Algebra - Baldor PDF

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ALGEBRA CON GRÁFICOS Y 6523 EJERCICIOS Y PROBLEMAS CON RESPUESTAS DR . AURELIO BALDOR Obra aprobada y recomendada como texto para los Institutos de Segunda Enseñanza de la Re- Fundador, Director y Jefe de la Cá- pública por el Ministerio de Educación, previo tedra de Matemáticas del Colegio Baidor, ...

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Algebra - Baldor albert einstein

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ALGEBR CON R~ G FICOS Y 6523 EJRCIOS Y PROBLEMAS CON RESPUTA

DR

.

AURELIO BALDOR

Ob~a a~~~bad ~ ~ec~~e~da c~~~ ~e~~~ ~a~a ~~~ I~~~i~~~~~ de Seg~~da E~~e~a~~a de ~a Re-

de ~a

F~~dad~~, i~D ec~~~ ~ Jef ~ed~a ed Ma~~ e ~~ica~ de~ C~~egi~ Baid~~, H a, C~~ ab~ b . Jef

ed ~a STEVN ACDEMY, H~b~ke~, Ne~-Je~~e~, U

C~~ed~a

de

P~~fe~~~ PETR'S COLEG Ne~-Je~~e~

.S .A de

C~~~b~ica ~~~ e~ Mi~i~~e~i~ de Ed~cai~~, ~~e~i~ i~f~~~e fa~~~ab~e de ~a J~~~a T~c~ica de Di-

Ma~e~~~ica~,

~ec~~~e~ de I~~~i~~~~~ de Seg~~da E~~e~a~~a

.

EDICON 1980 SAINT

Ma~e~~~ica~, . Je~~e~ Ci~~,

TOALMEN REVISAD POR EL AUTOR

.

De~~~i~~ ega~L I . S . B . N . : 84-357062

EDICONS

CULTRA

Y

CENTROAMI,

DISTRBUCONE

: M

. 9 .74-1980

S . A .

CODIE, S

. A . MADRI

.

E~ ~~~~ieda i~~e~ec~~a~ . Q~eda hec~ e~ de~~~i~~ ~~e ~~e~c~ibe ~a ~e~ ~~~hibda ~a ~e~~~d~ci~~ e~ ~~d~ ~ ~ e ~a~~e

I~~~e~~ ~~~ EDIM ORGANIZC GRAFIC, S P~~~g~~~ I~d~~~~ia~ de A~~~~~~~~i~~~, ~~~ Ca~~e D ~~~ . 12 MOST LES (Mad~id) I~~~e~~ e~ E~~a~a - P~i~~ed i~ S~ai~

; .

. A .1

.

Pa~a ~e~~~~de~

a

~a ge~~i~ def~e~cia ~~e ha~ ~e~id~ c~~

e~~a ~b~a ~~~ P~~fe~~~e~ ~ A~~~~~~ de ~a A~~~ica La~i~a, ~ie~~~ i~~~~d~cid~, e~ ~a ~~e~e~~e edic~~, ~~a ~e~ie de ~e~~~a~ ~~e ~ie~de~ a ~~e e~~e ~ib~~ ~ea ~~~ efica~ e i~~e~e~a~~e He~~~ ~~~c~~ad~ ~~e ~a ~~e~e~~aci~~ c~~~~i~~~a ~~~ ~~ ~~~a ~~a ~~de~~~a f~e~~e de ~~~i~aci~~ ~a~a e~ ~~aba~~ e~c~.

~a~ . E~ c~~~e~id~ ha ~id~ c~ida~~a~e~~e ~e~i~ad~ ~ ~e ha~ i~~~~d~cid~ di~e~~~~ c~ad~~~ ~ ~ab~a~ ~a~a ~~ a~~e~di~a~e ~~~ ~i~a~ ~ efc~i~~ . E~ ~~~ de~ ~c ~~~, e~ ~~ d~b~e a~~ec~~ e~~~~ic~ ~ f~~ci~~a~, hace~ de e~~a ~b~a, ~i~ ~~ga~ a ~d da~, e~

A~geb~a

~~~ ~edag~gica ~ ~~~ed~~a de ~a~ ~~b~icad~ ha~~a h~~ e~ id~~a e~~a~~~ . L~~ Edi~~~e~ ha~ e~~i~ad~ ~~~~~~~~ i~~~~d~ci~ a~g~~~~ a~adi~~ ~~e c~~~~ib~~a~ a c~~~~e~a~ e~ c~~~e~id~ de ~~~ ~~~g~a~a~ ~ige~~e~

. Ta~e~ a~adi~~ ~~~, ~a~a e~~~e~a~ ~~~~ a~g~~~~, ~a~

N~~a~ ~~b~e ~ e C~~ce~~~ de ~ N ~e~~ ; ~N ~a ~~b~e ~a~ ca~~idae~ c~~~~e~a~ e i~gi~ a a~ia~ ~ e~ C~d~ a ~ de ~~~ Ti~~~ B~~ic~~ ed De~c~~~~~ici~~ Fac~~~ia~

.

E~~e~a~~~ ~~e e~ P~~fe~~~ad~ de Hi~~a~~a~~~ica ~e~a a~~i~a~a~ e~ i~ge~~e

e~f~e~~~ ~e~di~ ~~~ ~~d~~ ~~~ ~~c~ic~~ ~~e

ha~ i~~e~~e~id~ e~ ~a c~~feci~~ de e~~a ~b~a . S~~~ ~~~ ~~eda ~ei~e~a~ ~~e~~~~ ~~~ ~~~f~~d~ ag~adeci~ie~~~ ~~~ ~a ac~gida ~~e ~e ha~ di~~e~~ad~ ~ie~~~e

. L~~ EDITORS

C~~

ace~d~ad de~~ci~~ ~ ~~~~~ ~~g~~~~, deic~ e~~e e~f~e~~~ di~ e ~~ia~, a ~a i~~~~idab~e ~e~~~ia de ~i P~~fe~~~a ~ D ~a A~a L~i~a Se~~a~~ ~ P~~ce~, ~~e f~e~a P~e~ide~~a de e~~a E~~~e~a d~~a~~e ~~~ a~~~ 192 a 1926

D~ . J~~~ A

~ad~e,

. . L~~e~

Se~~a~~

;

~~ ~ e~ ~~~~e~ de~ ~~~e~~ de a~i~a~e~ ~~e ~~~e~a~ a~~ ~~~gi~ ~a A~i~~~~ica . E~ ~~ige~ de~ Pa~a~~~ cie~~~~ de ~ig~~~ ~a~a ~~e e~ h~~~~~~e~i~~ . b~e a~ca~~a~a ~~ c~~ce~~~ ab~~~ac~~ de~ ~~~e~~, ba~e i~di~~e~~ab~e ~a~a ~a f~~~aci~~ de ~a cie~cia a~geb~aic

CONEPT ED NUMERO EN LOS PUEBLOS PRIMTIVOS (25,0- A . C .) Medi~ ~ c~~~a~ f~e~~~ ~a~ ~~i~e~a~ ac~i~idae~ ~a~e~~~ica~ de~ h~~b~e ~~i~i~i~~ . Hacie~d~ ~a~ca~ e~ ~~~ ~~~~c~~ de ~~~ ~~b~~e~ ~~g~ab~, e~~~~ ~~i~e~~~ ~~eb~~~, ~a ~edic~~ de~ ~ie~-

A~geb~a

.

PRELIMNAS O

~

O 2

e~ ~a ~a~a de ~a Ma~e~~~ica ~~e e~~~dia ~a ca~~ida c~~~i~LGEBRA de~ad de~ ~~d~ ~~~ ge~e~~ a ~~~ib~e . CAR~CTER ELD CON LA ARITMEC

Y SU DIFERNCA ALGEBR

O

E~ c~~ce~~~ de ~a ca~~ida e~ A~geb~a e~ ~~ch~ ~~~ a~~~i~ ~~e e~ A~i~~~~ica . E~ A~i~~~~ica ~a~ ca~~idae~ ~e ~e~~e~e~~a~ ~~~ ~~~e~~~ ~ ~~~~~ e~: ~ei~~e ; ~a~a ~~e~a~ ~a~~~e~ de~e~~i~ad~~ . A~~, 20 e~~~e~a ~~ ~~~~ ~a~~~ e~~~e~a~ ~~ ~a~~~ ~a~~~ ~ ~e~~~ ~~e ~~~e hab~~ ~~e e~c~ib~ ~~ ~~~e~~ di~~i~~~ de 20 . E~ A~geb~a, ~a~a ~~g~a~ ~a ge~e~a~i~aci~~, ~a~ ca~~idae~ ~e ~e~~e~e~~a~ ~~~ ~edi~ de ~e~~a~, ~a~ c~a~e~ ~~ede~ ~e~~e~e~~a~ ~~d~~ ~~~ ~a~~~e~ A~~, a ~e~~e~e~~a e~ ~a~~~ ~~e ~~~~~~~~ ~e a~ig~e~~~, ~ ~~~ ~a~~~ ~~ede ~e~~e~e~~a~ 20 ~ ~~~ de 20 ~ ~e~~~ de 20, a ~~e~~~a e~eci~~, a~~~~e c~~~ie~e ad~e~~i~ ~~e c~a~d~ e~ ~~ ~~~b~e~a a~ig~a~~~ a ~~a ~e~~a ~~ ~a~~~ de~e~~i~ad~, e~a ~e~~a ~~ ~~ede ~e~~e~e~~a~, e~ e~ ~i~~~ ~~~b~e~a, ~~~~ . ~a~~~ di~~i~~~ de~ ~~e ~e he~~~ a~ig~ad~ NOTACI ALGEBRIC L ~ ~ ~~~e~~~

~~~b~~~~ ~~ad~~ e~ ~ ~a~ ~e~~a~ .

A~geb~a

~a~a ~e~~e~e~~a~ ~a~ ca~~idae~ ~~~ ~~~

5

e~

.



6

~ ALGEBR

L~~ ~~~e~~~ ~e e~~~ea~ ~a~a ~e~~e~e~~a~ ca~~idae~ c~~~cida~ ~ e-d ~e~~i~ad~ . La~ ~e~~a~ ~e e~~~ea~ ~a~a ~e~~e~e~~a~ ~~da c~a~e de ca~~idae~, ~a ~ea~ c~~~cida~ ~ de~c~~~cida~ . La~ ca~~idae~ c~~~cida~ ~e e~~~e~a~ ~~~ ~a~ ~~i~e~a~ ~e~~a~ de~ a~fabe~~ : a, b, c, d . . . La~ ca~~idae~ de~c~~~cida~ ~e ~e~~e~e~~a~ ~~~ ~a~ ~~~i~a~ ~e~~a~ de~ a~fabe~~ : ~, ~, ~, ~, ~, ~ . U~a ~i~~a ~e~~a ~~ede ~e~~e~e~~a~ di~~i~~~~ ~a~~~e~ dife~e~ci~~d~~~~ ~~~ ~edi~ de c~~i~~a~ ; ~~~ e~e~~~~ : a', a", a"', ~~e ~e ~ee~ a ~~i~a, g~~da, a ~e~ce~a, ~ ~a~bi~~ ~~~ ~edi~ de ~~b~~dice~ ; ~~~ e~e~~~~ a 8 , ~~e ~e ~ee~ a ~~b~~~, a ~~bd~~, a ~~b~~e~ .

O

FORMULAS

C~~~ec~e~cia de ~a ge~e~a~i~aci~~ ~~e i~~~ica ~a ~e~~e~e~~aci~~ de ~a~ ca~~idae~ ~~~ ~edi~ de ~e~~a~ ~~~ ~a~ f~~~~~a~ a~geb~aic~ F~~~~~a a~geb~aic e~ ~a ~e~~e~e~~aci~~, ~~~ ~edi~ de ~e~~a~, de ~~a ~eg~a ~ de ~~ ~~i~ci~i~ ge~e~a~ . A~~, ~a Ge~~e~~~a e~~e~a ~~e e~ ~~ea de ~~ ~ec~~~g~~~ e~ ig~a~ a~ ~~~d~c~~ de ~~ ba~e ~~~ ~~ a~~~~a ; ~~eg~, ~~a~a~d~ A a~ ~~ea de~ ~ec~~~g~~~, b a ~a ba~e ~ h a ~a a~~~~a, ~a f~~~~~a/ ~e~~e~e~~a~~ de ~~ ~~d~ ge~e~a~' e~ ~~ea de c~a~~~ie~ ~ec~~~g~~~, ~~e~ e~ ~~ea de ~~ ~ec~~~g~~~ da~ ~e ~b~e~d~~ c~~ ~~~~ ~~~~i~~i~ b ~ h e~ ~a f~~~~~a a~~e~i~~ ~~~ ~~~ ~a~~~e~ e~ e~ ca~~ da~ . A~~, ~i ~a ba~e de ~~ ~ec~~~g~~~ e~ 3 ~ . ~ ~~ a~~~~a 2 ~ ., ~~ ~~ea ~e~~ E~ ~~ea de ~~~~ ~ec~~~g~~~ ~ c ~a a~b e f~e~a 8 ~ . ~ ~~ a~~~~a 31 ~

O

SIGNO

DEL

. ~e~~a

.

A = b ~ h

A=b~h=3

.=6

~..~ X2 2 ~

~. 2 .

:

:

/'

A = b ~ h =8

~4~ 34

~ .2 . (1)

~. = 28

ALGEBR

L~~ ~ig~~~ ~e ~~ead~~ e~ ~aci~~, Sig~~~ de Re~ci~a ~ ~ ig~S ~~ de Ag~~~aci~~

O

a ~e: a ~, 2a ,

A~geb~a

~~~ de ~~e~ c~a~e~

: Sig~~~ de O~e.

6 SIGNO DE OPERACIN

E ~ A~geb~a ~e ~e~ifca~ c~~ ~a~ ca~~idae~ ~a~ ~i~~a~ ~~e~aci~~e~ ~~e ~e A~i~~~~ica : S~~a, Re~~a, M~~~i~~icai~~, Di~i~i~~, E~~~aci~~ a P~~e~ia~c ~ E~~~aci~~ de Ra~ce~, ~~e ~e i~dica~ c~~ ~~~ ~ig~~~ ~ig~ie~~e~ ~~~ . A~~ a + E~ Sig~~ de ~a S~~a e~ +, ~~e ~e ~ee (I) E~ e~ f~~~~~a~ a~geb~aic~

Ca~

. XVI, ~~gi~a 270, ~e e~~~dia a~~~ia~e~~e ~~d~ ~~ ~e~aci~~ad~ c~~ ~a~ .

: b ~e ~ee "a ~~~

b"

.



~~

E~ Sig~~ de ~a Re~~a

PRELIMNAS

e~ -,

~~e ~e ~ee

~e~~~ .

A~~,

7

~ a- b

~e ~ee "a

~e-

~~~ b" E~ ig~S ~ de ~a M~~~i~~icai~~ e~ ~, ~~e ~e ~ee ~~~~i~~icad~ ~~~ . ~A ~, a ~ b ~e ~ee "a ~~~~i~~icad~ ~~~ b" . E~ ~~ga~ de~ ~ig~~ ~ ~~e~e e~~~ea~~e ~~ ~~~~~ e~~~e ~~~ fac~~~e~ ~ ~a~bi~~ ~e i~dica ~a ~~~~i~~icai~~ c~~~ca~d~ ~~~ fac~~~e~ e~~~e ~a~~~~e~i~ . A~~, a .b ~ (a)b e~~i~a~e~ a a ~ b . E~~~e fac~~~e~ ~i~e~a~e~ ~ e~~~e ~~ fac~~~ ~~~~~ic~ ~ ~~~ ~i~e~a~ e~ e~~i~a~e ~ig~~ de ~~~~i~~icai~~ ~~e~e ~~i~i~~e . A~~ abc a a ~ b ~ c ; 5~~ e~~i~a~e a 5 ~ ~ ~ ~. E~ Sig~~ de ~a Di~i~i~~ e~ -, ~~e ~e ~ee di~id~ e~~~e . A~~, a - b ~e b" . Ta~bi~~ ~e i~dica ~a di~i~i~~ ~e~a~a~d~ e~ di~ee "a di~id~ e~~~e ~ide~d~ ~ e~ di~i~~~ ~~~ ~~a ~a~a h~~i~~~~a~ . A~~, ~0 e~~i~a~e a ~ . - ~: E~ Sig~~ de ~a E~e~aci~~ a ~P ~e~cia e~ e~ e~~~~e~~e, ~~e ~e ~~ ~~~e~~ ~e~~e~~ c~~~cad~ a~~iba ~ a ~a de~echa de ~~a ca~~ida, e~ c~a~ i~dica ~a~ ~ece~ ~~e dicha ca~~ida, ~~a~ad ba~e, ~e ~~~a c~~~ fac~~~ . A~~,

a 3 = a

;b 6 = b



C~a~d~ ~~a ~e~~a ~~ ~ie~e e~~~~e~~e, ~~ e~~~~e~~e e~ ~a ~~ida a~ ; ~~~ e~~i~a~e a ~'~'~' . e~~i~a~e a E~ Sig~~ ed Ra~~ e~ V, ~~a~ad~ ~ig~~ ~adica~, ~ ba~~ e~~e ~ig~~ ~e c~~~ca ~a ca~~ida a ~a c~a~ ~e ~e e~~~ae ~a ~a~~ . A~~, -, ,~a- e~~i~a~e a ~a~~ c~ad~ad de a, ~ ~ea, ~a ca~~ida ~~e e~e~ad a~ c~ad~ad~ ~e~~~d~ce ~a ca~~ida a ; e~~i~a~e a ~a~~ c~bica de b, ~ ~ea ~a ca~~ida ~~e e~e~ad a~ c~b~ ~e~~~d~ce ~a ca~~ida b .

O

A~~,

7

.

a

COEFINT

E~ e~ ~~~d~c~~ de d~~ fac~~~e~, c~a~~~ie~a de ~~~ fac~~~e~ e~ ~~a~ad~ c~efic~~e de~ ~~~~ fac~~~ . a e i~dica A~~, e~ e~ ~~~d~c~~ 3a e~ fac~~~ 3 e~ c~efic~~e de~ fac~~~ ; ~e ~~e e~ fac~~~ a ~e ~~~a c~~~ ~~~a~d~ ~~e~ ~ece~, ~ ~ea 3a = a + a + a e~ ~~~d~c~~ 5b, e~ fac~~~ 5 e~ c~efic~~e de b e i~dica ~~e 5b=+-' . E~~~~ ~~~ c~efic~~e~ ~~~~~ic~~ . E~ e~ ~~~d~c~~ ab, e~ fac~~~ a e~ c~efic~~e de~ fac~~~ b, e i~dica ~~e e~ fac~~~ b ~e ~~~a c~~~ ~~~a~d~ a ~ece~, ~ ~ea ab = b + b + b + b . . .a ~ece~ . E~~e e~ ~~ c~efic~~e ~i~e~a~ . E~ e~ ~~~d~c~~ de ~~~ de d~~ fac~~~e~, ~~~ ~ ~a~i~~ de e~~~~ ~~~ e~ e~ e~ c~efic~~e c~efic~~e de ~~~ ~e~~a~~e~ . A~~, e~ e~ ~~~d~c~~ abcd, a ; ab e~ e~ c~efic~~e de cd ; abc e~ e~ c~efic~~e de d . de bcd C~a~d~ ~~a ca~~ida ~~ ~ie~e c~efic~~e ~~~~~ic~, ~~ c~efic~~e . e~ ~a ~~ida . A~~, b e~~i~a~e a ~b ; abc e~~i~a~e a ~abc

.



ALGEBR

~ 8

8O

SIGNO DE RELACION

Se e~~~ea~ ~e ~~~ ~ig~~~ ~a~a i~dica~ ~a ~e~aci~~ ~~e e~i~~e e~~~e d~~ . L~~ ~~i~ci~a~e~ ~~~ :

ca~~idae~

=, ~~e ~e ~ee ig~a~ a >, ~~e ~e ~ee ~a~~~ ~~e ~ . A~~, a < b + c "a

ig~a~ a b" . ~e ~ee ~" + ~ ~a~~~ ~~e ~e ~ee a" ~e~~~ ~~e

MOD

b ~+ c" .

AGRUPCION

~L ~ ~ig~~~ de ag~~~aci~~ ~~~ : e~ ~a~~~~e~i~ ~~di~a~i~ ~i~ a~g~~a~ ~ c~~che~e [ ], ~a~ ~~a~e~ ~~ ~ ~a ba~~a ~ ~~~c~~~ E~~~~ ~ig~~~ i~dica~ ~~e ~a ~~e~aci~~ c~~~cad e~~~e e~~~~ deb efc~~a~~e ~~i~e~~ . A~~, (a+ b)c i~dica ~~e e~ ~e~~~~ad~ de ~a ~~~a de deb ~~~~i~~ica~~e ~~~ c ; [a - b]~ i~dica ~~e ~a dife~e~cia e~~~e ~~~~i~~ica~~e ~~~ ~; ~ a + b 1 _ ~ c - d ~ i~dica ~~e ~a ~~~a de ~idi~~e e~~~e ~a dife~e~cia de c ~ d . 10

~" .

DE

RESOLV

LOS

EN ARITMEC

Y

EN

( ), e~ ~a~~~~e-

a ~ b a ~ b a ~ b

deb

deb di-

PROBLEMAS ALGEBR

~E ~~~e~~~ a c~~~i~~aci~~ ~~ e~e~~~~ ~a~a hace~ ~~~a~ ~a dife~e~cia e~~~e e~ ~~~~d~ a~i~~~~ic~ ~ e~ a~geb~aic~ e~ ~a ~e~~~~ci~~ de ~~~b~e~a~, f~~dad~ e~~e ~~~i~~ e~ ~a ~~~aci~~ a~geb~aic ~ e~ ~a ge~e~a~i~aci~~ ~~e ~~~a i~~~ica . La~ eda~ de A ~ B ~~~a~ 48 a~~~ . Si ~a eda de eda de A, ~~~~ eda ~ie~e cad ~~~?

B e~ 5 ~ece~ ~a

METOD ARITMECO

Eda de

Eda de O ~ea, eg~, 1

A ~~~ eda de B = 48 ~ a ~~ . e~ 5 ~ece~ ~a de A, ~e~d~e~~~ : A ~~~ 5 ~ece~ ~a eda de A = 48 a~~~ 6 ~ece~ ~a eda de A = 48 a~~~ ; daE de A = 8 a~~~ . R . daE de B = 8 a~~~ ~ 5 = 40 a~~~ . R .

B

C~~~ ~a eda de

.

METOD ALGEBRICO

C~~~ ~a eda de

~ ~ea,

A

e~ ~~a ca~~ida de~c~~~cida ~a ~e~~e~e~~~ ~~~ Sea ~ =eda de A . E~~~~ce~ 5~ =eda de B .

C~~~ a~ba~ eda~ ~~~a~ 48 a~~~, ~e~d~e~~~ ~ + 5~ = 84 a~~~ 6~ = 84 a~~~ .

;

:

~.



CANTIDES POSITVA

Si 6 ~ece~ ~ e~~i~a~e a

~ ~ea ~E ~~~ce~

~ = 8 a~~~, eda de

1

CANTIDES

Y NEGATIVS

* 9

~~ a~~~ . ~ ~a~d~~ ~a ~e~~a I~~~e (~e -1' a~~~,

5~ = 8 a~~~

A . ~ 5 = 40 a~~~, eda de

R . B .

R .

POSITVA Y NEGATIVS

E~ A~geb~a, c~a~d~ ~e e~~~dia~ ca~~idae~ ~~e ~~ede~ ~~~a~~e e~ d~~ ~e~~id~~ ~~~e~~~~ ~ ~~e ~~~ de c~~dic~~ ~ de ~~d~ de ~e~ ~~~e~~~~, ~e e~~~e~a e~ ~e~~id~, c~~dic~~ ~ ~~~d~ de ~e~ (~a~~~ ~e~a~i~~) de ~a ca~~ida ~~~ ~edi~ de ~~~ ~ig~~~ + ~ -, a~~e~~~ie~d~ e~ ~ig~~ + a ~a~ ca~~idade~ ~~~ada~ e~ ~~ ~e~~id~ de~e~~i~ad~ (ca~~idae~ ~~~i~i~a~) ~ a~~e~~~ie~d~ e~ ~ig~~ - a ~a~ ca~~idae~ ~~~ada~ e~ ~e~~id~ ~~~e~~~ a~ a~~e~i~~ (ca~~idae~ ~ega~i~a~) . A~~, e~ habe~ ~e de~ig~a c~~ e~ ~ig~~ + ~ ~a~ de~da~ c~~ e~ ~ig~~ - . Pa~a e~~~e~a~ ~~e ~~a ~e~~~~a ~ie~e $10 de habe~, di~e~~~ ~~e ~ie~e + $10, ~ ~a~a e~~~e~a~ ~~e deb $10, di~e~~~ ~~e ~ie~e - $10 . L~~ g~ad~~ ~~b~e ce~~ de~ ~e~~~~e~~~ ~e de~ig~a~ c~~ e~ ~ig~~ + ~ ~~~ g~ad~~ ba~~ ce~~ c~~ e~ ~ig~~ - . A~~, ~a~a i~dica~ ~~e e~ ~e~~~~e~~~ ~a~ca 10 ~~b~e ce~~ e~c~ib~e~~~ + 10 ~ ~a~a i~dica~ ~~e ~a~ca 8~ ba~~ ce~~ e~c~ib~e~~~ -8~ E~ ca~i~~ ~ec~~~id~ a ~a de~echa ~ haci a~~iba de ~~ ~~~~~ ~e de~ig~a c~~ e~ ~ig~~ + ~ e~ ca~i~~ ~ec~~~id~ a ~a i~~~ie~da ~ haci ab~~ de ~~ ~~~~~ ~e ~e~~e~e~~a c~~ e~ ~ig~~ - . A~~, ~i he~~~ ~ec~~~id~ 20 ~ . a ~a de~echa de ~~ ~~~~~ da~, di~e~~~ ~~e he~~~ ~ec~~~id~ +20 ~ ., ~ ~i ~ec~~~e~~~ 30 ~ . a ~a i~~~ie~da de ~~ ~~~~~ e~c~ib~e~~~ -30 ~ . E~ ~ie~~~ ~~a~~c~~~id~ de~~~~~ de C~i~~~ ~e c~~~ide~a ~~~i~i~~ ~ e~ ~ie~~~ ~~a~~c~~~id~ a~~e~ de C~i~~~, ~ega~i~~ . A~~, + 150 a~~~ ~ig~ifca 150 a~~~ D . C . ~ - 78 a~~~ ~ig~ifca 78 a~~~ A .C . E~ ~~ ~~~~e i~~~~d~cid~ e~ e~ ~~e~~, ~e~~e~e~~a~~~ c~~ e~ ~ig~~ + ~a ~~~ci~~ ~~e ~e ha~~a de~ ~~e~~ haci a~~iba ~ c~~ e~ ~ig~~ - ~a ~~~ci~~ ~~e ~e ha~~a de~ ~~e~~ haci ab~~ . A~~, ~a~a e~~~e~a~ ~~e ~a ~~~gi~~d de~ ~~~~e ~~e ~e ha~~a de~ ~~e~~ haci a~~iba ~ide 15 ~ ., e~c~ib~e~~~ + 15 ~ ., ~ ~i ~a ~~~ci~~ i~~~~d~cida e~ e~ ~~e~~ e~ de 8 ~ ., e~c~ib~e~~~ - 8 ~ . La ~a~i~~d ~~~~e ~e de~ig~a c~~ e~ ~ig~~ + ~ ~a ~a~i~~d ~~~ c~~ e~ ~ig~~ - ; ~a ~~~gi~~d e~~e ~e c~~~ide~a ~~~i~i~a ~ ~a ~~~gi~~d ~e~~e, ~ega~i~a . P~~ ~~ ~a~~~, ~~ ~~~~~ de ~a Tie~~a c~~a ~i~~aci~~ ge~g~~fica ~ea : + 45~ de ~~~gi~~d ~ -15~ de ~a~i~~d ~e ha~~a~~ a 45~ a~ e~~e de~ ~~i~e~ ~e~ida~~ ~ a 15~ ba~~ e~ Ec~ad~~ .

12

ELCION DEL SENTIDO POSITV

La fi~aci~~ de~ ~e~~id~ ~~~i~i~~ e~ ca~~idae~ ~~e ~~ede~ ~~~a~~e e~ d~~ ~e~~id~~ ~~~e~~~~ e~ a~bi~~a~ia, de~e~de de ~~e~~~a ~~~~~~ad

; e~ deci~,



~~ ALGEBR

~~e ~~de~~~ ~~~a~ c~~~ ~e~~id~ ~~~i~i~~ e~ ~~e ~~e~a~~~ ; ~e~~ ~~a ~e~ fi~ad~ e~ ~e~~id~ ~~~i~i~~, e~ ~e~~id~ ~~~e~~~ a ~~~e ~e~~ e~ ~ega~i~~ . A~~, ~i ~~~a~~~ c~~~ ~e~~id~ ~~~i~i~~ e~ ca~i~~ ~ec~~~id~ a ~a de~echa de ~~ ~~~~~, e~ ca~i~~ ~ec~~~id~ a ~a i~~~ie~da de e~e ~~~~~ ~e~~ ~ega~i~~, ~e~~ ~ada ~~~ i~~ide ~~~a~ c~~~ ~~~i~i~~ e~ ca~i~~ ~ec~~~id~ a ~a i~~~ie~da de~ ~~~~~ ~ e~~~~ce~ e~ ca~i~~ ~ec~~~id~ a ~a de~echa de~ ~~~~~ ~e~~a ~ega~i~~ . A~~, ~i ~~b~e e~ ~eg~e~~~ AB ~~~a~~~ c~~~ ~~~i~i~~ e~ ~e~~id~ de haci B, e~ ~e~~id~ de B haci A ~e~~a ~ega . + + ~e~~ ~i fi~a~~~ ~i~~, c~~~ ~e~~id~ ~~~i~i~~ A B A B i c a h , A de e~ ~e~~id~ de A haci B ~e~~a ~ega~i~~ . N~ ~b~~a~~e, e~ ~a ~~~c~ica ~e ace~~a~ ge~e~a~~e~~e ~~~ ~e~~id~~ ~~~i~i~~~ de ~~e ~e ~~a~~ e~ e~ ~~~e~~ a~~e~i~~ . A

B

13 CERO e~

. A~~, ~e~~e~e~~a~ e~ e~~ad~ ec~~~~i~a a~~e~cia de ca~~ida c~ de ~~a ~e~~~~a ~~~ 0 e~~i~a~e a deci~ ~~e ~~ ~ie~e habe~ ~i de~da~ . ~La ca~~idae~ ~~~i~i~a~ ~~~ ~a~~~e~ ~~e 0 ~ ~a~ ~ega~i~a~ ~e~~~e~ ~~e 0 . A~~, + 3 e~ ~~a ca~~ida ~~e e~ ~~e~ ~~idae~ ~a~~~ ~~e 0 ; + 5 e~ ~~a ca~~ida ~~e e~ ci~c~ ~~idae~ ~a~~~ ~~e 0, ~ie~~~a~ ~~e - 3 e~ ~~a ca~~ida ~~e e~ ~~e~ ~~idae~ ~e~~~ ~~e 0 ~ - 5 e~ ~~a ca~~ida ~~e e~ ci~c~ ~~idae~ ~e~~~ ~~e 0 . De d~~ ca~~idae~ ~~~i~i~a~, e~ ~a~~~ ~a de ~a~~~ ~a~~~ ab~~~~~~ ; a~~, + 5 e~ ~a~~~ ~~e + 3, ~ie~~~a~ ~~e de d~~ ca~~idae~ ~ega~i~a~ e~ ~a~~~ ~a de ~e~~~ ~a~~~ ab~~~~~~ : - 3 e~ ~a~~~ ~~e - 5 ; - 9 e~ ~e~~~ ~~e - 4 . EJRCIOS

SOBRE Y

CANTIDES

POSITVA

NEGATIVS

1) U~ h~~b~e c~b~a $130 . Pag ~~a de~da de $80 ~ ~~eg~ hace c~~~~a~ ~~~ ~a~~~ de $95 . ~C~~~~~ ~ie~e? Te~ie~d~ $130, ~ag~ $80 ; ~~eg~, ~e ~~ed~ c~~ $50 ga~~~ de $95 ~ c~~~ ~~~~ ~ie~e $50 ~i c~~~e e~ ~~a de~da de $45 ~a~~~, ~ie~e ac~~a~~e~~e - $45 . R . EJRCIO

IF

. De~~~~~ hace ~~ . P~~ ~~

1

1 .

Ped~~ deb~a 60 ~b ~~~a~e~ ~ ~ecib~ 320 . E~~~e~a~ ~~ e~~ad~ ec~~~~ic~ ~U h~~b~e ~~e ~e~~a 170 ~~c~e~ i~h ~ ~~a c~~~~a ~~~ ~a~~~ de 15 E~~~e~a~ ~~ ~e ~ad~ ec~~~~ic~ . 3 . ~ Te ~a $20 . C~b~~ $56 ~ ~a~g ~ de~da~ ~~~ $189 . ~C~~~~~ ~e~g~? 2 .

. .



CANTIDES POSITVA

4 . 5 . 6 . 7 . 8 .

Y NEGATIVS

~

1

C~~~~~ ~~~a~ ~~~ ~a~~~ de 65 ~~~e~ ~ a~i~e~~~~ ~~~ 178 . Si de~~~~~ ~ecib~ 280, ~c~~~ e~ ~i e~~ad~ ec~~~~ic~? Te~~a $20 . Pag~~ $15 ~~e deb~a, de~~~~~ c~b~~ $40 ~ ~~eg~ hice ga~~~~ ~~~ $75 . ~C~~~~~ ~e~g~? E~~i~~e hace ~~a c~~~~a ~~~ $67 ; de~~~~~ ~ecibe $72 ; ~~eg~ hace ~~~a c~~~~a ~~~ $1( ; ~ de~~~~~ ~ecibe $2 . E~~~e~a~ ~~ e~~ad~ ec~~~~i~c . De~~~~~ de ~ecib~ 20 c~~~~e~ hag~ ~~e~ ga~~~~ ~~~ 78, 81 ~ 93 . Recib~ e~~~~ce~ 41 ~ ~~eg~ hag~ ~~ ~~e~~ ga~~~ ~~~ 59 . ~C~~~~~ ~e~g~? Ped~~ ~e~~a ~~e~ de~da~ de $45, $6 ~ $79 ~e~~ec~i~a~e~~e . E~~~~ce~ ~ecibe $20 ~ hace ~~ ga~~~ de $10 . ~C~~~~~ ~ie~e?

2) A ~a~ 6 a . ~ . e~ ~e~~~~e~~~ ~a~ca - 04 . A ~a~ 9 a . ~ . ha ~~bid~ 7~ ~ de~de e~~a h~~a ha~~a ~a~ 5 ~. ~ . ha ba~ad~ 1~ . E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 5 ~ . ~ . A ~a~ 6 a . ~ . ~a~ca -4~ . C~~~ a ~a~ 9 a . ~ . ha ~~bid~ 7~, c~~~a~~~ ~ie~e di~i~i~~e~ de ~a e~ca~a de~de -4~ haci a~~iba ~ ~e~d~e~~~ 3~ ~~b~e . ~i . ah ba~ad~ 1~, c~~~a~d~ ce~~ (+3~) ; c~~~ de~de e~~a h~~a ha~~a ~a~ 5 ~ 1 di~i~i~~e~ de ~a e~ca~a de~de +3~ haci ab~~ ~~ega~e~~~ a -8~ . L~eg~, a ~a~ 5 ~ . R . . ~ . ~a ~e~~e~a~~~a e~ de -8~ .

EJRCIO 2

1 .

A ~a~ 9 a . ~ . e~ ~e~~~~e~~~ ~a~ca +12~ ~ de e~~a h~~a a ~a~ 8 ~ . ~ . ha ba~ad~ 15 .~ . ~ . E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 8 ~ ~a~ 6 a . ~ . e~ ~e~~~~e~~~ ~a~ca -3~ . A ~a~ 10 a . ~ . ~a ~e~~e~a~~~a e~ 8 ~ ~~~ a~~a ~ de~de e~~a h~~a ha~~a ~a~ 9 ~ . ~ . ha ba~ad~ 6~ . E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 9 ~ .~ . A~a 1 ~~ ~ . e~ ~e~~~~e~~~ ~a~ca +15~ ~ a ~a~ 10 ~ 0 . . ~ . ~a~ca -3 ~C~~~~~~ g~ad~~ ha ba~ad~ ~a ~e~~e~a~~~a? ~a~ 3 a . ~ . e~ ~e~~~~e~~~ ~a~ca -8~ ~ a~ ~edi~d~a +5~ . ~C~~~~~~ g~ad~~ ha ~~bid~ ~a ~e~~e~a~~~a? ~a~ 8 a . ~ . e~ ~e~~~~e~~~ ~a~ca -4~ ; a ~a~ 9 a . ~ . ha ~~bid~ 7~ ;a ~a~ 4 ~ . ~ . ha ~~bid~ 2~ ~~~ ~ a ~a~ 1 ~ . E~~~e~a~ . ~ . ha ba~ad~ 1 ~ ~a ~e~~e~a~~~a a ~a~ 1 ~ .~ . ~a~ 6 a . i~ . e~ ~e~~~~e~~~ ~a~ca -8~ . De ~a~ 6 a . ~ . a ~a~ 1 a .~ . ~~be a ~a~~~ de 4~ ~~~ h~~a . E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 7 a . ~ ., a ~a~ 8 a . ~ . ~ a ~a~ 1 a .~ . ~a~ 8 a . ~ . e~ ~e~~~~e~~~ ~a~ca -1~ . De ~a~ 8 a . ~ . a ~a~ 1 a . ~ . ba~a a ~a~~~ de 2~ ~~~ h~~a ~ de 1 a . ~ . a 2 ~ . M . ~~be a ~a~~~ de 3~ ~~~ h~~a . E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 10 a . ~ ., a ~a~ 1 a . ~ ., a ~a~ 12 a .~ . ~ a ~a~ 2 ~ .~ . E~ d~a 10 de dice~b~e ~~ ba~c~ ~e ha~~a a 56~ a~ ~e~~e de~ ~~i~e~ ~e~ida~~ . De~ d~a 10 a~ 18 ~ec~~~e 7~ haci e~ e~~e . E~~~e~a~ ~~ ~~~gi~~d e~~e d~a . : 71~ de ~~~gi~~d E~ d~a ~~i~e~~ de feb~e~~ ~a ~i~~aci~~ de ~~ ba~c~ e~ ~e~~e ~ 15 . De~ d~a ~~i~e~~ a~ 26 ha ~ec~~~id~ 5~ haci ~ de ...