More complex examples of equations. Scheme for solving simple linear equations

How to learn to solve simple and complex equations

Dear parents!

Without basic mathematical training, education is impossible modern man. At school, mathematics serves as a supporting subject for many related disciplines. In post-school life, it becomes a real necessity continuing education, which requires basic school-wide training, including mathematical.

AT primary school not only knowledge is laid on the main topics, but also develops logical thinking, imagination and spatial representations, as well as an interest in this subject.

Observing the principle of continuity, we will focus on the most important topic, namely "The relationship of action components in solving compound equations."

By using this lesson you can easily learn how to solve complicated equations. In this lesson, you will get to know step by step instructions solutions of complicated equations.

Many parents are baffled by the question - how to get children to learn how to solve simple and complex equations. If the equations are simple - this is still half the trouble, but there are also complex ones - for example, integral ones. By the way, for information, there are also such equations, over the solution of which they are struggling the best minds of our planet and for the solution of which very significant cash prizes are issued. For example, if you rememberPerelmanand an unclaimed cash bonus of several million.

However, let's return to the beginning to simple mathematical equations and repeat the types of equations and the names of the components. Little warm-up:

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WARM-UP

Find the extra number in each column:

2) What word is missing in each column?

3) Match the words from the first column with the words from the 2nd column.

"Equation" "Equality"

4) How do you explain what “equality” is?

5) And the "equation"? Is it equality? What is special about it?

term sum

reduced difference

subtrahend product

factorequality

dividend

the equation

Conclusion: An equation is an equality with a variable whose value must be found.

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I suggest that each group write the equation on a piece of paper with a felt-tip pen: (on the board)

One of the group should read their equation in mathematical language and comment on their solution, i.e., pronounce the operation being performed with known action components (algorithm).

Conclusion: We are able to solve simple equations of all kinds according to the algorithm, read and write literal expressions.

I propose to solve a problem in which a new type of equations appears.

Conclusion: We got acquainted with the solution of equations, one of the parts of which contains a numerical expression, the value of which must be found and a simple equation obtained.

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Consider another version of the equation, the solution of which reduces to solving the chain simple equations. Here is one of the introduction of compound equations.

I suggest everyone write down a sentence in mathematical language:

The product of the difference between the numbers x and 4 and the number 3 is 15.

CONCLUSION: problem situation motivates the goal of the lesson: to learn how to solve equations in which the unknown component is an expression. Such equations are compound equations.

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Or maybe the already studied types of equations will help us? (algorithms)

Which of the known equations is similar to our equation? X * a = in

VERY IMPORTANT QUESTION: What is the expression on the left side - sum, difference, product or quotient?

(x - 4) * 3 = 15 (product)

Why? (because the last action is multiplication)

Conclusion:Such equations have not yet been considered. But we can decide if the expressionx - 4superimpose a card (y - y), and you get an equation that can be easily solved using a simple algorithm for finding an unknown component.

When solving compound equations, it is necessary at each step to select an action at an automated level, commenting, naming the components of the action.

(y - 5) * 4 = 28
y - 5 = 28: 4
y - 5 = 7
y = 5 +7
y = 12
(12 - 5) * 4 = 28
28 = 28 (and)

Conclusion:In classes with different backgrounds, this work can be organized in different ways. In more prepared classes, even for the primary fixation, expressions can be used in which not two, but three or more actions, but their solution requires more steps with each step simplifying the equation until a simple equation is obtained. And each time you can observe how the unknown component of actions changes.

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CONCLUSION:

When it comes to something very simple, understandable, we often say: "The matter is clear, as two times two - four!".

But before you think of the fact that two times two is four, people had to study for many, many thousands of years.

Many rules from school textbooks of arithmetic and geometry were known to the ancient Greeks more than two thousand years ago.

Wherever you need to count, measure, compare something, you can’t do without mathematics.

It is hard to imagine how people would live if they did not know how to count, measure, compare. Mathematics teaches this.

Today you have plunged into school life, have been in the role of students and I suggest you, dear parents, evaluate your skills on a scale.

You sit in a restaurant and flip through the menu. All dishes look so delicious that you don't know what to choose. Maybe order them all?

Surely you have encountered such problems. If not in food, then in something else. We spend great amount time and energy to make a choice between equally attractive options. But, on the other hand, the options cannot be the same, because each of them is attractive in its own way.

Once you make a choice, you are faced with a new choice. This is an endless series of important decisions, which are the fear of making the wrong choice. These three methods will help you make better decisions at all levels of your life.

Make habits to avoid everyday decisions

The point is, if you get in the habit of eating salad for lunch, you won't have to decide what to order at a coffee shop.

By developing habits that deal with such simple everyday tasks, you save energy for making more complex and important decisions. In addition, if you get into the habit of eating salad for breakfast, you won't have to waste your willpower not to eat something fatty and fried instead of a salad.

But this applies to predictable cases. What about unexpected decisions?

"If - then": a method for unpredictable decisions

For example, someone constantly interrupts your speech and you are not sure how to react to this and whether to react at all. According to the if-then method, you decide: if he interrupts you two more times, then you will make him a polite remark, and if this does not work, then in a more rude form.

These two methods help most decisions that confront us every day. But when it comes to questions strategic planning For example, how to respond to the threat of competitors, which products to invest more in, where to cut the budget, they are powerless.

These are decisions that can be delayed for a week, a month or even a year, hindering the development of the company. They cannot be dealt with through habit, and the if-then method will not work here either. As a rule, there are no clear and correct answers to such questions.

Often the leadership team delays the adoption of such decisions. He collects information, weighs the pros and cons, continues to wait and observe the situation, hoping that something will appear that will point to the right decision.

And if we assume that there is no right answer, will this help to make a decision quickly?

Imagine that you need to make a decision in the next 15 minutes. Not tomorrow, not next week, when you gather enough information, and not in a month, when you talk to everyone involved in the problem.

You have a quarter of an hour to make a decision. Take action.

This is the third way, which helps to take complex decisions concerning long-term planning.

Use the time

If you've researched a problem and found that the options for solving it are equally attractive, accept that there is no right answer, set yourself a time limit, and just choose any option. If checking one of the solutions requires minimum investment, choose it and check it out. But if this is not possible, then choose any and as soon as possible: the time you spend on useless thoughts can be better used.

Of course, you may disagree: "If I wait, the correct answer may appear." Maybe, but, firstly, you are wasting valuable time waiting for the situation to be clarified. Secondly, waiting causes you to procrastinate and put off other decisions related to it, reduces productivity and slows down the company's development.

Try it right now. If you have a question that you have been putting off for a long time, give yourself three minutes and do it. If you have too many similar ones, write a list and set a time for each solution.

You'll see, with each decision you will feel a little better, anxiety will decrease, you will feel that you are moving forward.

So, you choose a light salad. Was it the right choice? Who knows... At least you ate and didn't sit hungry over the menu of dishes.

Scientists have studied the rhythms of brain activity and identified the one that is best suited for creative insight and search useful ideas

Scientists have studied the rhythms of brain activity and identified the one that is best suited for creative insight and the search for useful ideas.

There is. Sleep. Solve problems. Repeat. Chances are, apart from a night's sleep, you spend most of your time solving various problems - especially at work.

Not that it was bad. Many of the best entrepreneurs world, from Sarah Blakely to Richard Branson, owe their success to the ability to detect problems (in this case, unmet consumer needs) and come up with solutions.

But whichever important part our life hasn't been about solving problems, it's still stress, and some people seem to deal with it better than others.

Therefore, for those who want to become more successful in this game, you can try something new: seek solutions in a dream. Literally. It is called "Catch Your Theta Rhythm". No, it's not about self-hypnosis or meditation: it's pure science and it works.

But let's first understand:

What are brain rhythms?

As professor Ned Herrmann explains, this is rhythms that control the electrical activity of the brain. Depending on your activity level four different rhythms can be distinguished. We list them in order of decreasing wave frequency.

• During periods of maximum activity (for example, during an important job interview), your brain works in beta rhythm.
• When you are relaxed - for example, just finished big project and you can finally exhale - the brain switches to alpha rhythm.
• Now let's jump ahead: the fourth rhythm is denoted by the letter "delta" and is fixed when you are in deep sleep.

We skipped the third stage, the theta rhythm, because it is the one that is best suited for problem solving. Herrmann says:

“People who spend a lot of time behind the wheel often come up with good ideas during these periods when they are in theta rhythm ... This can happen in the shower or bath, and even while shaving or combing your hair. This is the state in which problem solving becomes so automatic that you can mentally disengage from it. With theta rhythm, it often seems that the flow of thoughts is not limited by anything - neither by internal censorship, nor by guilt.

The brain comes into this state, including during falling asleep or waking up, when you are balancing between wakefulness and deep sleep. Herrmann explains:

“During awakening, the brain can maintain the theta rhythm for a long period, say 5 to 15 minutes, and this time can be used to freely reflect on yesterday’s events or what has to be done in the new day. This period can be very productive and bring many meaningful and creative ideas.”

Is there any real evidence that this works?

Catch the moment when your brain is ready to give you best ideas, - technique, which successful people have been going on for hundreds of years.

Artists, writers and great thinkers have long noticed that those moments when we "nod" - that is, exactly when the theta rhythm prevails in the brain - best time to awaken creativity.

Albert Einstein and Thomas Edison had a habit of solving complex problems while half asleep. A quick, creative mind is built to solve problems, which is why even a brief reflection on the tasks of the new day early in the morning while you are still in this state (or even at night when you start to fall asleep) can bring amazing results. What worked for Einstein might work for you - though we don't promise you'll be an author. new theory relativity.

How to use your theta rhythm?

It will take some time. But if you turn to this practice regularly, you will have good habit which will boost your productivity by new level. Here's what you need for this:

1. Choose a task

In the morning, when you have already begun to wake up, but your eyes are still closed, and your brain is still half asleep, think about the most pressing problem or task that you will have to face today. Maybe it will be a tricky conversation, important negotiations with a client, writing a report, or developing a new marketing campaign. But no matter how many tasks hover in your mind, you must choose one - and let your brain work on it.

Do not try to direct or limit your thoughts in any way, just make sure that they do not go too far from given topic. Most likely, your brain will unconsciously begin to pick up a solution.

Often you will get a couple of useful ideas as a result. Sometimes - even a brilliant insight. Most likely, at first you will forget to use this method every day, but over time it will become another habit, part of your morning rituals.

2. Take notes

Perhaps the most frustrating part of problem solving with Theta Rhythm will be that you will forget these inspired ideas as soon as your head leaves the pillow. You will torment your brain in the shower, trying to extract from it the brilliant three-point plan that you just mentally sketched out. This is why you should write down your decisions as soon as you are awake enough to open your eyes.

Grab your smartphone (it's still charging at the head, isn't it?) and immediately record your thoughts - in text or on a voice recorder. Don't waste time. Limit yourself to keywords, descriptions, and phrases that will kick-start your memory later when you're ready to use the information.

An added benefit: The blue light on your phone's screen will help you wake up. And if you want to resort to the same method in the evening, in the process of falling asleep, it is better to use a pen and paper - so artificial light will not disturb your sleep.

3. Analyze experience

Keep a journal of your "theta thoughts" - over time, this will help you find typical solutions and their areas of application. You may find that this method is most effective for you at solving creative problems, or you may find that it gives you an advantage in dealing with people or planning. This will help you understand what tasks should be solved using theta rhythm in the future.

Inspiration can come from anywhere.

But the same is true for obstacles.

Theta Thinking uses the brain's universal ability to solve problems so that you can remember those solutions and use them. Often it helps to get around another obstacle in the way or bridge the gap between a half-baked idea and a really useful solution, and why not take advantage of this? You don't even have to get out of bed to do this! published

There are moments in life when a seemingly hopeless situation appears before you - or a problem, any solution of which promises to be not in your favor. Do not rush to give up on the realization of your dreams, achieve your goal, or panic. One wise man of antiquity said: "Choose time to think - this is the source of strength." Well, it's hard to disagree with him, because the mind - powerful weapon. Even the most complex problem has dozens of solutions, and it is only out of sight because people are used to thinking in certain frames. To solve a complex problem, it is necessary to coordinate the work of consciousness and subconsciousness - this will expand your "horizon" and allow you to see new opportunities.

Technique "100 ideas"

To master the 100 Ideas technique, you will need only 1-2 hours of free time, a comfortable personal corner where no one will disturb you, as well as paper and a pencil. Ask relatives and friends in advance not to involve you during the “meditation”, turn off the phone and just relax. At the top of a piece of paper, formulate and write down your question or dilemma. Number the list from one to 100 and start generating ideas.

At first, ideas come one after another, although, alas, they are not new - you will describe all your "trump cards", including skills, acquaintances, connections, financial resources, time that you can devote to solving the problem. Then it will still seem incredible to find a hundred answers, and if you stop at 20-30 points, you will feel empty. A small hitch awaits you, naturally formed when the consciousness, walking in a vicious circle, has exhausted the options available to it and went through everything that it has already encountered in personal experience.

The second phase of your journey to your subconscious is another 40 points where you are still using your consciousness but your hidden powers start to wake up and open a second wind. At this stage, the image of your thinking emerges. You will notice that your ideas start to repeat themselves, and there are all sorts of clichés and attitudes in them. Your goal is not to dismiss them, but to carefully write them down on paper, and here's why: it is these stamps that are the frames that you cannot go beyond and look around. It could be public opinion, dissatisfaction with the authorities, lack of self-confidence and any other “burrs” in your psyche. At the same time, you can discover your hidden problems or fears that prevent you from moving forward. This stage will require the greatest endurance from you - after all, it is not at all easy to brush aside the first thirty points that are clearly in your comfort zone and take on new, unknown and therefore sometimes frightening ideas - this is normal, the main thing is not to give up. In addition, this internal struggle only helps to move on to the third phase of the journey.

It is the last 30 points that will open Pandora's box in front of you, because the number 100 was not chosen by chance. It is it that allows your intuition to fully open up and surprise yourself with unexpected “insights from above” - impromptu of your awakening subconscious, from where ideas appear without any processing and filtering by the mind. In your search, you have already abandoned logic, noticing how square it really is, and you understand that your way of thinking lay only in one plane - and the world, it turns out, is three-dimensional (not counting time). Now, when the mind stops dictating to you what is “possible” and what is “not”, the door to the subconscious is thrown open. You can easily invent something out of the ordinary and at first glance completely absurd. It may even seem to you that you should not write down an idea that is obviously inappropriate for you, it is not clear what images appeared in your head. However, it is precisely strange, sometimes stupid phrases that can turn out to be unrefined diamonds. Remember how people thought the Earth was flat and were afraid to fall off its edge, and how the idea that the planet was round and spinning was once called heresy. Crazy ideas may not be clear to you at first, but you will feel that there is something in them - this will serve as a straw that will tell you the right direction.

It can also happen that, after laying out so many ideas, you suddenly realize that this was not a problem at all - or you saw only the tip of the iceberg, so you need to make a new list to answer a completely different question.

There are a few more rules that must be observed when working with this technique. First of all, the list must be compiled in one go, without interruptions - otherwise your dormant brilliant ideas will remain dormant under the weight of everyday thinking. While working, do not reread the list and evaluate how much has already been done and how many points are left - this will distract you, and prevent your thoughts from repeating naturally - and therefore, will not allow you to see your own stumbling blocks. Tune in right away: you will evaluate and criticize your ideas after drawing up all the hundreds of points - and while the process is underway, you need to write down any thoughts (after all, you are not obliged to show this paper to anyone if you do not want to). If the work is in full swing, shorten the words, the main thing is that you can then read what you meant. You can, of course, use a laptop instead of a pencil and paper, but remember: the source electromagnetic waves, at least theoretically, prevents your brain, aura and, if you like, chakras from connecting to the universal mind - and in general it is great to function. But this is at personal discretion.

The “tasty” bonuses of the “100 Ideas” technique are not only in the possibility of deep introspection and finding original solutions to their difficult situations, but also in the fact that with it you can develop diversified and plan your future, find new incentives for self-development and grow above yourself . To do this, at your leisure, reflect on the answers to the following (and any of your own) topics:

• How to educate yourself
• How to improve relationships
• How to improve your life
• How to make money
• How to improve business
• How to help people
• How to increase personal effectiveness
• How to become healthier
• Things I keep putting off until tomorrow
• The things I do best
• Things that demotivate me
• Qualities I want to develop in myself
• Questions I need to find answers to
• Values ​​I believe in
• Things I appreciate in life
• Professions in which I want to try myself
• Things (people) that slow me down in achieving my goal
• Things that cheer me up
• Conclusions that life has taught me
• Things to get rid of
• Places I would like to visit
• Mistakes for which I forgive myself (others)
• Ways to think more creatively

52. More complex examples equations.
Example 1 .

5 / (x - 1) - 3 / (x + 1) \u003d 15 / (x 2 - 1)

The common denominator is x 2 - 1, since x 2 - 1 \u003d (x + 1) (x - 1). Multiply both sides of this equation by x 2 - 1. We get:

or, after reduction,

5(x + 1) - 3(x - 1) = 15

5x + 5 – 3x + 3 = 15

2x=7 and x=3½

Consider another equation:

5 / (x-1) - 3 / (x + 1) \u003d 4 (x 2 - 1)

Solving as above, we get:

5(x + 1) - 3(x - 1) = 4
5x + 5 - 3x - 3 = 4 or 2x = 2 and x = 1.

Let's see if our equalities are justified if we replace x in each of the considered equations with the found number.

For the first example, we get:

We see that there is no room for any doubts here: we have found such a number for x that the required equality is justified.

For the second example, we get:

5/(1-1) - 3/2 = 15/(1-1) or 5/0 - 3/2 = 15/0

Doubts arise here: we meet here with division by zero, which is impossible. If in the future we manage to give a certain, albeit indirect, meaning to this division, then we can agree that the found solution x - 1 satisfies our equation. Until then, we must admit that our equation does not have a solution at all that has a direct meaning.

Such cases can occur when the unknown is somehow included in the denominators of the fractions in the equation, and some of these denominators, when the solution is found, vanish.

Example 2 .

You can immediately see that this equation has the form of a proportion: the ratio of the number x + 3 to the number x - 1 is equal to the ratio of the number 2x + 3 to the number 2x - 2. Let someone, in view of this circumstance, decide to apply here to free the equation from fractions are the main property of proportion (the product of the extreme terms is equal to the product of the averages). Then he will get:

(x + 3) (2x - 2) = (2x + 3) (x - 1)

2x 2 + 6x - 2x - 6 = 2x 2 + 3x - 2x - 3.

Here it may raise fears that we will not cope with this equation, the fact that the equation includes terms with x 2 . However, we can subtract 2x 2 from both sides of the equation - this will not break the equation; then the members with x 2 will be destroyed, and we get:

6x - 2x - 6 = 3x - 2x - 3

Let's move the unknown terms to the left, the known ones to the right - we get:

3x=3 or x=1

Remembering this equation

(x + 3)/(x - 1) = (2x + 3)/(2x - 2)

we will immediately notice that the found value for x (x = 1) vanishes the denominators of each fraction; we must abandon such a solution until we have considered the question of division by zero.

If we also note that the application of the property of proportion has complicated matters and that a simpler equation could be obtained by multiplying both parts of the given by a common denominator, namely by 2(x - 1) - after all, 2x - 2 = 2 (x - 1) , then we get:

2(x + 3) = 2x - 3 or 2x + 6 = 2x - 3 or 6 = -3,

which is impossible.

This circumstance indicates that this equation does not have solutions that have a direct meaning, which would not turn the denominators of this equation to zero.
Let's solve the equation now:

(3x + 5)/(x - 1) = (2x + 18)/(2x - 2)

We multiply both parts of the equation 2(x - 1), i.e. by a common denominator, we get:

6x + 10 = 2x + 18

The found solution does not nullify the denominator and has a direct meaning:

or 11 = 11

If someone, instead of multiplying both parts by 2(x - 1), would use the property of proportion, he would get:

(3x + 5)(2x - 2) = (2x + 18)(x - 1) or
6x 2 + 4x - 10 = 2x 2 + 16x - 18.

Here already the terms with x 2 would not be annihilated. By transferring all unknown terms to the left side, and known ones to the right, we would get

4x 2 - 12x = -8

x 2 - 3x = -2

We cannot solve this equation now. In the future, we will learn how to solve such equations and find two solutions for it: 1) we can take x = 2 and 2) we can take x = 1. It is easy to check both solutions:

1) 2 2 - 3 2 = -2 and 2) 1 2 - 3 1 = -2

If we remember the initial equation

(3x + 5) / (x - 1) = (2x + 18) / (2x - 2),

we will see that now we get both of its solutions: 1) x = 2 is the solution that has a direct meaning and does not turn the denominator to zero, 2) x = 1 is the solution that turns the denominator to zero and does not have a direct meaning .

Example 3 .

Let's find the common denominator of the fractions included in this equation, for which we factorize each of the denominators:

1) x 2 - 5x + 6 \u003d x 2 - 3x - 2x + 6 \u003d x (x - 3) - 2 (x - 3) \u003d (x - 3) (x - 2),

2) x 2 - x - 2 \u003d x 2 - 2x + x - 2 \u003d x (x - 2) + (x - 2) \u003d (x - 2) (x + 1),

3) x 2 - 2x - 3 \u003d x 2 - 3x + x - 3 \u003d x (x - 3) + (x - 3) \u003d (x - 3) (x + 1).

The common denominator is (x - 3)(x - 2)(x + 1).

Multiply both sides of this equation (and we can now rewrite it as:

to a common denominator (x - 3) (x - 2) (x + 1). Then, after reducing each fraction, we get:

3(x + 1) - 2(x - 3) = 2(x - 2) or
3x + 3 - 2x + 6 = 2x - 4.

From here we get:

–x = –13 and x = 13.

This solution has a direct meaning: it does not set any of the denominators to zero.

If we were to take the equation:

then, proceeding in exactly the same way as above, we would get

3(x + 1) - 2(x - 3) = x - 2

3x + 3 - 2x + 6 = x - 2

3x - 2x - x = -3 - 6 - 2,

where would you get

which is impossible. This circumstance shows that it is impossible to find a solution for the last equation that has a direct meaning.