Representative - definition of representative by The Free Dictionary /representative
representative Also found in: , , , , , , .
rep·re·sen·ta·tive
(rĕp′rĭ-zĕn′tə-tĭv)n.1.
One that serves as an example or type for others of the same classification.2.
One that serves as a delegate or agent for another.3. a.
A member of a governmental body, usually legislative, chosen by popular vote.b.
A member of the US House of Representatives or of the lower house of a state legislature.adj.1.
Representing, depicting, or portraying or able to do so.2.
Authorized to act as an official delegate or agent.3.
Of or characteristic of government by representation.4.
Like or typical of others of the same class.rep′re·sen′ta·tive·ly adv.rep′re·sen′ta·tive·ness n.representative (?r?pr?'z?nt?t?v) n1. a person or thing that represents another or others2. a person who represents and tries to sell the products or services of a firm, esp a travelling salesman. Often shortened to: rep 3. a typical example4.
(Government, Politics & Diplomacy) a person representing a constituency in a deliberative, legislative, or executive body, esp (capital) a member of the House of Representatives (the lower house of Congress). See also 5.
(General Sporting Terms) NZ a rugby player, football player, etc, chosen to represent a province in interprovincial sportsadj6.
symbolic7. a. exemplif typical: a representative example of the species. b. containing or including examples of all the interests, types, etc, in a group: a representative collection. 8. acting as deputy or proxy for another or others9.
(Government, Politics & Diplomacy) acting for or representing a constituency or the whole people in the process of government: a representative council. 10.
(Government, Politics & Diplomacy) of, characterized by, or relating to the political principle of representation of the people: representative government. 11. of or relating to a mental picture or representation ?repre'sentatively adv ?repre'sentativeness nrep•re•sent•a•tive
(ˌrɛp rɪˈzɛn tə tɪv)
a person or thing that represents another or others.
an agent or deputy:
a legal representative.
a person who represents a constituency or community in a legislative body, esp. a member of the U.S. House of Representatives or a lower house in certain state legislatures.
a typical example or specimen.
representing.
made up of representatives.
of, characterized by, or founded on representation of the people in government:
a representative democracy.
exemplif typical.
rep`re•sent′a•tive•ly, adv.
Switch to Noun1.representative - a person who represents others & - a representative who acts on behalf of other persons or organizations - a member of a municipal legislative body (as a city council); "aldermen usually represent city wards" - someone who is a member of a legislative assembly - a woman assemblyman,
- a person who is in a position to give yo "he used his business contacts to get an introduction to the governor" - a person appointed or elected to represent others,
- someone sent on a mission to represent the interests of someone else,
- the chief public representative of a country who may also be the head of government, ,
- someone who negotiates (confers with others in order to reach a settlement) - a person who manages the affairs of another - the representative of Puerto Rico in the United States House of Representatives - a representative for a labor union2.representative - an advocate who represents someone else' "the meeting was attended by spokespersons for all the major organs of government", , , , ,
- a person who pleads for a cause or propounds an idea - an inf "an ambassador of good will", , ,
- a slick spokesperson who can turn any criticism to the advantage of their employer,
- a spokesperson (as a lawyer) - a male spokesperson - a female spokesperson, , , , ,
- a salesman who travels to call on customers3.representative - a member of the United States House of Representatives,
- informal abbreviation of `representative' - someone who makes or enacts laws4.representative - an item of information that is typical "this patient provides a typical example of the syndrome"; "there is an example on page 10", ,
- knowledge acquired through study or experience or instruction,
- "it was an apology for a meal"; "a poor excuse for an automobile" - an instance that does not conform to a ru "all her chi the only exception was her last child"; "an exception tests the rule",
- an example that is used to justify similar occurrences at a later time - the most typical example or representative of a type - a small part of something intended as representative of the whole - an example regarded as typical of its classAdj.1.representative - serving to "representative moviegoers"; "a representative modern play" - exhibiting the qualities or characteristics that identify a group "a typical American girl"; "a typical suburban community"; "the typical car owner drives 10,000 miles a year"; "a painting typical of the Impressionist school"; "a typical romantic poem"; "a typical case of arteritis"2.representative - standin "the bald eagle is representative of the United States",
- not standing for something else3.representative - being or characteristic of government by representation in which citizens exercise power through elected officers "representative government as defined by Abraham Lincoln is government of the people, by the people, for the people" - characterized by or advocating or based upon the principles of democrac "democratic government"; "a democratic country"; "a democratic scorn for bloated dukes and lords"- George du Maurierrepresentativenoun1. , , , , , , ,
(Scot.), spokesman or spokeswoman trade union representatives2. ,
the representative for Eastleigh3. , , , ,
She was a sales representative.adjective1. , , , , , ,
a representative government2. , , , , ,
fairly representative groups of adults typical , , 3. , , ,
images chosen as representative of English liferepresentativenoun1. One that is representative of a group or class:, , , , , .2. One who stands for another:, .adjective1. Serving as a symbol:, , , .2. Serving to describe:, , .3. Having the nature of, constituting, or serving as a type:, , , , , , , , , , , , .
????????? ????????????????? ?????????????? ??????reprezentativnízástupcezastupitelsk?repraesentativrepresentativsaelgertypiskkarakteristisktyypillinenreprezentativanképviseletitipikusfulltrúa-fulltrúisem er daemiger?urumbo?sma?ur, fulltrúi代表する????obchodn? zástupcareprezentatívnyzastupite?sk?predstavnikrepresentativ??????????temsil eden?i?n hìnhrepresentative [&#x2reprɪˈzentətɪv]A. ADJ →
(of de) these figures are more representative → estas
representative government →
m a person not fully representative of the group → una
el B. N1. (gen) →
mf2. (esp Brit) (Comm) →
mf3. (US) (Pol) Representative → diputado/a m/fthe House of Representatives → la
→ el representative [&#x2rɛprɪˈzɛntətɪv] n [person, worker] → (e) m/ftrade union representatives →
(= sales person) → (e) m/f (US) (= politician) → (e) m/f adj [sample, cross-section] → représentatif/iverepresentative of sth → / de qchrepresentative adj (→ für) (= typical) cross section, sample → ; attitude, game → ; (= symbolic) →
(= acting for) → ; a representative body → eine
(Parl) government → , ; representative assembly → Abgeordnetenversammlung f n (Comm) → (in) m(f); (Jur) → (r), (r) mf; (US Pol) → (r) mf; authorized representative → (r) mf ?
N brepresentative [&#x2rɛprɪˈzɛntətɪv]1. adj representative (of) → rappresentativo/a (di)2. n (gen) →
m/f, delegato/a (Comm) →
m/f (di commercio) (Am) (Pol) Representative → represent (repr?'zent)
to speak or act on behalf of. You have been chosen to represent our association at the conference. verteenwoordig
представям
representar
zastupovat
repraesentere , εκπροσωπ?
???????????? ????, ?????? ?? ??? ??? ????
predstavljati, zastupati
vera fulltrúi (e-s)
atstovauti, reprezentuoti
reprezentēt; pārstāvēt
bercakap bagi pihak
???? (???????) ????? ???? ???????: ???????? ???? ??? ??? ???? ???? ????
a re-pre-zenta
zastupova?
predstavljati
representera
????????????
бути представником, репрезентувати
???????? ????
代表，作为…的代言人 2.
to be a sign, symbol, picture etc of. In this play, the man in black represents Death and the young girl Life. verteenwoordig
??????? ???
изобразявам
representar
p?edstavovat
symbolisere ,
???? ????? ???? ??? ????
???????? ????, ????? ????
glumiti, prikazivati
melambangkan
standa fyrir, tákna
simbolizuoti, vaizduoti
simbolizēt; nozīmēt
melambangkan , ,
a simboliza
predstavova?
predstavljati
predstavljati
symbolisera
????????????????
simgesi olmak,
символ?зувати; уособлювати
????? ??? ????
t??ng tr?ng
象征，表示 3.
to show or illustrate. What he said represents the feelings of many people. illustreer
????????? ????????? ????? ??????
представям
representar
vyjad?ovat
vaere udtryk for ,
heaks n?iteks olema, v?ljendama
???? ????? ??????? ????
??????? ????, ????????? ????, ?????? ??????? ????
zastupati, predo?iti
menunjukkan
…? ??(??, ??)? ??
rodyti, i?reik?ti
atspogu? izteikt
sebagai contoh ,
vyjadrova?
predstavljati
beteckna, illustrera
??????????? ,
代表，顯示出
зображати; в?дбивати
体现，反映 ?represen'tation noun1.
the act of representing or the state of being represented.
verteenwoordiging
представяне
representa??o
reprezentace
die Vertretung
repraesentation , εκπροσ?πηση
esindamine, esitamine
????????????
zastupanje, predstavljanje
képviselet
perwakilan
?a? a? standa fyrir e-?
atstovavimas
pārstāvē? pārstāvība
mewakili , , , ,
????????: ?????: ?????: ???? ????? ?????: (??) ????? ???: ??????: ?? ????
re-pre-zen-tare
reprezentácia
predstavljanje, zastopanje
predstavljanje
representation
?????????????
temsil etme/edilme
зображення
s? ??i di?n 2.
a person or thing that represents. These primitive statues are intended as representations of gods and goddesses. verteenwoordiger
изображение
representa??o
ztělesnění
die Darstellung
fremstilling ,
zastupnik, predstavnik
ábrázolás(i mód)
e-?/e-r sem stendur fyrir e-?
(pa)vaizdavimas, vaizdas
pārstāvis
wakil ; bilde, skildring
reprezentare
stelesnenie
predstavitev
reprezentacija
??????; ????????? ,
образ
ng??i ??i di?n 3.
(often in plural) a strong appeal, demand or protest.
verteenwoordiging
????????? ??????
протест
representa??o
prohlá?ení, protest, tvrzení
die Vorhaltung
indsigelse ,
p??rdumine, kinnitus
??????? ?????
prigovor, prosvjed
parei?kimas, reikalavimas, protestas
kehendak , , , , roszczenia, za?alenia
reclama?ie ;
prehlásenie, protest
zvani?na ?alba
inv?ndning, gensaga, protest
????????????
resm? ?ik?yetler
抗議，投訴
твердження, заява
l?i ph?n ??i
抗议，请求 ?repre'sentative (-t?tiv)
adjective1.
being a good example (of something); typical. We need opinions from a representat Is this poem representative of his work? verteenwoordigend
????????? ??????
представителен
representativo
reprezentativní
tüüpiline
??????? ?????????
??????????
reprezentativan, tipi?an
khas, contoh yang tepat
sem er daemiger?ur
būdingas, reprezentatyvus
reprezentatī raksturīgs
contoh yang baik ,
reprezentatywny
???????? ????? ????: ???????? ??????: ?????
re-prezentativ
показательный; типичный
reprezentatívny
representativ
????????????
有代表性的
характерний; показовий
tiêu bi?u 2.
carried on by elected people. representative government. verteenwoordigend
????????? ????????
представителен
representativo
zastupitelsk?
repraesentativ
rahvaesinduslik
edustuksellinen
??????????
koji zastupa
képviseleti
fulltrúa-
(kam nors) atstovaujantis
pārstāvniecisks
perwakilan
przedstawicielski
zastupite?sk?
predstavni?ki
predstavni?ki
representativ
??????????????
temsil eden,
представницький
(also rep (rep) ) a person who r a travelling salesman. Our representative will call on you this afternoon. verteenwoordiger
представител
representante
der/die Vertreter(in)
(müügi)esindaja
???????? ????
représentant/-ante
trgova?ki putnik, predstavnik
megbízott
perwakilan
umbo?sma?ur, fulltrúi
セールスマン
?galiotinis
pārstāvis
wakil urus niaga
reprezentant
obchodn? zástupca
trgova?ki putnik
representant
представник
ng??i ??i di?n c?a doanh nghi?p 2.
a person who represents a person or group of people. A Member of Parliament is the representative of the people in his constituency. verteenwoordiger
????????? ?????
представител
representante
der/die Vertreter(in)
repraesentant
edustaja , , député/-ée
(???? ??? ???) ?????????, ?????
képvisel?
pārstāvis
perwakilan sesuatu kumpulan
re-pre- deputat
predstavnik
predstavnik
representant
уповноважений; делегат
ng??i ??i di?n cho m?t nhóm ng??i representative →
reprezentativní repraesentativ
tyypillinen
reprezentativan
代表する ????
representativ ??????????
?i?n hình representative n representante mf; patient service — (PSR) representante de servicio(s) al paciente
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Amy never would pet him like the others, but she was very glad to see him now, and quite clung to him, feeling that he was the representative of the dear family for whom she longed more than she would confess. As Duncan could only act as the representative of the commandant of the fort, the ceremonies which should have accompanied a meeting between the heads of the adverse forces were, of course, dispensed with. At two or three epochs, when the fortunes of the family were low, this representative of hereditary qualities had made his appearance, and caused the traditionary gossips of the town to whisper among themselves, "Here is the old Pyncheon come again At all events, I, the present writer, as their representative, hereby take shame upon myself for their sakes, and pray that any curse incurred by them -- as I have heard, and as the dreary and unprosperous condition of the race, for many a long year back, would argue to exist -- may be now and henceforth removed. The families had all been of different nationalities--there had been a representative of several races that had displaced each other in the stockyards. I meet this American government, or its representative, the State government, directly, and face to face, once a year--no more--in the person of its tax- this is the only mode in which a man situated as I am
and it then says distinctly, R and the simplest, the most effectual, and, in the present posture of affairs, the indispensablest mode of treating with it on this head, of expressing your little satisfaction with and love for it, is to deny it then. A representative could not be prouder of his election to a seat in the American Congress, than a slave on one of the out-farms would be of his election to do errands at the Great House Farm. He said it was a very old name i that the ancestors of th that all Morton had o that even now he considered the representative of that house might, if he liked, make an alliance with the best. Necessity, which knows no law, either in the drama or out of it, accepted a lad of eighteen as the representative of "Sir Anthony Absolute"; the stage-manager undertaking to supply the necessary wrinkles from the illimitable resources of theatrical art. Any one of these scouts used to think nothing of politely assisting an old lady in black out of a vehicle, killing any proctor whom she inquired for, representing his employer as the lawful successor and representative of that proctor, and bearing the old lady off (sometimes greatly affected) to his employer's office. Which side he was on, I couldn't make out, for he seemed to me to be grinding the w I only know that when I stole out on tiptoe, he was not on t for, he was making the legs of the old gentleman who presided, quite convulsive under the table, by his denunciations of his conduct as the representative of British law and justice in that chair that day. Similarly, everything, however trifling, that has been written about, so long as it has been written about sufficiently well, becomes relatively enduring and representative of the country in which it is found.
▲representative▼
representative
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官方公共微信A L G E B R A
ALGEBRAIC EXPRESSIONS
ALGEBRA IS A METHOD OF WRITTEN CALCULATIONS that help us reason about numbers. At the very outset, the student should realize that algebra is a skill. &And like any skill -- driving a car, baking cookies, playing the guitar -- it requires practice. A lot of practice. Written practice. &That said, let us begin.
The first thing to note is that in algebra we use letters as well as
numbers. But the letters represent numbers. We imitate the rules of arithmetic with letters, because we mean that the rule will be true for any numbers.
Here, for example, is the rule for adding fractions:
The letters a and b mean:
The numbers that are in the numerators. The letter c means:
The number in the denominator. The rule means:
"Whatever those numbers are, add the numerators and write their sum over the common denominator."
Algebra is telling us how to do any problem that looks like that. That is one reason why we use letters.
(The symbols for numbers, after all, are nothing but written marks. And so are letters
As the student will see, algebra
depends only on the patterns that the symbols make.)
are the numerical symbols, while the letters are called literal symbols.
Question 1. & are the four operations of arithmetic, and
their operation signs?
To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload").Do the problem yourself first!
Addition: &a + b. & The operation sign is + , and is called the plus sign. &Read a + b as "a&plus&b."
For example, if a represents 3, and b represents 4, then a + b represents 7.
Subtraction: &a & b. & The operation sign is & , and is called the minus sign. &Read a & b as "a&minus&b."
If a represents 8, for example, and b represents 2, then a & b represents 6.
3) & Multiplication: &a&&b. &Read a&&b as "a&times&b."
The multiplication sign in algebra is a centered dot. &We do not use the multiplication cross &, because we do not want to confuse it with the letter x.
And so if a represents 2, and b represents 5, then
a&&b = 2&&5 = 10.
"2 times 5 equals 10."
Do not confuse the centered dot -- 2&5, which in the United States means multiplication -- with the decimal point: &2.5.
However, we often omit the multiplication dot and simply write ab. &Read "a, b." &In other words, when there is no operation sign between two letters, or between a letter and a number, it always means multiplication. &2x &means& 2 times x.
Division: &
&as "a divided by b."
In algebra, we
use the horizontal division bar. &If a&represents 10, for example and b represents 2, then
"10 divided by 2 is 5."
Note: &In algebra we call &a + b &a "sum" even though we do not name an answer. &As the student will see, we name something in algebra
simply by how it looks. &In fact, you will see that you do algebra with your eyes, and then what you write on the paper, follows.
Similarly, we call &a & b& a difference, &ab &
&a product, and&
&a quotient.
This sign =
of course is the equal sign, and we read this --
-- as "a&equals (or is equal to) b."
That means that the number on the left that a represents, is equal to the number on the right that b represents. &If we write
a + b = c,
and if a represents 5, and b&represents 6, then c must represent 11.
&What is the function of parentheses () in algebra?
3 + (4 + 5) &&&&&&3(4 + 5)
Parentheses signify that we should treat what they
encloseas one number.
3 + (4 + 5) = 3 + 9 = 12. &&&&3(4 + 5) = 3&& 9 = 27.
Note: &When there is no operation sign between 3 and (4 + 5), it means multiplication.
Problem 1.&&&In algebra, how do we write
a) &5 times 6? &5&& 6
b) &x times y? &xy
&&c)&&x divided by y? &
d) &x plus 5 &plus &x minus 2?
+ 5) + (x & 2)
e) &x plus 5 &times &x minus 2?
+ 5)(x & 2)
Problem 2.&&&Distinguish the following:
a) &8 & (3 + 2) &&&&&&&&b) &8 & 3 + 2
a) &8 & (3 + 2) = 8 & 5 = 3.
b) &8 & 3 + 2 = 5 + 2 = 7.
In a), we treat 3 + 2 as one number. In b), we do not. We are to first subtract 3 and then add 2. (But see the
There is a common misconception that parentheses always signify multiplication. In , in fact, we will see that we use parentheses to separate the operation sign from the algebraic sign. &8 + (&2).
3. &Terms versus factors.
When numbers are added or subtracted, they are called terms.
When numbers are multiplied, they are called factors.
is a sum of four terms: &a & b + c & d.
In algebra we speak of a "sum" of terms, even though there are subtractions. &In other words, anything that looks like what you see above, we call a sum.
Here is a product of four factors: &abcd.
The word factor always signifies multiplication.
And again, we speak of the "product" abcd, even though we do not name an answer.
Problem 3. &In the following expression, how many terms are there? &And each term has how many factors?
2a + 4ab + 5a(b + c)
There are three terms. &2a is the first term. It has two factors: 2 and&a. 4ab is the second term. It has three factors: 4, a, and b. And 5a(b + c) is all one term. It also has three factors: 5, a, and (b + c). The parentheses mean that we should treat whatever is enclosed as one number.
When all the
are equal -- 2&&2&&2&&2 -- we call the product a power of that factor. &Thus, a&&a is called the second power of a, &or "a&squared." a&&a&&a is the third power of a, &or "a cubed."& aaaa is
a to the fourth power, and so on. &We say that a itself is the first power of a.
Now, rather than write aaaa, we write a just once
&and place a small 4:
a4 ("a to the 4th")
That small 4 is called an exponent. &It indicates the number of times to repeat a as a factor.
83&("8 to the third power" or simply "8 to the third")
means &8&&8&&8.
Problem 4.&&&Name the first five powers of 2. &
2, 4, 8, 16, 32.
Problem 5.&&&Read, then calculate each of the following.
"5 to the second power" or "5 squared" = 25.
"2 to the third power" or "2 cubed" = 8.
"10 to the fourth"
"12 to the first"
However, it is the style in algebra not to write the exponent 1.
a = a1 =1a.
The student must take care not to confuse 3a, which means 3&times a, with a3, which means a times a times a.
a + a + a, &
Question 4. &
are several operations,
8 + 4(2 + 3)2 & 7,
what is the order of operations?
Before answering, let us note that since skill in science is the reason students are requi and since orders of operations appear only in certain forms, then in these pages we present only those forms that the student is ever likely to encounter in the actual practice of algebra. &The division sign & is never used in scientific formulas, only the . And the multiplication cross & is used only in scientific notation -- therefore the student will never see the following:
3 + 6 & (5 + 3) & 3 & 8.
Such a problem would be purely academic, which is to say, it is an exercise for its own sake, and is of no practical value. It leads nowhere.
order of operations is as follows:
Evaluate the parentheses, if there are any, and if they require evaluation.
Evaluate the powers, that is, the exponents.
Multiply or divide -- it does not matter.
Add or subtract.
In Examples 1 and 2 below, we will see in what sense we may add or subtract. &And in Example 3 we will encounter multiply or divide.
Note: &To "evaluate" means to name and write a number.
Example 1.&& &8 + 4(2 + 3)2 & 7
First, we will evaluate the parentheses, that is, we will replace 2 + 3 with 5:
= 8 + 4&&52 & 7
Since there is now just one number, 5, it is not necessary to write parentheses.
Notice that we transformed one element, the parentheses, and rewrote all the rest.
Next, evaluate the exponents:
= 8 + 4& 25 & 7
Now multiply:
= 8 + 100 & 7
Finally, add or subtract, it will not matter. &If we add first:
= 108&& 7 = 101.
While if we subtract first:
8 + 100 & 7 = 8 +&93 = 101.
Example 2.&&& & 60 + 3.
100 & 60 + 3 &does not mean &100 & 63.
Only if there were parentheses --
100 & (60 + 3)
-- could we treat 60 + 3 as one number. &In the absence of parentheses, the problem means to subtract 60 from 100, then add 3:
100 & 60 + 3 = 40 + 3 = 43.
In fact, it will not matter whether we add first or subtract first,
100 & 60 + 3 = 103 & 60 = 43.
When we come to , we will see that
100 & 60 + 3 = 100 + (&60) + 3.
The order in which we "add" those will not matter.
Example 3.&& &
11&&35&&&&5
There are no parentheses to evaluate &and no exponents. &Next in the order is multiply or divide.& We may do either -- we will get the same answer.& But it is usually more skillful to divide first, because we will then have smaller numbers to multiply.& Therefore, we will first divide 35 by 5:
11&&35&&&&5
.&&&½(3 + 4)12 &= ½&&7&&12.
The order of factors does not matter: &abc = bac = cab, and so on. &Therefore we may first do ½&&12. &That is, we may first divide 12 by 2:
½&&7&&12 = 7&&6 = 42.
Example 5. &The division bar.&& &
8 + 2010 & 3
In any problem with the division bar, before we can divide we must evaluate the top and bottom according to the order of operations.
In other words, we must interpret the top and bottom as being in parentheses.
8 + 2010 & 3
& &means &&
(8 + 20)(10 & 3)
Now we proceed as usual and evaluate the parentheses first. &The&answer is 4.
Problem 6.&&&Evaluate each of the following according to the order of operations.
3 + 4& 5 =
2 + 3& 4 + 5 =
3 + 20 = 23
2 + 12 + 5 = 19
4 + 5(2 + 6) =
(4 + 5)(2 + 6) =
4 + 5& 8 = 4 + 40 = 44
&&9& 8 = 72
½(3 + 4)8 =
11& 2 = 22
We may divide first.
½& 7& 8 = 7&&4 = 28. (½&&8 = 4) &
3214 & 3&&22&
&2 + 2&&9&14 & 3&&4
&2 + 18&14 & 12
Question 5. & do we mean by the value of a letter?
The value of a letter is a number. It is the number that will replace the letter when we do the order of operations.
Question 6. &What does it mean to evaluate an expression?
It means to replace each letter with its value, and then do the order of operations.
Example 6.&&&Let x = 10, &y = 4, &z = 2. &Evaluate the following.
& &a) &x + yz
& &b) &(x + y)z
In each case, copy the pattern. &Copy the + signs and copy the parentheses (& ). &When you come to x, replace it with 10. &When you come to y, replace it with 4. &And when you come to z, replace it with 2.
Problem 7.&&&Let x = 10, &y = 4, &z = 2, and evaluate the following.
x + 2(y + z) =
(x + 2)(y + z) =
10 + 2(4 + 2) = 10 + 2&&6= 10 + 12 = 22.
(10 + 2)(4 + 2) = 12&&6 = 72
x & 3(y & z) =
(x & 3)(y & z) =
10 & 3(4 & 2) = 10 & 3&&2 = 10 & 6 = 4
(10 &3)(4 & 2) = 7&&2 = 14
x & y + z =
x & (y + z) =
10 & 4 + 2 = 6 + 2 = 8
10 & (4 + 2) = 10 & 6 = 4
g) & x2 & y2 + 3z2 =
&100 & 16 + 3&&4 = 100 & 16 + 12 = 84 + 12 =96.
, 100 & 16 + 12 does not mean 100 & (16 + 12).
10y² + z³& & &x2
10&&16 + 8&&&&&100
160 + 8&&&100
That is 168 divided by 100. &.
Question 7. & is a literal symbol -- x, y, z --
called a variable?
Because its
A variable, such as x, is a kind of blank or empty symbol.
&It is therefore available to take any value we might give it: &a positive number or, as we shall see, a whole number or a fraction.
Numerical symbols -- 2, 3, 4 -- are called constants. The value of those symbols does not vary.
Problem 8. & variables.&&&Let the value of the variable y depend on the value of the variable x as follows:
y = 2x + 4.
Calculate the value of y that corresponds to each value of x:
When x = 0, &y = 2&&0 + 4 = 0 + 4 = 4.
When x = 1, &y = 2&&1 + 4 = 2 + 4 = 6.
When x = 2, & y = 2&&2 + 4 = 4 + 4 = 8.
When x = 3, & y = 2&&3 + 4 = 6 + 4 = 10.
When x = 4, & y = 2&&4 + 4 = 8 + 4 = 12.
expressions
Real problems in science or in business occur in ordinary language.& To do such problems, we typically have to translate them into algebraic language.
Problem 9.&&&Write an algebraic expression that will symbolize each of the following.
a) & Six times a certain number. &
6n, or 6x, or 6m. Any letter will do.
b) &Six more than a certain number.
c) &Six less than a certain number.
d) &Six minus a certain number.
e) &A number repeated as a
three times.
f) &A number repeated as a term three times.
g) & The sum of three consecutive whole numbers. The idea, for example,
g) & of &6 + 7 + 8. &[Hint: &Let x be the first number.]
x + (x + 1) + (x + 2)
h) &Eight less than twice a certain number.
i) &One more than three times a certain number.
Now an algebraic expression is not a sentence, it does not have a verb, which is typically the equal sign&= .& An algebraic statement has an equal sign.
Problem 10.&&&Write each statement algebraically.
a) &The sum of two numbers is twenty.
x&+&y = 20.
b) &The difference of two numbers is twenty.
x & y = 20.
c) &The product of two numbers is twenty.
d) &Twice the product of two numbers is twenty.
e) &The quotient of two numbers is equal to the sum of those numbers.
A formula is an algebraic rule for evaluating some quantity.& A formula is a statement.
Example 7.&&&Here is the formula for the area A of a rectangle whose
base is b and whose height is h.
"The area of a rectangle is equal to the base times the height."
And here is the formula for its perimeter P -- that is, its boundary:
P = 2b + 2h.
"The perimeter of a rectangle is equal to two times the base plus two times the height."
For, in a rectangle the opposite sides are equal.
Problem 11.&&&Evaluate the formulas for A and P when b = 10 in, and h&=&6 in.
A = bh = 10&&6 = 60 in2.
P = 2b + 2h = 2&&10 + 2&&6 = 20 + 12 = 32 in.
Problem 12.&&&The area A of trapezoid is given by this formula,
A = ½(a + b)h.
Find A when a = 2 cm, b = 5 cm, and h = 4 cm.
A = ½(2 + 5)4 = ½&&7& 4 = &7&&2 = 14 cm2.
When 1 cm is the unit of length, then 1 cm² ("1 square centimeter") is the unit of area.
Problem 13.&&& formula for changing temperature in degrees Fahrenheit (F) to degrees Celsius (C) is given by this formula:
Find C if F = 68&.
Replace F with 68:
(68 & 32) =&
&&36 = 5&&4 = 20&.
&&36 &means "Five ninths of 36."
"One ninth of 36 is 4. &So five ninths is five times 4: &20."
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Lawrence Spector
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