the circle.

X2 + (m (x + 1)j2 = 1 X2 + M 2( x2 + 2x + 1) = 1 X2 + M2 X2 + 2m2 X + m2 — 1 = 0 (1 + m2 )x2 + 2m2 X + (m2 — 1) = 0

The equation you end up with after substitution is a quadratic equation of the form Ax2 + Bx + C = 0 where the A = 1 + M2, b = 2M2, And C = M - 1. Use the quadratic formula to solve for X.

2m2 +/(2m2)2 — 4 (1 + M2)(m2 — 1)

2 (1 + M2) 2m2 +7 4m4 — 4m4 + 4

2 (1 + m2)

= — 2m2 + J4 = — 2m2 + 2 = — m2 + 1 2 (1 + m2) 2 (1 + m2) 1 + m2

You only need the + part of the ± portion of the formula, because the – part gives you the point (-1,0), which you already have. So the x-coordinate of a

Point on the graph of the unit circle is X = —Mm++~\L. You get the y-coordinate

By putting this X-coordinate into the equation for the unit circle and solving for Y. I’ll leave that little goodie for you to do.

X=

Part IV