Ecuaciones de Primer Grado

5/22/2018 Ecuaciones de Primer Grado

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LICEO N1 JAVIERA CARRERA 2012 MATEMATICA Benjamn Rojas F.

ECUACIONES DE PRIMER GRADO

Una ecuacin es una igualdad que tiene un trmino desconocido llamado incgnita y que generalmen-

se simboliza con la letra x. Resolver una ecuacin es encontrar el valor de esa nica incgnita !ara esto de-bemos a!licar las !ro!iedades de una igualdad vistas anteriormente.

ecuaciones algebaicas" #e resuelven a!licando los conce!tos b$sicos del algebra.

%&em!lo" Resuelve las siguientes ecuaciones "

1' ( ) ( )1x2(2x( +=+

2' ( )[ ] ( )[ ](x21(1x211x) =++ (' ( ) ( ) x*1x(x

22=+

+' ( ) ( )2x+,1x)x2 +=+

)' ( ) ( ) ( )222

x2x+x2 =

,' ( )( ) ( )( )(x2,x1x22x ++=++

( ) ( ) ( )[ ] ( )[ ]

[ ] [ ]

10 x

10x

211(x,x2x))- x

x,(2x211x),2(x2(x

(x21(2x211x)2x2(,(x

(x21(1x211x)2'1x2(2x(1'

=

=

++=+=

+=++=

+=+++=+

=+++=+

( ) ( ) ( ) ( )

( )

1 x

((x x

),x+x)x2*x1x2xx, x

x+,)x)x2x*1x2xx, x

2x+,1x)x2+'x*1x(x'(

22

22

22

=

==

+=+=+++

+=+=+++

+=+=+

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

) xx

1,10x(,+x

1 /1,x10+(2,+1,x+xx1,

121)x)x+,x1,x+x+x(2x1,2x

1x1)2x2x)2x,+x1,x+x+x1,xx2

(x2,x1x22x,'x2x+x2')

222

22222

222

==

==

=+=++

+=+=++

++=+++=++

++=++=

ecuaciones con coe!icien"es !accionaios" #e resuelven calculando el entre todos los denominado-

res y multi!licando este valor !or todos los trminos de la ecua-cin de tal 3orma de eliminar los denominadores de la ecuacin.

4uego se resuelve la ecuacin que se obtiene como cualquier o-

tra ecuacin con coe3icientes enteros.

%&em!lo" Resuelve las siguientes ecuaciones"

1' x2(x =

2' ,x(

1

,

x

=+

('

(x)

,

1x2

(x +=

+

+'+

10x)

(x2

,

1x +=


2/9

)' ( ) ( ) 1(x2(

11x2 =+

,' ( )12

1x

(

1x+

,

1)

+

(x21x( ++

=+

( ) ( ) ( )

x(x

,(*x,2x

1 /(,x**

(+ x1 /,x2

122*2+x+0x2+x(+*x,x(x

2+x+012x2+2*x2(,x, xx(, x

(x)1x212(x(,x,2xx(2((

x(

/*2

(x)

,

1x2

(x('/,,x

(

1

,

x2'x /(2

(

x'1

==

==

===

++=+==

+=++=+=

+=+=+=

+=

+

=+=

( ) ( )

( ) ( ) ( ) ( ) ( )

1*) x

2

( x1*)x

,+x1 /1*)x

,((x2,x120+,0x,x,+x

((x2,,x,0x,120x,++x

((x211x,10x,120(x2(1x+

/(1(x2(

11x2)'/2+

+

10x)

(x2

,

1x'+

=

==

==

=++=

=+++=+

=++=

=++

=

( )

( ) ( ) ( )

( x

,2x

(0(,1+x12x1,x,(,x

1x12+x1,(0x,(,(,x

1x121x++1)2(x2(1x(,

/12121x

(1x+

,1)

+(x21x(',

=

=

++=

++=++

++=+

++

=+

ecuaciones con inc#gni"a en el $eno%ina$o" #e utiliza el mismo mtodo usado !ara resolver una ecua-

cin 3raccionaria es decir encontrar el entre todos los

denominadores y multi!licar cada trmino de la ecuacin

!or este valor.


1'(x2

*

(x

)

+

=

+

2'1x

(

1x

2

1x

12

=

+

('( ) 2x2

+

2x2

+

1x

12

=

+

+

+' 112x*x

1

(x

1x,

+x

)x*

2

=

+

+

)'12xx,

**x()x2

+x(

)x

(x2

*x,2

2

+

+=

+

+

,' 2)x)

1x+

+x+

2x(

(x(

(x2

2x2

x)=

++

+

2


3/9

( ) ( )

( ) ( ) ( ) ( )( ) ( )

( ) ( )

( ) ( ) ( ) ( )

(

+ x2 x

+(x,(x

1 /+(x1 /,(x

12(x2x1)21x*x10

(2x21x21x*1)x10

(1x21x1(x*(x2)

1x1x /1x1x

(

1x

2

1x

1

(x2

*(x2(x

(x

)(x2(x

1x

(

1x

2

1x

12'(x2(x /

(x2

*

(x

)'1

2

==

==

==

+==

=++=+

=++=+

+

+

=

++

++=

+

++

=

+

++

+

=

+

( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )

2 x

(21,x

1+1)12x*x2)x1,

12x*x1+x2)x,1)x1,x*

12x*x1+x2)x,1)x1,x*

(x+x11x,+x)x*(x

(x+x /1(x+x

1

(x

1x,

+x

)x*

112x*x

1

(x

1x,

+x

)x*+'

(

) x

)(x

1 /)x(

212x+ x

2x22x+x21 x

1x21x2x21 x

1x1x21x21x

1x1x /1x2

+

1x2

+

1x

1

2x2

+

2x2

+

1x

1('

222

222

2

22

22

2

2

2

2

=

=

+++=++

+=+

+=+

=+

=

+

=

+

+

=

=

=

=

=+++

=+++

+=++

+

=

+

+

=

+

+

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

) x

0x1

1)2**x()x1+x(

**x()x21)x1+x1,2x(x1

**x()x21)x1+x1,2x(x1

**x()x2)x(x2*x,+x(

+x((x2 /+x((x2

**x()x2

+x(

)x

(x2

*x,

12xx,

**x()x2

+x(

)x

(x2

*x,')

222

222

2

2

2

2

=

=

+=+

+=+

+=+

+=++

+

+

+=

+

+

+

+=

+

+

(


4/9

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1(

102 x

1021(x

1 /1021(x

12(0,0120x120x+x+)x+01)0x

120x12012x+(0x+),0x+01)0x

1x1201x+122x(1)(x220x)(0

1x /,021x)

1x+

1x+

2x(

1x(

(x2

1x2

x)

2)x)

1x+

+x+

2x(

(x(

(x2

2x2

x)',

=

=

=

+=+

=+++

=+++

=

++

+

=

++

+

ecuaciones li"eales" #on ecuaciones en las que algunos o todos los coe3icientes numricos son letras se re-

suelven como cualquier ecuacin vista anteriormente.


1' a2xa =+

2' +xb2xa =

(' a2

1

a

x1

a2

x2 =

+' ( ) ( )axbbabxa 22 ++=+

)' (2

(

ba2

x+=+

+

,' ( )

aa

b

b

xba

a

bxa 22=+

*' x(mxm 2 =+

' ( ) ( ) xa12x2axax ((((

+=++

'a

+x

+

ax =

10' 222bxa

ba

bxa

xa

bxa

bxa

=

+

( )

( )

( )

1 x

2a

2a x

2a2a x

2ax2xaba

2 x

ax22xa2ba xa

2ax

ax12xa+2xbxa2axa

/2aa2

1

a

x1

a2

x('+xb2xa2'a2xa1' 2

2

=

+

+=

+=+

+=+

=

=+=

=

===

=

==+

( ) ( ) ( )

( ) ( )

( ) ( )

( )

( )

ba2( xba x

b(a, x

ba

ba x

b(a,b,a21xbaba xb,a12(ba,xbababaxbxa

ba2,ba2(xbaxbbabaxa

ba2 /2(2

(

ba2

x+)'axbbabxa+'

2

2

22

22

22

+==

+=

=

+==

+=+++=

+=++++=+

+=+

+

++=+

+


5/9

( )

( ) ( )( ) ( )( )

( )( )

( )

( )

ba

2ab x

baa

ba2 x

(m xba2baxa

(m

(m(mxbbabbaxaxab

(m(m(m xbabxababxab

mx(xmbabxbabxab

x(mxm*'/abaa

b

b

xba

a

bxa',

2

2

(2(22

2(22(

22(22

222

+

=

+

=

==+

+

+=++=+

++=+=++

=+=+

=+=+

( ) ( )

( )

0 x

a12a(a(

0 x

0a12a(a(x

aaxa12xa(xa(

xa12x2axa(xa(xaxa(xa( x

xa12x2axax'

(22

(22

(((22

(((22((22(

((((

=

+

=

=+

+=+

+=+++++

+=++

( ) ( )

( )

( ) ( )

+a x

+a

+a+a x

1,a+a x

1,ax+xa

1,x+axa

+x+axa

/+aa

+x

+

ax'

2

2

2

+=

+=

=

=

=

=

=

( )( ) ( )( )

( ) ( )

( )

(a

ab x

(ab

abb x

abb(abx

1 /bab(abxbababxabx2

ababxxababx2xa

abbxaxabxa

bxabxa /bxabxa

ba

bxa

xa

bxa

bxa

bxa

ba

bxa

xa

bxa

bxa'10

2

2

2

22222

2

222

=

=

=

=

=

=+

=+

+

+

=

+

=

+

ecuaciones co%&ues"as" #e resuelven am!li3icando un lado de la igualdad o ambos lados !or el cal-

culado entre los denominadores de las 3racciones com!uestas con el !ro!sito de

trans3ormar la ecuacin en una ecuacin m$s sim!le.


)


6/9

( )

( ) ( )

1)

)2 x

)21)x

1 /)2x1)

(,*2(0x1)x

*2x(0(,1)x

12x),12x)(

12x) /+

(

12x)

11

(

1).esy1)y)(resdenominadolosentreculado+

(

/1))

+x

(

1

/1)1)

11

(

-calel!or

)

+x

(

11)

11

3raccinlaam!li3icase*)0

)

+x

(

11)

11

('1

=

=

=

++=

=

=

=

=

=

( )

( ) ( )

,

(x

(,x

+x2x

x2x+

x21x,

2.esy+y2resdenominadolosx2 /

1

x2

x,

entrecalculadoel!or+

2

1

3raccinlaam!li3icase

1

x2

x,

(.esy(y1resdenominadolosentre/2+

/22

1

/((

x2(

/((

x2

calculadoel!or

(

x2(

(

x2

3raccinlaam!li3icase+

2

1

(

x2(

(

x2

'2

=

=

=

=+

=+

=

+

=

+

=

+

+

=

+

,


7/9

( ) ( )( ) ( )

( )

( ) ( )

1 /(0)x2)0

22)0x1,0x0

0x1,022)x0

x1,10*)x(0(

x1, /((

10

x1,

*)x(0

(

10

x1,

*)x(0

(

10

x2+++0+0x

1()x0,0,0x

(

10

x22+1x+0

(x2+)1x,0

(,0.esy1)yentre,culado(

10

/(,01)

x2

1x

/(,0

(x2

,

1x

-calel!or

1)

x2

1x

(x2

,

1x

3raccinlaam!li3icase(

1(

1)

x2

1x

(x2

,

1x

'(

=

+=

+=

+=

+=

+

=

+

=

+

+

=

+

++

=

+

+

+

+

+

+

=

+

+

+

2)0

(0) x

(0)2)0x

=

=

( )

( )

2 x

21

x

2x1x

x

x.esyresdenominadolosentre2x1x

x

calculadoel!or

x

x11

13raccinlaam!li3icase2

/xx

x11

/x1

2

x

x11

1

x1esyresdenominadolosentre2

1x1

x11

1

calculadoel!or

x1

11

13raccinlaamli3icase2

x1 /x1

11

x1 /11

1

2

x1

1

1

11

1'+

=

=

=

=

+

+

=+

=+

+=

+

+

+

=

+

+

+

=

+

*


8/9

( )

( )

( )

2

2

2

x

1

x1x

1x1

11x

x

.1xesyresdenominadolosentrex

1

1x /1x

x1

1x /11

1

1x

x

calculadoel!or

1x

x1

13raccinlaamli3icase

x

1

1x

x1

11

11x

x

')

=

+

=

+

=

+

( )

( )

( )

( )

( ) ( ) ( )

( )

( )

(

2 x

2(x

2x(x x

2x(x1 x

2x(x /x

x

1

2x(x

1

x

1

2x(x

1xx

.1xesyresdenominadolosentrex

1

x21x

1x1x

calculadoel!orx2

11x

x

3raccinlaam!li3icasex

1

1xx /2

1x /11x

x

x

1

x2

1

1x

x

x

1

1x1

11x

x

x

1

1x1

11x

x

x1

1

1x1

11x

x

22

22

22

22

22

2

2

2

2

2

2

=

=

+=

+=

+=

+

=

+

+

=

=

=

=

+

=

=

+


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Ecuaciones de Primer Grado