Equations and Identities - раздел Образование, Grammar: verb “to be”, there is/are. Possessive and Personal Pronouns An Equation Is A Statement Of Equality Between Two Algebraic Expressions. The...
An equation is a statement of equality between two algebraic expressions. The two expressions are called members, or sides of the equation. If the two members of an equation are finite and are exactly the same, or become the same, for every value of the symbols or variables involved, the equation is called an identical equation or an identity, for example
(x - 2)2 = x2 - 4х + 4, (x + 3) (x - 2) = x2 + x - 6
If the two members of an equation are equal for certain particular values of the symbols involved, but not for all values, the equation is called a conditional equation, or briefly, an equation. An equation in one unknown, say x, is a way of describing one or more numbers by stating a condition the numbers must satisfy. To solve an equation is to find values of the unknowns that make the two members equal. Such values of the unknowns are called roots or solutions of the equation.
The following rules aid in finding the root.
1. The roots of an equation remain unchanged if the same expression is added to or subtracted from both sides of the equation.
2. The roots of an equation remain the same if both sides of the equation are multiplied or divided by the same expression other than zero and not involving the letter whose value is in question.
The equation 2x = 4, where x is the unknown, is true for x = 2. To illustrate the first of the above two rules, add 5x to both sides of the equation 2x = 4. We get 2x + 5x = 4+5x which, like equation 2x = 4, is true for only x = 2. To illustrate the importance of the restriction in the second of the above two laws, multiply both sides of the equation by x and get
(2x) x = (4x) x which is true not only for x = 2, but also for x = 0.
It is always a good plan to check the accuracy of one's work by substituting the result in the original equation to see whether the equation is true for this value.
These numbers or values of the unknown x actually satisfy the equation upon substitution.
For convenience equations are classified in various ways; according to degree they may be linear, quadratic, cubic, etc.; in form, integral or fractional, rational or irrational. Regardless of the form the equation is in at first, the process of solving will involve transformations which will finally put it in the form:
the unknown = one or more definite values
Those transformations when applied to an equation will give a new or derived equation. A derived equation is said to be equivalent with respect to an original equation if it contains all the roots of that equation and no others. The following operations will always lead to equivalent equations, i.e.
1. Adding to or subtracting the same finite quantity from both members.
2. Multiplying or dividing both members by the same quantity provided this quantity is not zero and does not contain the unknown.
If the equation is fractional it may be changed into an integral equation by multiplying both sides by the Least Common Denominator. This process is called clearing the equation of fractions. The integral equation will have all the roots of the original fractional equation but sometimes will include additional roots. These extraneous roots will be values of the unknown for which the Least Common Denominator is zero and they can readily be recognized and discarded.
An equation in which the variable is raised to the first power only is usually called a linear, or first degree, equation.
To solve an equation containing fractions, first reduce each fraction to its lowest terms. Then multiply each side of the equation by the Least Common Denominator of all the denominators. This process is called clearing of fractions.
A quadratic equation is one which can be reduced to the form
2ax + bx + с = 0 (a 
0) where a, b and с are known and x is unknown.
Все темы данного раздела:
Ex. 1. Analyze the following sentences and translate them into Russian.
1. They live together with their parents. 2. Oxford is famous for its University. 3. I didn’t have an
C) Preparatory “it”.
It
is
was
will be
easy/difficult/impossible/dangerous/safe/
expensive/interesting/nice/wonderful/
terrible/a pleasure
Ex. 4. Make the following sentences interrogative and negative.
a) 1. Twenty miles is a long way to walk. 2. My native city is very large. 3. The pair of black trousers is cheap.4. Phonetics is a branch of linguistics. 5. The family are fond of their house. 6.
Ex. 6. Analyze the following sentences and translate them into Russian.
a) 1. There are some big trees in the garden. 2. There is a seminar on philosophy today. 3. There are comfortable apartments in this block of flats. 4. There are a lot of accidents on this road. 5.
Read and learn the basic vocabulary terms.
sense (n) [sens]
ощущение, чувство
support (v, n) [sq'pLt]
содержать (семью); поддержка
seem (v)
About My Family and Myself
I believe that everything has its beginning in the family. Family is very important for every person, because it gives you a sense of stability and tradition, a feeling of having support and unders
Looks and Appearance
beautiful (adj) ['bjHtIful]
красивый (о женщинах)
blond/fair [feq]/ginger [GInGq]/ dark hair
светлые/русые/рыжие/черные в
Features of Character
be in good/bad mood [mu:d]
быть в хорошем (плохом) настроении
brave (adj) [breIv]
смелый
devoted
Interests and Ambitions
ambition (n) [xm'bISqn]
стремление
be keen [kJn] on smth
увлекаться чем-либо
desire (n), (v) [di'
Family Members and Relations in the Family
average ['xvqrIG] /small/large family
средняя/маленькая/большая
семья
consist of (v) [kqn'sIst]
состоят
Ex. 12. Translate into English making use of the words from the box.
slender
beard
shy
short
fat
hazel eyes
pale skin
kind-hearted
plump cheeks
honest
ginger-haired
curly hair
broad shoulders
Ex. 1. Make the sentences interrogative and negative.
a) 1. I usually have a sandwich for my lunch. 2. Students have one English class a week. 3. We have a shop next to the post office. 4. Most cars have got four wheels. 5. An insect has got six legs.
Present Simple
Positive
Negative
Interrogative
I
we
you
they
Ex. 4 . Make the following sentences interrogative and negative.
1. We do a lot of things in our free time. 2. The shops open at 9 o’clock and close at 5.30. 3. It costs a lot of money to stay in luxury hotels. 4. The Moon goes round the Earth. 5. They usually s
Past Simple
Positive
Negative
Interrogative
I
we
you
they
he
she
Future Simple
Positive / Negative
Interrogative
I/we
you/they
he/she/it
(shall(‘ll)/shall not/shan’t) will(‘ll)/wil
General
Auxiliary verb
Subject
Predicate or part of it
Obje
Read and learn the basic vocabulary terms.
delay (n) [dI`leI]
задержка
recent (adj) [`rJs(q)nt]
последний, недавний
manage (v) [`mxnIG]
Memorise the following word combinations.
to be busy
быть занятым
the students’ hall of residence
студенческое общежитие
as a matter of fac
A Letter to a Friend
Dear Linda,
I’m very sorry for the delay in answering your recent letter. I was so awfully busy. In the spring I passed my A-Level examinations and tests at school and managed to get good
Ex. 10. Find the Russian equivalents for the following English word combinations.
1. to belong to a group
2. to describe the routine
3. to depend on the weather
4. to have a late night
5. on the other hand
Pronouns some, any, no
Affirmative sentences
Examples
Some and its compounds (somebody, something, somewhere, someone) is used with the meaning “small amo
Read and learn the basic vocabulary terms.
number (n) [`nAmbq]- число, количество, номер
date back to (v) [`deIt] – датироваться, относиться к определенному времени
antiquity (n) [xn`tIkwItI]- древность, античность
Numbers
The beginning of our number system dates back to antiquity where symbols, which we call positive integers, were used as an aid in counting, and only in the nineteenth century the system, which we k
Ex. 8. Answer the following questions.
1. What were positive integers used for? 2. When was the number system completed? 3. Are the first numbers which we use the positive integers or the negative ones? 4. Is unity a first or a last int
Ex. 1. Analyze the following sentences and translate them into Russian.
1. She could do sums in her head when she was 6 years old. 2. Peter was able to carry out the experiment successfully, but Nick couldn’t finish it without any help.
Read and learn the basic vocabulary terms.
number (n) [`nAmbq] число, количество; v перечислять
numeral (n) [`nju:mqrql] цифра, символ, число
digit (n) [`dIGIt] цифра
value (n) [`vxlju:] величина, значение; (v) це
Four Basic Operations of Arithmetic
Many thousands of years ago this was a world without numbers. Today, using the same numbers in many different ways, man can build bridges, skyscrapers, fly off the Earth like a bird, even measure t
Ex. 14. Choose the phrase closest in meaning to the given statement.
1. Dan can’t be a teacher.
a) I’m sure Dan isn’t a teacher.
b) I think Dan isn’t a teacher.
2. Need I take the tablets every day?
a) Is it a good idea to take th
Ex. 4. Translate the following sentences into Russian.
1. Mathematicians have used mathematical formulas in solving these problems. 2. By the end of the lesson we’ll have been able to obtain the modified definition of the function. 3. Scientific theori
Read and learn the basic vocabulary terms.
part [pa:t] n
часть
proper ['prOpq] a
правильный
improper [im'prOpq] a
неправильный
Rational numbers and decimal numerals
In mathematics, a fraction is a concept of a proportional relation between a part and a whole. In other words, it is an example of a specific type of ratio, in which the two number
Ex.7. Give the proper English equivalents for the Russian expressions.
the greatest common factor; to invert; the quotient; performed operations; the numerator and denominator; decimal numerals; proper fractions; the minuend, subtrahend and r
Ex. 10. Choose the correct tense form.
1. (Have you ever heard/ Did you ever hear) about continued fractions?
2. We (have started/ started) learning fractions when we were at school.
3. By the end of the lecture the st
Ex. 11. Ask special questions.
1. By 5 o’clock the experiment will have been over. (by what time) 2. By the age of 41 Sophia Kovalevskaya had won recognition of mathematicians all over the world. (whose) 3. Zero concept has got
Ex. 12. Translate into English.
1. Дробь представляет часть целого. Она показывает, что что-то разделили на несколько равных частей. 2. В дроби число, стоящее над чертой называется числителем, число, стоящее под чертой называетс
Degrees of Comparison
short words:
-er, -est
big – bigger – the biggest
thin – thinner – the thinnest
short – shorter – the shortest
Ex. 1. Analyze these sentences and compare the adjectives given there. Translate them into Russian.
1. He has a difficult test. I have a more difficult test. Her test is the most difficult of all. 2. Your problem is easy. His problem is easier than yours. That student’s problem is the easiest. 3.
Types of Comparisons
There are a number of different sentence patterns with comparative and superlative forms:
Than чем
This book is more interesting
Ex. 5. Give the proper English equivalents for the Russian expressions.
the best, greater, as difficult as, less, longer… that one, as interesting as, not so important as, the youngest, better, not so famous as, the sooner… the better, the old
Facts to be remembered
a. Present Perfect Continuous is used:
For actions started and finished in the past and lasted for some time. The result of the actions is visible
Note the difference in translation between the Present Perfect Continuous and the Present Perfect Tenses.
1. – You look hot. – I’ve been running all the way.
Я бежала всю дорогу.
2. I’ve been learning irregular verbs all afternoon.
Я учил неправильные глаголы ве
Read and learn the basic vocabulary terms.
compute [kqm`pjHt]
вычислять
deal (dealt) with [dJl]
иметь дело с; рассматривать
apply [q`plaI]
The Nature of Algebra
Algebra is a generalization of arithmetic. Each statement of arithmetic has been dealing with particular numbers for years: the statement (20 + 4)2 = 202 + 2 • 20 • 4 + 4
Ex. 13. Give the proper English equivalents for the Russian expressions.
computed, instead of, simpler, ordinary letters, replace, hold, generalization, relations, concerning, multinomials
1. Algebra is обобщение
Monomials and Polynomials
1. Algebraic expressions are divided into two groups according to the last operation indicated. 2. A monomial is an algebraic expression whose last operation is neither addition not subtraction. 3.
Ex. 1. Read these sentences. Compare the predicates in these pairs of sentences
a
1. I attend all lectures at the University.
2. We often use the Internet.
3. He applies a new method in his research.
&n
Read and learn the basic vocabulary terms.
equation (n) [i`kweiSqn] – уравнение
statement (n) [`steItmqnt] – утверждение, формулировка
finite (a) [`faInaIt] – конечный
variable (a) [`vFqriqbl] – переменная
Ex. 5. Give the proper English equivalents for the Russian expressions.
to solve, the unknown quantity, substituted into the expression, the value, roots, have been transposed, the Least Common Denominator, is checked, clearing, to satisfy
Ex. 9. Translate the sentences from English into Russian.
1. In solving problems by means of algebraic equations, the first and the most difficult step which must be done is to translate the words into the algebraic language. 2. Definite rules cannot be g
Ex. 10. Translate the sentences from Russian into English.
1. Уравнение – это утверждение, выражающее равенство двух алгебраических выражений. 2. Корень уравнения остаётся прежним, если к обеим частям уравнения прибавить или от обеих частей уравнения вычес
Ex.1. Analyze these pairs of sentences and compare the predicates given there.
1. They are solving the equation at the moment.
2. He was dividing these numerals at 2 o’clock yesterday.
3. We have already reduced the fr
Guess the meaning of the following words.
Algebraic adj. ["xlGI`breIIk]; polynomial n. ["pOlI`nqumIql]; integral adj. [`IntIgr(q)l]; constant n. [`kOnstqnt]; coefficient n. ["kquI`fISqnt]; exponent n. [eks`pqunqnt]; fundamen
Polynomials
A number represented by algebraic symbols is referred to as an algebraic expression.
An algebraic expression whose parts are not separated by + or – is called a term; as 2
Ex. 6. Give the proper English equivalents for the Russian expressions.
a trinomial, descending, subtraction, obtain, a fraction, coefficients, terms, exponents, sum, remainder, fractional
1. Each of these polynomi
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