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» Solving Equation 5. Online Equations. Multiplying and dividing equations

Solving Equation 5. Online Equations. Multiplying and dividing equations

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Biology– life science is one of the oldest sciences. Man has accumulated knowledge about living organisms over thousands of years. As knowledge accumulated, biology differentiated into independent sciences (botany, zoology, microbiology, genetics, etc.). The importance of border disciplines connecting biology with other sciences - physics, chemistry, mathematics, etc., is increasingly increasing. As a result of integration, biophysics, biochemistry, space biology, etc. arose.

Currently, biology is a complex science, formed as a result of the differentiation and integration of different disciplines.

Used in biology various methods research: observation, experiment, comparison, etc.

Biology studies living organisms. They are open biological systems that receive energy and nutrients from environment. Living organisms respond to external influences, contain all the information they need for development and reproduction, and are adapted to a specific habitat.

All living systems, regardless of the level of organization, have common features, and the systems themselves are in continuous interaction. Scientists distinguish the following levels of organization of living nature: molecular, cellular, organismal, population-species, ecosystem and biosphere.

Chapter 1. Molecular level

The molecular level can be called the initial, deepest level of organization of living things. Every living organism consists of molecules of organic substances - proteins, nucleic acids, carbohydrates, fats (lipids), called biological molecules. Biologists are investigating the role of these essential biological compounds in the growth and development of organisms, storage and transmission hereditary information, metabolism and energy conversion in living cells and in other processes.


In this chapter you will learn

What are biopolymers;

What structure do biomolecules have?

What functions do biomolecules perform?

What are viruses and what are their features?

§ 4. Molecular level: general characteristics

1. What is a chemical element?

2. What are called an atom and a molecule?

3. What organic substances do you know?


Any living system, no matter how complexly organized it may be, manifests itself at the level of functioning of biological macromolecules.

By studying living organisms, you learned that they consist of the same chemical elements, as inanimate. Currently, more than 100 elements are known, most of them are found in living organisms. The most common elements in living nature include carbon, oxygen, hydrogen and nitrogen. It is these elements that form molecules (compounds) of the so-called organic matter.

The basis of all organic compounds carbon serves. It can come into contact with many atoms and their groups, forming chains that differ in chemical composition, structure, length and shape. Molecules are formed from groups of atoms, and from the latter - more complex molecules that differ in structure and function. These organic compounds that make up the cells of living organisms are called biological polymers or biopolymers.

Polymer(from Greek policies- numerous) - a chain consisting of numerous links - monomers, each of which is relatively simple. A polymer molecule can consist of many thousands of interconnected monomers, which can be the same or different (Fig. 4).


Rice. 4. Scheme of the structure of monomers and polymers


The properties of biopolymers depend on the structure of their molecules: on the number and variety of monomer units that form the polymer. All of them are universal, since they are built according to the same plan for all living organisms, regardless of species.

Each type of biopolymer is characterized by a specific structure and function. Yes, molecules proteins are the main structural elements cells and regulate the processes occurring in them. Nucleic acids participate in the transfer of genetic (hereditary) information from cell to cell, from organism to organism. Carbohydrates And fats They are the most important sources of energy necessary for the life of organisms.

It is at the molecular level that the transformation of all types of energy and metabolism in the cell occurs. The mechanisms of these processes are also universal for all living organisms.

At the same time, it turned out that the diverse properties of biopolymers that make up all organisms are due to different combinations of just a few types of monomers, forming many variants of long polymer chains. This principle underlies the diversity of life on our planet.

The specific properties of biopolymers appear only in a living cell. Once isolated from cells, biopolymer molecules lose biological entity and are characterized only physical and chemical properties the class of compounds to which they belong.

Only by studying the molecular level can one understand how the processes of the origin and evolution of life on our planet proceeded, what are the molecular basis of heredity and metabolic processes in a living organism.

Continuity between the molecular level and the next cellular level is ensured by the fact that biological molecules are the material from which supramolecular - cellular - structures are formed.

Organic substances: proteins, nucleic acids, carbohydrates, fats (lipids). Biopolymers. Monomers

Questions

1. What processes do scientists study at the molecular level?

2. What elements predominate in the composition of living organisms?

3. Why are molecules of proteins, nucleic acids, carbohydrates and lipids considered as biopolymers only in the cell?

4. What is meant by the universality of biopolymer molecules?

5. How is the diversity of properties of biopolymers that make up living organisms achieved?

Tasks

What biological patterns can be formulated based on the analysis of the text of the paragraph? Discuss them with class members.

§ 5. Carbohydrates

1. What substances related to carbohydrates do you know?

2. What role do carbohydrates play in a living organism?

3. As a result of what process are carbohydrates formed in the cells of green plants?


Carbohydrates, or saccharides, is one of the main groups of organic compounds. They are part of the cells of all living organisms.

Carbohydrates are made up of carbon, hydrogen and oxygen. They received the name “carbohydrates” because most of them have the same ratio of hydrogen and oxygen in the molecule as in the water molecule. The general formula of carbohydrates is C n (H 2 0) m.

All carbohydrates are divided into simple, or monosaccharides, and complex, or polysaccharides(Fig. 5). From monosaccharides highest value for living organisms have ribose, deoxyribose, glucose, fructose, galactose.


Rice. 5. The structure of molecules of simple and complex carbohydrates


Di- And polysaccharides are formed by combining two or more monosaccharide molecules. So, sucrose(cane sugar), maltose(malt sugar), lactose(milk sugar) – disaccharides, formed as a result of the fusion of two monosaccharide molecules. Disaccharides are similar in properties to monosaccharides. For example, both horony are soluble in water and have a sweet taste.

Polysaccharides consist of a large number of monosaccharides. These include starch, glycogen, cellulose, chitin etc. (Fig. 6). With an increase in the number of monomers, the solubility of polysaccharides decreases and the sweet taste disappears.

The main function of carbohydrates is energy. During the breakdown and oxidation of carbohydrate molecules, energy is released (with the breakdown of 1 g of carbohydrates - 17.6 kJ), which ensures the vital functions of the body. When there is an excess of carbohydrates, they accumulate in the cell as reserve substances (starch, glycogen) and, if necessary, are used by the body as a source of energy. Increased breakdown of carbohydrates in cells can be observed, for example, during seed germination, intense muscle work, and prolonged fasting.

Carbohydrates are also used as building material . Thus, cellulose is an important structural component of the cell walls of many unicellular organisms, fungi and plants. Due to its special structure, cellulose is insoluble in water and has high strength. On average, 20-40% of the material in plant cell walls is cellulose, and cotton fibers are almost pure cellulose, which is why they are used to make textiles.


Rice. 6. Scheme of the structure of polysaccharides


Chitin is part of the cell walls of some protozoa and fungi; it is also found in certain groups of animals, such as arthropods, as an important component of their exoskeleton.

Complex polysaccharides are also known, consisting of two types of simple sugars, which regularly alternate in long chains. Such polysaccharides perform structural functions in the supporting tissues of animals. They are part of the intercellular substance of the skin, tendons, and cartilage, giving them strength and elasticity.

Some polysaccharides are part of cell membranes and serve as receptors, allowing cells to recognize each other and interact.

Carbohydrates, or saccharides. Monosaccharides. Disaccharides. Polysaccharides. Ribose. Deoxyribose. Glucose. Fructose. Galactose. Sucrose. Maltose. Lactose. Starch. Glycogen. Chitin

Questions

1. What composition and structure do carbohydrate molecules have?

2. What carbohydrates are called mono-, di- and polysaccharides?

3. What functions do carbohydrates perform in living organisms?

Tasks

Analyze Figure 6 “Structure diagram of polysaccharides” and the text of the paragraph. What assumptions can you make based on a comparison of the structural features of the molecules and the functions performed by starch, glycogen and cellulose in a living organism? Discuss this issue with your classmates.

§ 6. Lipids

1. What fat-like substances do you know?

2. What foods are rich in fat?

3. What is the role of fats in the body?


Lipids(from Greek lipos- fat) is a large group of fat-like substances that are insoluble in water. Most lipids consist of high molecular weight fatty acids and trihydric alcohol glycerol (Fig. 7).

Lipids are present in all cells without exception, performing specific biological functions.

Fats- the simplest and most widespread lipids - play an important role as energy source. When oxidized, they provide more than twice as much energy as carbohydrates (38.9 kJ when breaking down 1 g of fat).


Rice. 7. Structure of the triglyceride molecule


Fats are the main form lipid storage in a cage. In vertebrates, approximately half of the energy consumed by cells at rest comes from fat oxidation. Fats can also be used as a source of water (the oxidation of 1 g of fat produces more than 1 g of water). This is especially valuable for arctic and desert animals living in conditions of scarcity of free water.

Due to their low thermal conductivity, lipids perform protective functions, i.e. they serve for thermal insulation of organisms. For example, many vertebrates have a well-defined subcutaneous fat layer, which allows them to live in cold climates, and in cetaceans it also plays another role - it promotes buoyancy.

Lipids perform and construction function, since their insolubility in water makes them essential components of cell membranes.

Many hormones(eg, adrenal cortex, gonads) are lipid derivatives. Therefore, lipids are characterized regulatory function.

Lipids. Fats. Hormones. Functions of lipids: energy, storage, protective, construction, regulatory

Questions

1. What substances are lipids?

2. What structure do most lipids have?

3. What functions do lipids perform?

4. Which cells and tissues are richest in lipids?

Tasks

After analyzing the text of the paragraph, explain why many animals before winter, and migratory fish before spawning, tend to accumulate more fat. Give examples of animals and plants in which this phenomenon is most pronounced. Is excess fat always good for the body? Discuss this problem in class.

§ 7. Composition and structure of proteins

1. What is the role of proteins in the body?

2. What foods are rich in proteins?


Among organic substances squirrels, or proteins, are the most numerous, most diverse and of paramount importance biopolymers. They account for 50–80% of the dry mass of the cell.

Protein molecules have big sizes, that's why they are called macromolecules. In addition to carbon, oxygen, hydrogen and nitrogen, proteins may contain sulfur, phosphorus and iron. Proteins differ from each other in the number (from one hundred to several thousand), composition and sequence of monomers. Protein monomers are amino acids (Fig. 8).

An infinite variety of proteins is created by different combinations of just 20 amino acids. Each amino acid has its own name, special structure and properties. Their general formula can be represented in the following form:



An amino acid molecule consists of two parts identical to all amino acids, one of which is an amino group (-NH 2) with basic properties, the other is a carboxyl group (-COOH) with acidic properties. The part of the molecule called the radical (R) has a different structure for different amino acids. The presence of basic and acidic groups in one amino acid molecule determines their high reactivity. Through these groups, amino acids are combined to form proteins. In this case, a water molecule appears, and the released electrons form peptide bond. That's why proteins are called polypeptides.


Rice. 8. Examples of the structure of amino acids - monomers of protein molecules



Protein molecules can have different spatial configurations - protein structure, and in their structure there are four levels of structural organization (Fig. 9).

The sequence of amino acids in a polypeptide chain is primary structure squirrel. It is unique to any protein and determines its shape, properties and functions.

Most proteins have a spiral shape as a result of the formation of hydrogen bonds between CO and NH groups of different amino acid residues of the polypeptide chain. Hydrogen bonds are weak, but together they provide a fairly strong structure. This spiral is secondary structure squirrel.

Tertiary structure– three-dimensional spatial “packaging” of a polypeptide chain. The result is a bizarre, but specific configuration for each protein - globule. The strength of the tertiary structure is ensured by the various bonds that arise between amino acid radicals.


Rice. 9. Scheme of the structure of a protein molecule: I, II, III, IV – primary, secondary, tertiary, quaternary structures


Quaternary structure not typical for all proteins. It arises as a result of the combination of several macromolecules with a tertiary structure into a complex complex. For example, human blood hemoglobin is a complex of four protein macromolecules (Fig. 10).

This complexity of the structure of protein molecules is associated with the diversity of functions inherent in these biopolymers.

Violation of the natural structure of a protein is called denaturation(Fig. 11). It can occur under the influence of temperature, chemical substances, radiant energy and other factors. With a weak impact, only the quaternary structure disintegrates, with a stronger impact, the tertiary, and then the secondary, and the protein remains in the form of a polypeptide chain.


Rice. 10. Scheme of the structure of the hemoglobin molecule


This process is partially reversible: if the primary structure is not destroyed, then the denatured protein is able to restore its structure. It follows that all structural features of a protein macromolecule are determined by its primary structure.

Except simple proteins, consisting only of amino acids, there are also complex proteins, which may include carbohydrates ( glycoproteins), fats ( lipoproteins), nucleic acids ( nucleoproteins) and etc.

The role of proteins in the life of a cell is enormous. Modern biology has shown that the similarities and differences of organisms are ultimately determined by a set of proteins. The closer organisms are to each other in systematic position, the more similar their proteins are.


Rice. 11. Protein denaturation

Proteins, or proteins. Simple and complex proteins. Amino acids. Polypeptide. Primary, secondary, tertiary and quaternary structures of proteins

Questions

1. What substances are called proteins or proteins?

2. What is the primary structure of a protein?

3. How are secondary, tertiary and quaternary protein structures formed?

4. What is protein denaturation?

5. On what basis are proteins divided into simple and complex?

Tasks

Do you know that protein chicken egg consists mainly of proteins. Think about what explains the change in the protein structure of a boiled egg. Give other examples you know of where protein structure can change.

§ 8. Functions of proteins

1. What is the function of carbohydrates?

2. What functions of proteins do you know?


Proteins perform extremely important and diverse functions. This is possible largely due to the variety of forms and composition of the proteins themselves.

One of essential functions protein molecules - construction (plastic). Proteins are part of all cell membranes and cell organelles. The walls are composed predominantly of protein blood vessels, cartilage, tendons, hair and nails.

Of great importance catalytic, or enzymatic, protein function. Special proteins - enzymes are capable of accelerating biochemical reactions in cells tens and hundreds of millions of times. About a thousand enzymes are known. Each reaction is catalyzed by a specific enzyme. You will learn more about this below.

Motor function perform special contractile proteins. Thanks to them, cilia and flagella move in protozoa, chromosomes move during cell division, muscles contract in multicellular organisms, and other types of movement in living organisms are improved.

It is important transport function proteins. Thus, hemoglobin carries oxygen from the lungs to the cells of other tissues and organs. In muscles, in addition to hemoglobin, there is another gas transport protein - myoglobin. Serum proteins promote the transport of lipids and fatty acids, biologically diverse active substances. Transport proteins in the outer membrane of cells carry various substances from the environment into the cytoplasm.

Specific proteins perform protective function. They protect the body from invasion of foreign proteins and microorganisms and from damage. Thus, antibodies produced by lymphocytes block foreign proteins; fibrin and thrombin protect the body from blood loss.

Regulatory function inherent in proteins - hormones. They maintain constant concentrations of substances in the blood and cells, participate in growth, reproduction and other vital functions. important processes. For example, insulin regulates blood sugar.

Proteins also have signaling function. Proteins are built into the cell membrane that can change their tertiary structure in response to factors external environment. This is how signals are received from the external environment and information is transmitted into the cell.

Proteins can perform energy function, being one of the sources of energy in the cell. When 1 g of protein is completely broken down into final products, 17.6 kJ of energy is released. However, proteins are used extremely rarely as a source of energy. Amino acids released when protein molecules are broken down are used to build new proteins.

Functions of proteins: construction, motor, transport, protective, regulatory, signaling, energy, catalytic. Hormone. Enzyme

Questions

1. What explains the diversity of protein functions?

2. What functions of proteins do you know?

3. What role do hormone proteins play?

4. What function do enzyme proteins perform?

5. Why are proteins rarely used as a source of energy?

§ 9. Nucleic acids

1. What is the role of the nucleus in a cell?

2. What cell organelles are the transmission of hereditary characteristics associated with?

3. What substances are called acids?


Nucleic acids(from lat. nucleus– nucleus) were first discovered in the nuclei of leukocytes. Subsequently, it was found that nucleic acids are contained in all cells, not only in the nucleus, but also in the cytoplasm and various organelles.

There are two types of nucleic acids - deoxyribonucleic(abbreviated DNA) And ribonucleic(abbreviated RNA). The difference in names is explained by the fact that the DNA molecule contains a carbohydrate deoxyribose, and the RNA molecule is ribose.

Nucleic acids are biopolymers consisting of monomers - nucleotides. The nucleotide monomers of DNA and RNA have a similar structure.

Each nucleotide consists of three components connected by strong chemical bonds. This nitrogenous base, carbohydrate(ribose or deoxyribose) and remainder phosphoric acid (Fig. 12).

Part DNA molecules There are four types of nitrogenous bases: adenine, guanine, cytosine or thymine. They determine the names of the corresponding nucleotides: adenyl (A), guanyl (G), cytidyl (C) and thymidyl (T) (Fig. 13).


Rice. 12. Scheme of the structure of nucleotides - monomers of DNA (A) and RNA (B)


Each DNA strand is a polynucleotide consisting of several tens of thousands of nucleotides.

The DNA molecule has a complex structure. It consists of two helically twisted chains, which are connected to each other along their entire length by hydrogen bonds. This structure, characteristic only of DNA molecules, is called double helix.


Rice. 13. DNA nucleotides


Rice. 14. Complementary connection of nucleotides


When a DNA double helix is ​​formed, the nitrogenous bases of one chain are arranged in a strictly defined order opposite the nitrogenous bases of the other. In this case, an important pattern is revealed: thymine of another chain is always located opposite the adenine of one chain, cytosine is always located opposite guanine, and vice versa. This is explained by the fact that the nucleotide pairs adenine and thymine, as well as guanine and cytosine, strictly correspond to each other and are complementary, or complementary(from lat. complementum- addition), each other. And the pattern itself is called principle of complementarity. In this case, two hydrogen bonds always arise between adenine and thymine, and three between guanine and cytosine (Fig. 14).

Consequently, in any organism the number of adenyl nucleotides is equal to the number of thymidyl nucleotides, and the number of guanyl nucleotides is equal to the number of cytidyl nucleotides. Knowing the sequence of nucleotides in one DNA chain, the principle of complementarity can be used to establish the order of nucleotides in another chain.

By using four types DNA nucleotides contain all the information about the body, which is passed on to subsequent generations. In other words, DNA is the carrier of hereditary information.

DNA molecules are mainly found in the nuclei of cells, but small amounts are found in mitochondria and plastids.

An RNA molecule, unlike a DNA molecule, is a polymer consisting of a single chain of much smaller dimensions.

RNA monomers are nucleotides consisting of ribose, a phosphoric acid residue, and one of four nitrogenous bases. Three nitrogenous bases - adenine, guanine and cytosine - are the same as those of DNA, and the fourth - uracil.

The formation of an RNA polymer occurs through covalent bonds between ribose and the phosphoric acid residue of neighboring nucleotides.

There are three types of RNA, differing in structure, molecular size, location in the cell and functions performed.

Ribosomal RNA (rRNA) are part of ribosomes and participate in the formation of their active centers, where the process of protein biosynthesis occurs.

Transfer RNAs (tRNA) - the smallest in size - transport amino acids to the site of protein synthesis.

Information, or template, RNA (mRNA) are synthesized on a section of one of the chains of the DNA molecule and transmit information about the structure of the protein from the cell nucleus to the ribosomes, where this information is implemented.

Thus, Various types RNAs are a single functional system, aimed at implementing hereditary information through protein synthesis.

RNA molecules are found in the nucleus, cytoplasm, ribosomes, mitochondria and plastids of the cell.

Nucleic acid. Deoxyribonucleic acid, or DNA. Ribonucleic acid, or RNA. Nitrogen bases: adenine, guanine, cytosine, thymine, uracil, nucleotide. Double helix. Complementarity. Transfer RNA (tRNA). Ribosomal RNA (rRNA). Messenger RNA (mRNA)

Questions

1. What is the structure of a nucleotide?

2. What is the structure of the DNA molecule?

3. What is the principle of complementarity?

4. What are the similarities and differences in the structure of DNA and RNA molecules?

5. What types of RNA molecules do you know? What are their functions?

Tasks

1. Outline your paragraph.

2. Scientists have found that a fragment of a DNA chain has the following composition: C-G G A A A T T C C. Using the principle of complementarity, complete the second chain.

3. During the study, it was found that in the DNA molecule under study, adenines make up 26% of the total number of nitrogenous bases. Count the number of other nitrogenous bases in this molecule.

We solve the fractional rational equation 5/x = 100. This equation can be solved in two ways. Let's look at each of them.

Plan for solving the equation 5/x = 100

  • find the range of acceptable values ​​for a given equation;
  • the first way to solve the equation is by considering it as a proportion;
  • The second way to solve the equation is by finding the unknown divisor.

Finding the unknown term of the proportion

First, let's find the ODZ equation. There is a fraction sign on the left side of the equation and it is equivalent to the division sign. It is known that you cannot divide by zero. This means that from the ODZ we must exclude values ​​that turn the denominator to zero.

ODZ: x belongs to R\(0).

Now let's look at our equation as a proportion.

The main property of proportion.

The product of the extreme terms of a proportion is equal to the product of its middle terms.

For proportion a: b = c: d or a/b = c/d the main property is written like this: a · d = b · c.

Let's apply it and get a linear equation:

100 * x = 5 * 1;

Let's divide both sides of the equation by 100, thereby getting rid of the coefficient in front of the x variable:

Finding an unknown divisor

Let's look at the equation as a quotient. Where the dividend is 5, the divisor is x, and the result of division is the quotient is 100.

Let's remember the rule for finding an unknown divisor - you need to divide the dividend by the quotient.

The found root belongs to the ODZ equation.

Let's check the found solution to the equation. To do this, substitute the found roots into the original equation and perform the calculations:

The solution was found correctly.

An equation is an equality in which there is an unknown term - x. Its meaning must be found.

The unknown quantity is called the root of the equation. Solving an equation means finding its root, and to do this you need to know the properties of the equations. The equations for grade 5 are not difficult, but if you learn to solve them correctly, you will not have problems with them in the future.

The main property of the equations

When both sides of an equation change by the same amount, it continues to be the same equation with the same root. Let's solve some examples to better understand this rule.

How to Solve Equations: Addition or Subtraction

Suppose we have an equation of the form:

  • a + x = b - here a and b are numbers, and x is the unknown term of the equation.

If we add (or subtract from them) the value c to both sides of the equation, it will not change:

  • a + x + c = b + c
  • a + x - c = b - c.

Example 1

Let's use this property to solve the equation:

  • 37+x=51

Subtract the number 37 from both sides:

  • 37+x-37=51-37

we get:

  • x=51-37.

The root of the equation is x=14.

If we look closely at the last equation, we can see that it is the same as the first one. We simply moved term 37 from one side of the equation to the other, replacing plus with minus.

It turns out that any number can be transferred from one part of the equation to another with the opposite sign.

Example 2

  • 37+x=37+22

Let's carry out the same action, move the number 37 from the left side of the equation to the right:

  • x=37-37+22

Since 37-37=0, we simply reduce this and get:

  • x =22.

Identical terms of an equation with the same sign, located in different parts equations can be reduced (crossed out).

Multiplying and dividing equations

Both sides of the equality can also be multiplied or divided by the same number:

If the equality a = b is divided or multiplied by c, it does not change:

  • a/c = b/c,
  • ac = bс.

Example 3

  • 5x = 20

Let's divide both sides of the equation by 5:

  • 5x/5 = 20/5.

Since 5/5 = 1, we reduce these multiplier and divisor on the left side of the equation and get:

  • x = 20/5, x=4

Example 4

  • 5x = 5a

If both sides of the equation are divided by 5, we get:

  • 5x/5 = 5a/5.

The 5 in the numerator and denominator of the left and right sides are canceled, resulting in x = a. This means that identical factors on the left and right sides of the equations cancel.

Let's solve another example:

  • 13 + 2x = 21

We move term 13 from the left side of the equation to the right with the opposite sign:

  • 2x = 21 - 13
  • 2x = 8.

Dividing both sides of the equation by 2, we get:

  • x = 4.

One of the most important skills when admission to 5th grade is the ability to solve simple equations. Since 5th grade is not yet so far from primary school, then there are not so many types of equations that a student can solve. We will introduce you to all the basic types of equations that you need to be able to solve if you want enter a physics and mathematics school.

Type 1: "bulbous"
These are equations that you are almost likely to encounter when admission to any school or 5th grade club as separate task. They are easy to distinguish from others: in them the variable is present only once. For example, or.
They are solved very simply: you just need to “get” to the unknown, gradually “removing” everything unnecessary that surrounds it - as if peeling an onion - hence the name. To solve it, just remember a few rules from the second class. Let's list them all:

Addition

  1. term1 + term2 = sum
  2. term1 = sum - term2
  3. term2 = sum - term1

Subtraction

  1. minuend - subtrahend = difference
  2. minuend = subtrahend + difference
  3. subtrahend = minuend - difference

Multiplication

  1. factor1 * factor2 = product
  2. factor1 = product: factor2
  3. factor2 = product: factor1

Division

  1. dividend: divisor = quotient
  2. dividend = divisor * quotient
  3. divisor = dividend: quotient

Let's look at an example of how to apply these rules.

Note that we are dividing on and we receive . In this situation, we know the divisor and the quotient. To find the dividend, you need to multiply the divisor by the quotient:

We have become a little closer to ourselves. Now we see that is added and it turns out . This means that to find one of the terms, you need to subtract the known term from the sum:

And another “layer” has been removed from the unknown! Now we see a situation with a known value of the product () and one known multiplier ().

Now the situation is “minuend - subtrahend = difference”

And the last step - famous work() and one of the multipliers ()

Type 2: equations with brackets
Equations of this type are most often found in problems - 90% of all problems for admission to 5th grade. Unlike "onion equations" the variable here can appear several times, so it is impossible to solve it using the methods from the previous paragraph. Typical equations: or
The main difficulty is opening the brackets correctly. After you have managed to do this correctly, you should reduce similar terms (numbers to numbers, variables to variables), and after that we get the simplest "onion equation" which we can solve. But first things first.

Expanding parentheses. We will give a few rules that should be used in in this case. But, as practice shows, the student begins to open the brackets correctly only after 70-80 completed problems. The basic rule is this: any factor outside the brackets must be multiplied by each term inside the brackets. And the minus sign in front of the bracket changes the sign of all the expressions inside. So, the basic rules of disclosure:










Bringing similar. Here everything is much easier: you need, by transferring the terms through the equal sign, to ensure that on one side there are only terms with the unknown, and on the other - only numbers. The basic rule is this: each term transferred through changes its sign - if it was with, it will become with, and vice versa. After a successful transfer, it is necessary to count the total number of unknowns, the total number on the other side of the equality than the variables, and solve a simple "onion equation".