Chuyen de He Phuong Trinh Dai So

8/3/2019 Chuyen de He Phuong Trinh Dai So

1/70

NHOM THC HIEN

Dng Minh Ch(Bin tp)

V Hoi BoHunh Nguyn Minh T

Phm Anh ThH Ngc Trm

Nguyn Bch Vi

L Hu Tr(Kthut)

Trn Ch ThinNguyn Phc HinNguyn Thanh ViTrn Th Anh Th

Nguyn Th Qu Chi

Gio Vin: Hunh Ch Ho


2/70

http://www.ebook.edu.vn1

Mng ton h phng trnh l mng ton khng kh

lm, nhng n li c nhiu dng v nhiu cch gii khc

nhau, thi gian gn y (t 2002 n 2010) cc thi Cao

ng v i hc thng cho m nh im l nm 2010

thi khi A v khi B c cc cu ny, a s cc hc sinh

u bngv cha tng luyn gii loi ny.

Nay nhm mnh son quyn chuyn H phngtrnh nhm gip cc bn c ti liu tham kho v n tp.

Trong cun chuyn ny nhm mnh chia ra tng loi

cc bn s thun tin hn trong vic tham kho.

Trong quyn chuyn ny cc bn sc gp mt

s cch gii hay, ngn gn v chnh xc. Mc d c gng

khi bin son, nhng khng th trnh khi vi thiu st,

mong cc bn gp v thng cm quyn chuyn ny

c hon thin hn.

Nhm 2

vntoanhoc.com


3/70

2

& &Muc Luc

Trang

Phn 1 CC H PHNG TRNHTHNG THNG

1. H phng trnh bc nht hai n...4

2. H phng trnh bc nht ba n9

3. H phng trnh i xng loi I.16

4. H phng trnh i xng loi II245. H phng trnh ng cp bc hai30

Phn 2 CC H PHNG TRNH KHC

1. H phng trnh bc cao hai n..36

2. H phng trnh v t......40

3. H phng trnh khng mu mc...47

4. H phng trnh dng phng php hnh hc vect...56

5. H phng trnh trong cc k thi.60


4/70


1Cac He Phng Trnh

Thong Thng


5/70

4

1. H PHNG TRNH BC NHT HAI N

1/ Dng:

1 1 1

2 2 2

a x b y c

a x b y c+ =+ =

Cch gii:php th, php cng...

Php th:

V d 1: ( )2 1

12 3

+ =

=

x y

x y

Gii

( )1 2 1 2

12 3 2(1 2 ) 3

1 71 21 2

5 51

15 5

x y x y

x y y y

xx y

yy

= =

= =

= == =

=

Vy ( )7 1

; ;5 5

x y x y

= = =


6/70


Php cng:

V d 2:2 2 5 2 2 5

3 6 2 6 12

x y x y

x y x y

+ = + =

= =

Cng 2 v phng trnh li, ta c:4 17

174

y

y

=

=

Gii h phng trnh (1) ta c27

4x

=

Vy ( )27 17

; ;4 4

x y x y

= = =

2/ Gii v bin lun phng trnh:

Bc 1: Tnh cc nh thc:

1 1

1 2 2 1

2 2

a ba b a b

a bD = =

(gi l nh thc ca h)

1 1

1 2 2 1

2 2

x

c bc b c b

c bD = =

(gi l nh thc ca x)1 1

1 2 2 1

2 2

y

a ca c a c

a cD = =

(gi l nh thc ca y)


7/70

6

Bc 2:Bin lun:

NuD 0 th h c nghim duy nht:

;yx

x yDD

D D= =

NuD = 0 vDx 0 v Dy0 th h v nghim

NuD = Dx= Dy= 0 th h c v s nghim

hoc v nghim.

V d 1: Gii h phng trnh:

5 2 9

4 3 2

x y

x y

=

+ =

5 215 8 23

4 3Ta c: D

= = + =

9 2

27 4 232 3

xD

= = + =

5 9

10 36 464 2

yD

= = + =

23 46

1 223 23

;yx

x yDD

D D

= = = == =

Vy nghim ca h phng trnh l (1;2)


8/70


V d 2: Gii v bin lun h phng trnh:

1

2

mx y m

x my

+ = +

+ =

Gii

2

2

1 1 ( 1)( 1)1

1 12 ( 1)( 2)

2

12 1 1

1 2

x

y

mm m m

m

mm m m m

m

m mm m m

D

D

D

= = +

+= + = +

+= =

=

=

=

Bin lun:NuD 0 m 1 th h phng trnh

c nghim duy nht l

2

11

1

mx

m

ym

+ = + = +

NuD = 0 m = 1 th:

Khi m = 1 ta c h phng trnh:2

22

x yx y

x y

+ = + =

+ =

H c v s nghim: ( )0 00

2

=

=

x xx

y x


9/70

8

Khi m = 1 ta c h phng trnh:

0

2

0

(VN)2

x y

x y

x y

x y

+ =

=

=

=

3/ Gii h phng trnh bng phng php t n ph:


( )3( ) 9 2( )

22( ) 3( ) 11

x y x y

x y x y

+ + =

+ =

Gii

( )3( ) 2( ) 9

22( ) 3( ) 11

x y x y

x y x y

+ = + =

t u = x + y; v = x y

Ta c h phng trnh:

3 2 9 12 3 11 3

1 1

3 2

u v uu v v

x y x

x y y

= = = =

+ = =

= =

Vy h phng trnh c nghim ( 1; 2)


10/70


2. H PHNG TRNH BC NHT BA N

1/ Cc phng php chung: Nguyn tc chung gii h phng trnh nhiu n

vn l bin i h phng trnh cho thnh nhng htng ng hoc nhng h phng trnh h qu d gii

hn, (trong c nhng phng trnh vi sn ngy cngt). t c iu ny ta thng dng:-Phng php cng i s-Phng php th

Nu c dng php bin i khng tng ng thcn phi th li cc gi tr tm c ca n.V d 1: Gii h phng trnh :

2 2

3 3

3

( ) 5

9

x y z

I x y z

x y z

+ =

+ = + =

Gii

2 2

3 23

2 2

3 2

3 3

( ) ( ) 2 5 9 5 2

( ) 3 ( ) 9 9 527 9 9

2

3 3

2 9 5 2 9 5

23 5 2 0 0 13

+ = + =

+ = = + + = =

+ = + =

= = + = = = =

x y z x y z

I x y xy z z z xy

x y xy x y z z z z z z

x y z x y z

xy z z xy z z

z z z z z z


11/70

10

Vi 0z= ta c :

0

0

0

x y

xy

z

+ =

= =

H ny c nghim ( )0;0;0

Vi 1z= ta c :

3

2

1

x y

xy

z

+ =

= =

Gii h ta c hai nghim : ( ) ( )1;2;1 , 2;1;1

Vi 23z= ta c :

2

1

32

3

x y

xy

y

+ =

=

=

H c hai nghim :3 6 3 6 2 3 6 3 6 2

; ; , ; ;3 3 3 3 3 3

+ + .

Vy h (I) c 5 nghim :

( ) ( ) ( )3 6 3 6 2 3 6 3 6 2

0;0;0 , 1;2;1 , 2;1;1 , ; ; , ; ;3 3 3 3 3 3

+ +


12/70


V d 2: Gii h phng trnh :

( )

12

III 20

15

xy

yz

zx

=

= =

Nh

n xt: Nghin cu cch gii. Hy th gii h ny bngphng php th.

Tuy nhin c th nhn xt v du ca , ,x y z ramt cch gii cch.

R rng , ,x y z khc 0 v cng du.

Gii

Nhn v vi v ca ba phng trnh ta c mtphng trnh; kt hp vi hai trong ba phng trnh cho

ta c h:

( )2

12

20

3600

xy

yz

xyz

=

=

=

Ta d thy rng ( ) ( )III IV

V ( )2 60

360060

xyzxyz

xyz

== =

Nn h ( )IV tng ng vi hai h :

( )

12

V 20

60

xy

yz

xyz

=

= =

v ( )

12

VI 20

60

xy

yz

xyz

=

= =


13/70

12

Gii ( )V : Thay 12xy = vo phng trnh th bat a c

5=z ; thay 20yz= vo phng trnh th ba ta c 3=x .Thay 3, 5= =z vo phng trnh th ba ta c 4=y .

H ( )V c nghim: ( ).3;4;5

Gii ( )IV : Tng t h ( )IV c nghim ( )3; 4; 5 . Vy h ( )III c hai nghim: ( ) ( )3;4;5 , 3; 4; 5 .

2/ p dng h thc Vit i vi phng trnh bc ba:

H thc Vit i vi phng trnh bc ba c phtbiu nh sau :

NH L:Nu phng trnh 3 2 0+ + + =ax bx cx d c

ba nghim 1 2 3, ,x x th :

1 2 3

1 2 2 3 3 1

1 2 3

+ + =

+ + =

=

bx x x

a

cx x x x x x

a

dx x x

a

Ngc li, nu ba s 1 2 3, ,x x x tha mn cc ng thc :

1 2 3 1

1 2 2 3 3 1 2

1 2 3

x x x S x x x x x S

x x x P

+ + = + + = =

th chng l ba nghim ca phng trnh :

3 21 2 0 + =x S x S x P


14/70


Theo nh l trn y c thgii h phng trnh ba n

1 2 3 1

1 2 2 3 3 1 2

1 2 3

+ + =

+ + =

=

x x x S

x x x x x x S

x x x P

bng cch gii mt phng trnh bc ba

3 21 2 0 + =x S x S x P


1 2 3

1 2 2 3 3 1

1 2 3

1

10

8

x x x

x x x x x x

x x x

+ + =

+ + = =

GiiTheo nh l Vit , ,x y z l ba nghim ca phng trnh

3 2 10 8 0x x x+ + = Ta thy 1=x l mt nghim ca phng trnh trn. Do

phng trnh trn c vit thnh : 2( 1)( 2 8) 0x x x + =

Gii phng trnh ny ta c : 1 2 31, 4, 2x x x= = =

V h phng trnh ny i xng i vi , ,y z nn h c

3! 6= nghim sau :

( ) ( ) ( ) ( ) ( ) ( )1; 4;2 , 1;2; 4 , 2; 4;1 , 4;1;2 , 2;1; 4 , 4;2;1

V d 4:Gii h phng trnh :

2 2 2

4

( ) 22

18

x y z

II x y z

xyz

+ + =

+ + = =


15/70

14

GiiBin i h ny thnh h c dng ( )I . Bnh phng hai v

phng trnh th nht ri tr tng v vi phng trnh thhai ta c :

4 4

2( ) 4 hay 3

18 18

+ + = + + =

+ + = + + = = =

x y z x y z

xy yz zx xy yz zx

xyz xyz

Theo nh l Vit, , ,y z l ba nghim ca phng trnh :3 24 3 18 0x x x+ + =

Ta c th thy 2x = l mt nghim. Do :( )( )22 6 9 0x x x + + =

Gii phng trnh ny ta c : 1 2 32, 3x x x= = =

Vy h c ba nghim: ( ) ( ) ( )2; 3; 3 , 3;2; 3 , 3; 3;2


( ) 2 2 2

3 3 3

2

III 6

8

x y z

x y z

x y z

+ + =

+ + = + + =

Gii

( ) ( ) ( )

( ) ( ) ( )

2

2 2 2 2 2 2 2 2 2

2

III 2 6

8

x y z

x y z xy yz zx

x y z x y z xy x y yz y z zx z x

+ + =

+ + + + =

+ + + + + + + + + =

2

2 2 2

2

( ) 2( ) 6

( )( ) ( )( ) 3 8

x y z

x y z xy yz zx

x y z x y z xy yz zx x y z xyz

+ + =

+ + + + = + + + + + + + + + =


16/70


2

1

2

x y z

xy yz zx

xyz

+ + =

+ + = =

Vy , ,y z l ba nghim ca phng trnh bc ba2 2 2 0t t t+ = hay 2( 2)( 1) 0t t+ =

Phng trnh ny c nghim l : 1 2 31, 1, 2t t t= = =

Do h phng trnh c 6 nghim l hon v ca ba s1,1, 2 :

( 1;1; 2), ( 1; 2;1), (1; 1; 2), (1; 2; 1), ( 2; 1;1), ( 2;1; 1)

BITPRNLUYN

Gii cc h phng trnh :2 2 2

2 2 2 2 2 2

2 2 2 3 3 3

( ) 2 2 1

a) ( ) 3 b) 3 c) 9

( ) 4 4 1

+ = + = + + + =

+ = + = + + + = + = + = + + + =

y z x zx xy x x y z

z x y xy yz y x y z

x y z yz zx z x y z

2

2

2

1 1 13

6 ( ) 13( )( )1 1 1

d) ( )( ) 2 e) 3 f) 3 ( ) 5

( )( ) 3 6 ( ) 51

1

+ + =

+ =+ + =

+ + = + + = + = + + = + =

=

x y z x y z yz x y z x x

y z x y y y z x zxxy yz zx

z x y z z z x y xy

xyz


17/70

16

3. H PHNG TRNH I XNG LOI I

TM TT L THUYTV PHNG PHP GII TON:

1/ Dng tng qut ca hi xng loi I:nh ngha: Hi xng loi I l h cha 2 n x,y

m khi ta thay i vai tr x,y cho nhau th h phng trnhkhng thay i.

( )

( )

, 0

, 0

f x y

g x y

=

=, trong

( ) ( )

( ) ( )

, ,

, ,

f x y f y x

g x y g y x

=

=

Phng php gii tng qut:

i) Bc 1: t iu kin (nu c)ii) Bc 2: tS = x + y;P = xy (viS2 4P) .

Khi , ta a h v h mi chaS,P.

iii) Bc 3: Gii h mi tmS,P. ChnS,Ptha mn

S2 4P.

iiii) Bc 4: ViS,Ptm c thx,y l nghim caphng trnh:

X2

SX + P = 0

( nh l Vit o)*Ch :

i) Cn nh:2 2 2

3 3 3

2

3

x y S P

x y S SP

+ =

+ =


18/70


ii) i khi ta phi t n ph:

( )

( )

u u x

v v x

=

=v

S u v

P uv

= +

=

iii) C nhng h phng trnh trthnh

hi xng loi I sau khi ta t n ph.

2/ Mt s v d minh ha:

V d 1:Gi h phng trnh sau:

2 2

2 2 7 8x y xyx y x y

+ + = + + + =

(1)

GII

t:S x y

P xy

= +=

, vi S2 4P.

Khi , h (1) trthnh:

( )

22

2 22

2

2

2

77

2 7 82 8

77

36 0

2

= =

+ = + =

= =

= = =

P SS P

S S SS P S

P SP S

SS S

S


19/70

18

Vi: 3 2S P= = . Khi , x v y l nghim ca phng

trnh: 2 3 2 0X X + =

1

21

2 21

x

yX

X x

y

= ==

= = =

Vi: 2 3S P = = . Khi , x v y l ngim ca

phng trnh: 2 2 3 0X X+ =

1

31

3 31

= = =

= = =

x

yX

X xy

Vy h cho c 4 nghim

(x,y) = (1;2), (2;1), (1;3), (3;1).


2 22 2

1 1 5

1 19

x yx y

x yx y

+ + + = + + + =

( H Ngoi Thng TPHCM - Khi A,D nm 1997)


20/70


GII

t:

2

2

22

22

1 12

1 1 2

u x x ux x

v y y vy y

= + + =

= + + =

Khi , h (1) trthnh:

( )

2 2

2

5

13

5

2 13

5

6

u v

u v

u v

u v uv

u v

uv

+ =

+ =

+ =

+ =

+ =

=

u, v l nghim ca phng trnh: X2 5X+ 6 = 0

3

2

2

3

3

2

= =

= =

=

=

X

X

u

v

u

v


21/70

20

Trng hp 1: u = 2; v = 3

12

13

1 1

3 5 3 5

2 2

xx

y

yx x

y y+

+ =

+ =

= =

= =

Trng hp 2: u = 3; v = 2

1 3

12

3 5 3 5

2 2

1 1

xx

yy

x x

y y

+

+ =

+ =

= =

= =

Vy h cho c 4 nghim (x,y) l

3 5 3 5 3 5 3 51; , 1; , ;1 , ;1

2 2 2 2

+ +

.


22/70


3/ iu kin tham s hi xng loi I c nghim:

Phng php gii tng qut:

i) Bc 1: t iu kin (nu c).

ii) Bc 2: tS = x + y; P = xy vi iu kin caS,Pv S2 4P(*).

iii) Bc 3: Thayx,y biS,Pvo h phng trnh.Gii h tmS,Ptheo m, ri tiu kin (*) tm m.

(vi m l tham s)

V d 3: Tm iu kin m h phng trnh sau cnghim:

( )

4 1 4

13

+ =

+ =

x y

x y m

GII

t:4 0

1 0

=

=

u x

v y

Khi , h (1) trthnh:

2 2

4

3 5

u v

u v m

+ = + =

4

21 32

u v

muv

+ =

=

Suy ra u,v l nghim (khng m) ca phng trnh:

2 21 34 0 (*)2

mX X

+ =


23/70

22

Theo , h (1) c nghim Pt (*) c 2 nghim khng m.

3 13' 0 01320 7.

21 3 300

2

m

P mm

S

Vy 13 73

m l gi tr cn tm.

V d 4: Tm m h phng trnh sau c nghim thc:

1(1)

1 3

x y

x x y y m

+ =

+ =

GII

iu kin: 0; y 0x

Khi :

( ) ( )3 3

11

1 3 1 3

+ =+ =

+ = + =

x yx y

x x y y m x y m

t: ( )2

0; P= 0 4S x y xy S P = +

H phng trnh trthnh:

3

1 1

3 1 3

= =

= =

S S

S SP m P m


24/70


H (1) c nghim thc2 4

1 4 10 0 m

0 40

S Pm

Pm

S

Vy1

04

m l gi tr cn tm.

BI TP RN LUYN

Bi 1: Gii h phng trnh: 2 219

133

x y xy

x y xy

+ + =

+ + =.

Bi 2: Gii h phng trnh:2 2

2 2

1 14

1 14

x yx y

x yx y

+ + + = + + + =

.

Bi 3: Tm m h phng trnh c ng 2 nghim thc

phn bit.2 2

2

2 1

4

( )

( )

y m

x y

+ = + + =

Bi 4: Tm m h phng trnh sau c nhgim thc:2 2

4 4 104 4

( )( )

x y x yy x y m

+ + + =+ + =

Bi 5:Tm m h phng trnh c nghim thcx > 0,y > 0:

2 2

1x xy y m

x y xy m

+ + = + + =


25/70

24

4. H PHNG TRNH I XNG LOI II

1/ nh ngha:

H phng trnh i xng loi II l h cha hai nx,y mkhi ta thay i vai trx,y cho nhau th phng trnh ny tr

thnh phng trnh kia ca h.*Ch :Nu 0 0( ; )y l nghim ca h th 0 0( ; )y x cng l nghim ca h.

2/ Cc dng ca h phng trnh i xng loi II:

DDnngg 11::( , ) 0

( , ) 0

f x y

f y x

=

=(i v trx vy cho nhau th

phng trnh ny trthnh phng trnh kia).

Phng php gii chung:Tr v vi v hai phng trnh v bin i v dng

phng trnh tch s.

Kt hp mt phng trnh tch s vi mt phng trnhca h suy ra nghim ca h

V d1:Gii h phng trnh sau:2

2I 2( ) 2x y

y y x

= =

Nhn xt: Nu thay ng thix biy vy bix thphng trnh th nht s trthnh phng trnh th hai vngc li.


26/70


Gii

Tr tng v hai phng trnh trong h, ta c

( )( ) 2( ) ( )

( )( 1) 0

0

1 0

y x y x y x y

x y x y

x y

x y

+ =

+ =

=

+ =

Do , h phng trnh cho tng ng vi:

2

0(Ia)

2

x y

x x y

=

=hoc

2

1 0(Ib)

2

x y

x y y

+ =

=

Gii h (Ia) ta c nghim (0;0), (3;3).

Gii h (IIa) ta c nghim:

1 5 1 5 1 5 1 5; , ;2 2 2 2

+ +

Vy h phng trnh c 4 nghim l

(0;0), (3;3),1 5 1 5 1 5 1 5

; , ;2 2 2 2

+ +

DDnngg 22::( , ) 0

( , ) 0

f x y

g x y

= = (trong ch c 1 phng trnh

i xng loi I)

Cch gii:a phng trnh i xng v dng tch,giiy theox ri th vo phng trnh cn li.


27/70

26

V d 2: Gii h phng trnh:2

1 1(1)

2 1 0 (2)

x yx y

x xy

= =

Giiiu kin: 0; y 0x . Khi :

1(1) ( ) 1 0 1

x yx y

xy yx

= + = =

Vix = y th (2) 2 1 0 1x x = =

Vi1

y = th (2) v nghim

Vy h phng trnh c 2 nghim phn bit (1;1), (1;1).

3/ Mt s bi tp v phng trnh i xng loi II :

V d 3: Gii h phng trnh:2

2

3 2

3 2

x y

y y x

=

=

Gii

Tr v theo v ca hai phng trnh, ta c:2 2 3 3 2 2

0(x-y)(x+y-1) 01 0

x y x y y x

x yx y

+ =

= = + =

Vy h phng trnh cho tng ng vi:2 3 2

(I)0

x y

x y

=

=hoc

2 3 2(II)

1 0

x y

x y

=

+ =


28/70


Gii (I):2 ( 5) 03 2

( ) 0 5x xx x x

I x y x yx yx y

= = = = = =

==

Gii (II):2 23 2(1 ) 2 0

( ) 1 1

1 2

2 1

x x x x x

II y x y x

x x

y y

= =

= =

= =

= =

Vy h phng trnh c bn nghim(0;0), (5;5), (1;2), (2;1).

V d 4: Gii h phng trnh2 3 4 4 (1)

2 3 4 4 (2)

x y

y x

+ + =

+ + =

Gii

iu kin:

34

23

42

x

y

.

Ly(1) tr (2) ta c:

( ) ( )2 3 2 3 4 4 0

(2 3) (2 3) (4 ) (4 )

02 3 2 3 4 42 1

( ) 02 3 2 3 4 4

x y y x

x y y x

x y y x

x yx y y x

y

+ + + =

+ +

+ =+ + + +

+ = = + + + +

Thayx = y vo (1), ta c:2 3 4 4 7 2 (2 3)(4 ) 16x x x x x+ + = + + + =


29/70

28

22

39 02 2 5 12 9 11

9 38 33 09

= + + =

+ = =

xxx x x

x x x

Vy h phng trnh c 2 nghim phn bit

( ) ( )11 11

; 3;3 , ;9 9

x y

= .

V d 5: Gii h phng trnh

2

2

2

2

12

12

yy

x

xx

y

+=

+ =

Giiiu kin: , 0>x y Khi , h phng trnh cho tng ng

2 2

2 22 1 (1)2 1 (2)

yx y

xy x

= += +

Ly (1) tr (2) v theo v ta c:

( ) ( )

2 ( )

( ) 2

(3)

0 2 0

m

=

+ + = + + >

=

xy x y y x

x y xy x y xy x y

x y

Thay (3) vo (1) ta c:3 2

3 2

2

0

2 1

2 1 0

( 1)(2 1) 0 1x

x x

x x

x x x x

>

= +

=

+ + = =14243

Vy h phng trnh c nghim duy nht(x;y) = (1;1).

(thok)


30/70


BI TP RN LUYN

Bi 1:Gii h phng trnh:

) )

) )

22

2

2

321

3 12

4 43 3

4 43 3

x yy x yx

a by y xy x

y

y yx y x y

xc d

x xy x y x

y y

+ = + = +

+ = + + =

= = = =

Bi 2: Tm a h sau c nghim duy nht:

2

2

( 1)

( 1)

x xy a y

y xy a x

+ =

+ =

Bi 3: Chng minh rng vi 0a th phng trnh

sau c nghim duy nht:

22

22

2

2

ax y

y

ay x

x

= + = +


31/70

30

5. H PHNG TRNH NG CPBC HAI

1/ nh ngha:Biu thcf(x; y) gi l phng trnh ng cp bc 2nu

f(mx; my) = m2f(x; y)

H phng trnh ng cp bc hai c dng:

( )

( )

,

,

=

=

f x y a

g x y b

Trong :f(x; y) vg(x; y) l phng trnh ng cp bc 2;

vi a v b l hng s.

2/ Cch gii:

Xt x = 0 thay vo h kim tra.

Vix 0 ta ty = xtthay vo h ta c:

( )

( )

( )

( )

2

2

, 1,

, 1,

= =

= =

f x xt a x f t a

g x xt b x g t b

Sau , chia 2 v ca 2 phng trnh vi nhau ta c:( ) ( ) ( )1, 1, *=

af t g t

b

Gii phng trnh (*) ta tm c t.

Thtvo h ta tm c (x; y).


32/70


3/ Cc v d:

V d 1: Gii h phng trnh sau:( )

( )2 2

2 2

2 3 121

2 14

+ + =

=+

x y xy

yx y

Gii

D thyx = 0 khng l nghim ca h phng trnh

Vix 0 ta ty = xt. Khi h phng trnh trthnh:

( )

( )

( )

2 22 2 2 2

2 2 2 2 2 2

22

2

3 2 122 3 12

2 4 14 4 2 14

k : 4 2 03 2 62

4 2 7 2 2

+ + = + + =

+ + = + + =

+ + + + = + +

x t t x x t x t

x x t x t x t t

t tt t

t t t

Khi (2) 21

3 2 02

= + = =

tt t

t(tha)

Khi t = 1 th vo h ta c (x; y) = ( )2 2;

Khi t = 2 th vo h ta c (x; y)= (1; 2), (1; 2)

Vy nghim ca h l:(x; y) = ( )2 2; , (1; 2), (1; 2)


33/70

32

V d 2: Tm m h phng trnh sau c nghim:

( )

2 2

2 21

+ + =

+ + =

x xym y m

x m xy my m

Gii

D thyx = 0 khng l nghim ca h phng trnh

Vix 0 ta ty = xt. Th vo h phng trnh ta c

( )

( )

( )

( )( )

2 22 2 2 2

2 2 2 2 2 2

22

2

1

1 1

011 1 0

1 11

+ + = + + =

+ + = + + =

=+ + = + =

=+ + +

x t tm mx x tm x t m

x m x t x t m m x t m tm t m

tt tmm t t

m tt m tm t

Khi t = 0 th ( )2

k : 00

=

=

x mm

y

Khi (1m)t= 1 ( )2

2

1*

2 3 2

=

= +

xy

m

my m m

V2

2 732 3 2 2 084

m m m + = + >

nn (*) c ngha 1m

Vy vi 1m th h phng trnh trn c nghim.


34/70


V d3: Cho h phng trnh sau:2 2

2

4

3 4

+ =

=

x xy y m

y xy

Chng minh h phng trnh lun lun c nghim m .

GiiKhix = 0 khng l nghim ca h phng trnh.

Vix 0 ta ty = xt. Khi h phng trnh trthnh

( )

( )

( )

2 22 2 2 2

2 2 2 2 2

22

2

4 14

4 3 4

k : 3 04 1*3 4 3

+ = + =

+ = =

+ =

x t t mx x t x t m

x t x t x t t

tt t m

t t

Khi ( ) ( ) ( ) ( )24 16 3 4 0* ** + =m t m t

Vi m = 4 th (**) c dng 4 4 0 1 + = =t t (tho)

Vi m 4 th (**) c dng:

( ) ( )

24 16 3 4 0 + =m t m t

Vi2

2 40 1289 80 192 3 03 9

m m m = + = + >

Vy h phng trnh lun lun c nghim m .


35/70

34

BI TP RN LUYN

Bi 1: Gii cc h phng trnh sau:

)

)( )

)

)

)

2 2

2 2

2 2

2 2 2

2 2 2

2 2

2 2

2 2

2 2

3 2 11

2 3 17

5 2

10

2 2 2

2 2 1

2 2

2 2 2 0

2 3 3

2 2

x xy ya

x xy y

x y xyb

y x y

x y x xc

x y x y xy

x y xy y xd

x y y

x y x xy yd

x y x y

+ + =

+ + =

+ =

+ =

+ + =

+ =

+ + + + =

=

+ = + +

+ = +

Bi 2: Tm gi tr ca m phng trnh c nghim:

2 2

2 2

3 2 11

2 3 17

+ + =

+ + = +

x xy y

x xy y m


36/70


2Cac He Phng Trnh

Khac


37/70

36

1. H PHNG TRNH BC CAO HAI N

H phng trnh bc cao hai n l h gm hai phngtrnh hai n trong c t nht mt phng trnh c bc lnhn 1.

Phng php chung:Cc h phng trnh bc cao thng kh gii v khng

th nu ra tng phng php cu th gii.Do phngphp thng c s dng l chuyn chng v h phngtrnh bc hai bng mt trong hai phng php:

Phng php bin i tng ng.Phng php t n ph.

Cc dng h phng trnh thng gp:H phng trnh i xng loi 1H phng trnh i xng loi 22/ Mt s v d minh ha:

V d 1:

( )2 2

4 4

31

17

x xy y

x y

+ + =

+ =

GiiKhi (1) trthnh:

2 2 2

2 2 2 2 2 2 2 2

2 2

2 2 2 2 2 2

2

3 ( ) 3

( ) 17 2 (3 ) 17 2

( ) 3 ( ) 39 6 17 2 6 8 0

( ) 31

22

4

x y xy x y xy

x y x y xy x y

x y xy x y xyxy x y x y x y xy

x y xyx y

xyxy

xy

+ + + = +

+ = + = +

+ = + + = + + = + + + =

+ = ++ =

= =

=


38/70


Vi1

2x y

xy

+ ==

x,y l nghim ca phng trnh:

1 2

1 2

2 1 2

2 1

12 0

2

x x

y y

tt t

t

= = = =

= =

=

Vi

=

=+

2

1

xy

yx

x,y l nghim ca phng trnh:

3 42

3 4

1 212 0

2 2 1

x xtt t

t y y

= = = + = = = =

Vy h phng trnh c bn cp nghim l

(1;2), (2; 1), (1; 2), (2;1)

V d 2:Gii cc h phng trnh sau:

) ( )3 3 2 126+ =+ =

x yax y

Gii

t:x y S

xy P

+ =

=; iu kin 042 PS

Khi :

( )3 3

2 2 21

3( ) 3 ( ) 26 3 26

x y S S

Px y xy x y S SP

+ = = =

= + + = =

Vi S = 2 ; P = 32 1 3

3 3 1

x y x y

xy x y

+ = = = = = =

Vy h phng trnh c hai cp nghim l (1;3),(3; 1)


39/70

38

) 2 2 3 34

( )( ) 280(2)

+ =

+ + =

x yb

x y x y

Gii

( ) 2 3

2

2 [( ) 2 ][( ) 3 ( )] 280

(16 2 )(64 12 ) 2803

3( ) 40 93 0 31

3

x y xy x y xy x y

xy xy

xy

xy xyxy

+ + + =

==

+ = =

Vixy = 3 ta c:

=

=+

3

4

xy

yxx;y l nghim ca phng trnh:

2

x 1

y 3t 1t 4t 3 0

t 3 x 3

y 1

= == + = = =

=

Vi3

31=xy ta c:

=

=+

331

4

xy

yx

x;y l nghim ca phng trnh:

03

3142 =+ tt (v nghim)

Vy h c hai cp nghim l (1;3),(3;1)


40/70


BI TP RN LUYN

Bi 1: Gii h phng trnh:

)

)

)

)

3 3

4 4 2 2

2 22 2

2 2

4 4

7( ) 2

1

12

1 14

1 14

3

17

= =

+ =

+ =

+ + + =

+ + + =

+ + =

+ =

x ya

xy x y

x yb

x y x y

x yx y

c

x yx y

x xy yd

x y

Bi 2: Cho h phng trnh:

4 4 2 212

x y m

y x y

+ =

+ =

Tm m h phng trnh c ng hai nghim.

Bi 3: Tm m h sau c nghim duy nht:3 2 2

3 2 2

7

7

x y x mx

y x y my

= +

= +


41/70

40

2. H PHNG TRNH V T

NHNG PHNG PHP THNG DNG

GII H PHNG TRNH V T:

1/ Khcn thc a h cho v h hu t:

Mt vi nh l khi kh cn thc:

2 2

22

2 12 1

2 12 1

2 12 1

22

2

( ) ( ) ( ) ( ) 0

( ) 0( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) 0( ) ( )

( ) ( )

( ) 0( ) 0

( ) ( )(

++

++

++

= =

= =

= =

> >

<

Vy h (IV) c nghim khi1

0;4

m


46/70


4/ Phng php nh gi:

V d 5:Gii h phng trnh

1 (V)2 1xy y y

xy y y

+ = =

Gii

iu kin: 0 y 1, x 0, xy y 0 (*)

D thyy = 0 khng phi l nghim ca h.

Viy 0, (*) x 1.

Suy ra

1

1

1

1

xy y y

y y y

y

x

+

+ =

=

=

Thay (x;y) = (1;1) vo h, ta thy tha h.

Vy nghim ca h (V) l (x;y) = (1;1)


47/70

46

BI TP RN LUYN

Bi 1: Gii cc h phng trnh sau:

a)2 2 3

3 1 3 1 4

x y xy

x y

+ =

+ + + =

b)2 1 1

3 2 4

x y x y

x y

+ + + =

+ =

c)2 2 2 22 1 2 1

12 0

x x y x x y

xy y

+ + + + = +

=

d)2 2

3 2 2

4 3

4 0

x y x y

x x y

+ + =

=

Bi 2:

Tm cc gi tr ca m h phng trnh sau c nghim

1 2

2 1

y m

y m

+ + =

+ + =


48/70


3. H PHNG TRNH KHNG MU MC

Khi nim:

H phng trnh khng mu mc l h phng trnh

khng c cu trc (dng) c th, do cng khng c cchgii tng qut. Phi ty vo tng h phng trnh m c

cch gii ph hp.

Mt s cch gii cbn:

Phng php th,

Phng php t n s ph,

Phng php cng,

Phng php dng tnh n iu ca hm s,

Phng php dng bt ng thc,

Phng php nh gi,

Phng php a v h phng trnh cng bc

(ng cp).

Sau y l mt s v d c th cho cc phng php:


49/70

48

1/ Phng php th:

V d 1: Gii h phng trnh sau:

( )

( )

2

2 2

6 3 1 1

1 2

x xy x y

x y

+ + =

+ =

GiiTa bin i (1) thnh phng trnh bc hai theo n x:

( )26 1 3 1 0x y x y+ + =

Ta tnh bit s delta ca phng trnh trn:

( ) ( ) ( )2 2

1 3 24 1 3 5y y y = =

Ta tm dc nghim l 1 12 3

yx x

= =

Th 13

x = vo (2) 2 23

y =

Th 12

yx

= vo (2)

3 44 5

1 0

y x

y x

= =

= =

Vy nghim ca h l:

( ) ( )

1 2 2 1 2 2, ; , ;

3 3 3 3

3 4; 1;0 , ;

4 5x y

=


( )( ) ( )

( )

2 2

2

1 1 3 4 1 1

1 2

x y x y x x

xy x x

+ + + = +

+ + =


50/70


GiiD thyx = 0 khng tha mn phng trnh (2).

Vi x 0, t (2) ta c2 1

1x

yx

+ = . Thay vo (1) ta c:

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( ) ( )

2 22 2 2 2

3 2

2

3 4 11 1

1 2 1 1 3 1

1 2 2 1 1 3 1

2 2 1 0 1 2 do 0

x xx x x x x x x x

x x

x x x x x x

x x x x x x

+ = + =

+ =

+ = = =

Vi 1 1x y= = , Vi 52

2x y= =

Vy h c nghim l ( ) ( )5

21;1 , 2;;x y

=

V d 3: Gii h phng trnh:( )

( )

2 2 1

2

2 7

3

x x y

xy x y

+ + =

+ =

Gii

T ( ) ( )3

2 11

xy x

+ =

+, thay vo (1) ta c:

( )( )( )4 3 2 22 5 2 7 2 0 1 2 2 3 1 0

12 3 17 3 17

1 2 4 43 172 1 1 17 1 174

2 23 174

x x x x x x x x

x

xx x

x x

x y yy y

x

+ + = + + =

== +

= == = + = = = +

= =

=

( ) ( )3 17 1 17 3 17 1 17

1;2 , 2; 1 , ; , ;4 2 4 2

S

+ + +=


51/70

50

BI TP RN LUYN

Gii cc h phng trnh sau:

) )( )( )

) )

2 2

3 32 2

2 2 2

2 2 2

33 2 16

2 9 2 32 3 33

3 4 7 4 9 02 2 2 1 0 4 3

x xy yxy x x ya b

x y x y xyx y x y

xy y x y x yc dxy y x y x y x y

+ = =

= ++ =

+ + = =+ + = + = +

2/ Phng php t n s ph:


( )

( )( )( )

2

2

1 4I

1 2

x x y y

x x y

y

y

+ +

+ +

+ =

=

GiiD thyy = 0 khng tha h (I), nn ta c:

( )

( )

2

2

14

I1

2 1

x

y

xx y

y

x y

+

++

+ + =

=

t

2

1, 2u v x yxy= = + + , ta c: 12 1

1u v u

uv v

+ = == =

Khi , suy ra:

221 1 1 21

2 532 1

xx yy x

yx yy x

x y

+= = == +

= =+ =

+ =

Vy nghim ca h l: ( ) ( ) ( )1;2 , 2;5;x y = .


52/70



( )( )

( )

2 22

3

II1

4 4 7

2 3

yxy xx y

xx y

+ +

+ =+

+ =+

Giiiu kin:x + y 0. Khi :

( )

( ) ( )( )

2 2

2

3

II1

3 7

3

x y x yx y

x y x yx y

+

+

+ + =+

+ + =+

t 1u x yy

= + ++

(iu kin: 2u ), v x y=

( )( ) ( )

2 2

22II

2 133 131

loai3 3 3 132

u vv uu v

uu v u u

= == + =

= + = + =&

Suy ra:1

2 10

1

x y xx y

yx y

=

+ + =+

= =

Vy h c mt nghim duy nht ( ) ( )1;0;yx =

BI TP RN LUYNGii cc h phng trnh sau:2 2 2

3 32 2 2

2 2

52 2 2) b)

( 1) ( 1) 352 2 1

4 2 1 1) d)

( 1) ( 1) 2 3 2 4

x y xyx y x xa

x yx y x y xy

x y x y x y x yc

x x y y y x y

+ + =+ + =+ + + = + =

+ + + = + + + =+ + + + = + =


53/70

52

3/ Phng php cng:


1 1 4

6 4 6

x y

x y

+ + =

+ + + =

Gii

iu kin: 1, 1y

Cng v tr v theo v ca hai phng trnh, ta c:

( )*6 1 4 1 4

6 1 4 1 6

x x y y

x x y y

+ +

+ + + + =

+ + + + =

t5

6 1 6 1u ux x x x= + =+ + + +

54 1 4 1v

vy y y y= + =+ +

Khi h (*) trthnh

10 6 1 55 35 5 5 42 4 1 5

u vx xu x

v yy yu v

+

+

+ = + + == =

= =+ = + =

Vy nghim ca h l ( ) ( )3;4;yx =


( )

( )

2 2

2 2

1

2

91 2

91 2

x y y

y x x

+ = +

+ = + +


54/70


Gii

iu kin: , 2y >

Ly (1) tr (2) ta c:

( )

2 2 2 2

2 22 2

2 2

2 2

0 , 2

91 91 2 2

2 291 91

10

2 291 91

x y

x y y x y x

x y y xy x

y xx y

x yx y x y

y xx y

x y

> >

+

+

+ + = + +

= +

+ ++ +

+ + + + =

+ ++ +

=

144444444424444444443

Th y= vo phng trnh (1), ta c:

( )( )

( ) ( )

2 2 2 2

2

2

2

0 , 2

91 91 10 1 9

91

91

2 2

9 33 3

2 120

1 13 3 1 0

2 110

3 3x y

x x x x x x

x xx x

xx

x x

xx

x y

> >

+ = + + = +

= + +

++ +

+ =

++ +

= =

1444444442444444443

Vy h c m nghim duy nht: ( ) ( )3;3;yx =


55/70

54

BI TP RN LUYN

Gii cc h phng trnh sau:

2 2

2 3 2

4 2

13 3

1) b)

3( ) 2812 5

52 6 2

4) c)5

(1 2 ) 2 3 24

+ + + = + + =

+ + + =

+ + =

+ + + + = + =

+ + + = = +

x x yx y xy y

ax y x y

x yy

xx y x y xy xy y x yyb

x y xy x x x y x y

3/ Phng php dng bt ng thc:


1 1 1 6

9

x y z

x y z

+ + + + + =

+ + =

Giiiu kin: , , 1x y z

p dng bt ng thc Bunyakovsky, ta c:

( ) ( )2

1. 1 1. 1 1. 1 3 36x y z x y z + + + + + + + =

Suy ra: 1 1 1 6x y z + + + + +

ng thc xy ra 3x y z = = = tha mn phng trnh th

hai ca h.Vy h c mt nghim duy nht ( ) ( )3;3;3;;y zx =


56/70



2

2

3

4 2

4

6 4 2

2

1

3

1

4

1

xy

x

yz

y y

z

z z z

= +

=

+ +

=+ + +

Gii

V2

2

20

1

xy

x=

+nn xy ra hai trng hp sau:

Viy = 0, khi x = y = z = 0Vy ( ) ( )0;0;0;;y zx = l mt nghim ca h phng trnh.

Viy > 0, khi x > 0,z>0.

D thy 2 21 2x x+ nn2

2

2hay

1x

x y xx

+

.

Theo BT Cauchy, ta c:2

4 2 4 2 234 2

31 3 . .1 3 hay

1y

y y y y y y z yy y

+ + = + +

T phng trnh th 3 ca h suy ra x z . Vy x y z x ,iu ny xy ra x y z = = .

Thay vo phng trnh u ta c 1y z= = = (tho)

Vy nghim ca h l ( ) ( )( )0;0;0 1;1;1;;y zx =

BI TP RN LUYNGii cc h phng trnh sau:

) )( 1) ( 1) 2 4 1 4 1 4 1 9

61 1

x y y x xy x y z a b

x y z x y y x xy

+ = + + + + + =

+ + = + =


57/70

56

4. SDNG VCT GII

H PHNG TRNH

Mt vi btng thc vectthng dng:

Trong mt phng hoc trong khng gian cho hai vc t ,ur ura b ;

khi ta c:

( )| | | | | | 1+ +r r r ra b a b

Du " = " xy ra * :+ =r r r ra b k a kb hoc

mt trong hai vc tbng0r

( )| | | | | | 2

r r r ra b a b

Du " = " xy ra * : =r r r ra b k a kbhoc

mt trong hai vc tbng 0r

( )| | . | | . 3 r r r ru v u v

Du " = " xy ra * : =r r r ra b k a kbhoc


( ). | | . | | 4r r r ru v u v

Du " = " xy ra * :+ =r r r ra b k a kb hoc



58/70



( )2 2 2

2011 2011 2011

I

3

3

3

+ + =

+ + =

+ + =

x y z

x y z

x y z

Gii

Xt ( ; ; ), (1;1;1)u x y z v= =ur ur

Khi ta c:

2 2 2

2 2 2

3

1 1 1 3

= + + =

= + + =

ur

ur

x y z u

v

M 3 3. . .= = = =+ +ur ur ur ur ur uru v x y z u v u v

Vy uru cng phng vi urv 01 1 1x y z = = >

0x y z = = >

Kt hp vi (I) ta cx = y = z = 1 l nghim ca h.


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58


2 2

2

2 2

( )

2

3 8 8 8 2 4 2

+ = +

+ + =

+ + + = + +

x y y x z

x x y yz

x y xy yz x z

Gii

2 2 2 2

( ) ( ) 0( 1) (2 1) 0

4( ) 4( ) ( 1) (2 1)

+ + + = + + + =

+ + + = + + +

x x y y y z

x x y z

x y y z x z

Xt: ( ; ); ( ; ); ( 1;2 1)a x y b x y y z c x z = = + + = + +ur ur ur

2 2. 0; . 0;4 = = =

ur ur ur ur ur ura b a c b c

Nu th 10 0;2

= = = = ur ura x y z

Nu 0 th vur r ur ur

ba c cng phng 2 = urur

c b

Xt 2 trng hp 2 2v= = ur urur ur

c b c b

Ta cx = 0; y = 12; z = 1

2

Vy h c 2 nghim 1 1 10;0; v 0; ;2 2 2


60/70


BI TP RN LUYN

Bi 1: Gii h phng trnh sau:

2( ) 1 3 2 ( ) 1

2

2 1

x y x y x y x yx y

x y

+ + + + = + +

+

=

Bi 2: Chng minh rng h phng trnh sau v nghim:

4 4 4

2 2 2

1

2 7

x y z

x y z

+ + =

+ + =


2 2 2

2009 2009 2009

3

3

3

x y z

x y z

x y z

+ + =

+ + =

+ + =


1 2 2008

1 2 2008

20091 1 ... 1 2008

2008

20071 1 ... 1 2008

2008

x x x

x x x

+ + + + + + =

+ + + =


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60

5. H PHNG TRNH

TRONG CC K THI

V d 1: Gii h phng trnh

( )2 2 21 7

,

1 13

xy x yx y

x y xy y

+ + =

+ + =

(Tuyn sinh i hc, cao ng khi B 2009)

GiiD thyy = 0 khng l nghim nn h cho tng ngvi

22

2

11 77

1 1

13 13

xx xxy yy y

x x

x xy y y y

+ + =+ + =

+ + = + =

Suy ra2

1 120 0x x

y y+ + + =

151

5

12

xyx

yx y

+ = + =

=

(h v nghim)

141 4

3

x yxy

x y

+ =+ =

=

Vy trong trng hp ny, h c hai nghim

(x;y) =1

13

;

; (3;1).


62/70



( )( )

2 3 2

4 2

5

4 ,5

1 24

x y x y xy xy

x y

x y xy x

+ + + + =

+ + + =

(Tuyn sinh i hc, cao ng khi A 2008)Gii

H phng trnh cho tng ng

( )

2 2

22

5( )

45

4

x x xy x y xy

x y xy

+ + + + = + + =

Suy ra2 2 2 2

( ) ( )x y xy x y x y+ + + = + 2 2( )( 1 ) 0x y x y xy + + =

Vi

2 3

2

3

50

40 5

254

16

xx y

x yxy

y

=+ =+ =

= =

Vi

2

2

1

121 0 33

22

xx yx y xy

yxy

=+ = + = =

=

Vy nghim ca h l (x;y) = 3 35 25

;4 16

,3

1;2


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62

V d 3: Tm m h phng trnh c nghim thc:

3 33 3

1 15

1 115 10

x yx y

x y m

x y

+ + + = + + + =

(Tuyn sinh i hc, cao ng khi D 2007)

Gii

t ( )1 1

; 2, 2x a y b a bx y

+ = + =

H phng trnh cho trthnh

3 3

5 5

83( ) 15 10

a b a b

ab ma b a b m

+ = + =

= + + =

Do a, b l nghim ca phng trnh

( )2 5 8 0 *X X m + =

H phng trnh cho c nghim khi v ch khi PT (*) c

hai

nghimX1, X2 tha 1 22, 2x x

Vy7

2 224

m m


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1 22

2 33

2002 11

1 1

2

1 1

2

1 1

2

...

x

x

x

xx

xx

xx

= +

= +

= +

(Olympic 30/4/2002 ti THPT Chuyn L Hng Phong)

Gii

Nhn xt: Nu ( )1 2 2002, ,...,x x x l nghim th 1 2 2002, ,...,x x x

phi cng du v khc 0. ng thi ( )1 2 2002, ,...,x x x cng

l nghim, nn ta ch cn xt vi 1 2 2002, ,...,x x x dng.

Theo bt ng thc Cauchy: ( ) ( )1

2 1,2,...,2002 1,+ =ii

x ix

T cc phng trnh trong h v (1), ta c:

( )2 2 1 2hayi ix x

Mt khc cng cc phng trnh trong h th:

1 2 20021 2 2002

1 1 1

... ...x x x x x x+ + + = + + + ( )3

T (2) v (3) ta c: 1 2 2002 1...x x x= = = =

Kt qu: H c 2 nghim 1 2 2002

1 2 2002

1

1

...

...

x x x

x x x

= = = =

= = = =


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64

TUYN SINHI HC, CAONG

Tuyn sinh i hc, cao ng khi B 2002:

Gii h phng trnh3

2

y x y

x y x y

=

+ = + +

Kt qu: ( ) ( )3 1

; 1;1 , ;2 2

x y =

.

Tuyn sinh i hc, cao ng khi A 2003:

Gii h phng trnh3

1 1

2 1

x yx y

y x

=

= +

Kt qu:

( ) ( )1 5 1 5 1 5 1 5

; 1;1 , ; , ;2 2 2 2

x y + +

=

.

Tuyn sinh i hc, cao ng khi D 2004:

Tm m phng trnh sau c nghim:

( )1

,1 3

x yx y

x x y y m

+ =

+ =

Kt qu:1

04

m


66/70



Gii h phng trnh3

1 1 4

x y xy

x y

+ =

+ + + =

Kt qu: ( ) ( )1;1;x y =

Tuyn sinh i hc, cao ng khi B 2008:

Gii h phng trnh ( )4 3 2 2

2

2 2 9,

2 6 6

x x y x y xx y

x xy x

+ + = +

+ = +

Kt qu: ( )17

; 4;4

x y =

Tuyn sinh i hc, cao ng khi D 2009:

Gii h phng trnh( )

( )( )2

2

1 3 0;5

01

x x yx y

x yx

+ + =

+ = +

( ) ( )3

; 1;1 , 2;2

x y =


Gii h phng

trnh( ) ( )

( )2

2 2

4 1 2 5 2 0;

4 2 3 4 7

x x y yx y

x y x

+ + =

+ + =

Kt qu: ( )1

; ;22

x y =


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66

OLYMPIC 30/4

Olympic 30/4/1998 ti THPT Chuyn

L Hng Phong TP H Ch Minh:

Gii h phng trnh( )( )

( )

2

2

2

0 126 2

26 3

ax bx cbx cx a

cx ax b

=

+ ++ + =

+ + =

Hng dn: Cng (1), (2) v (3) v theo v ta c:

( )( )2 1 0 0a b c x x a b c+ + + + = + + = (V 2 1 0x x>+ + )

( )1 1c

x xa

= = (khng tho (2), (3) nn loi)

ax c = thay vo (2) tm ra a v bin lun tm b, c.

Kt qu: ( ) ( ) ( )1;4; 3 , 13;0; 13 , 26; 26;0 l 3 b s cn tm.



Gii h phng trnh

53 2 4

42

53 2

42

yy x

xy x

= +

+ = +


68/70


Hng dn:

Tm iu kin, sau vit li thnh:

( )

( )

5

42

1 21

1 23 2

y xx y

x y

+

=

+ =

1 2 15

42x y y x =

+

Tm y theox sau th vo (2) ta s tm dc nghim cah.

Kt qu: ( )5 2 6 5 2 6

; ;27 9

x y + +

=

Olympic 30/4/2005 ti THPT ChuynL Qu n TP Nng:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

22 2 2 2

22 2 2 2

22 2 2 2

3 1 1

3 1 2

3 1 3

x y z x x y z

y z x y y z x

z x y z z x y

=

=

=

+ + +

+ + +

+ + +

Hng dn: Chia lm 2 trng hp:

Trng hp 1:xyz= 0

Trng hp 2:xyz 0

Kt qu: H c nghim l:

( ) ( ) ( ) ( )9 3 9 5 5

0 0 0 0 0 0 113 4 11 6 4

; ; , ; ; , ; ; , , ; ; , ; ;vx y z x y z


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68

Olympic 30/4/2007 ti Hu:

Gii h phng trnh2 2

2

816

xyx y

x y

x y x y

+ + = + + =

Kt qu: ( ) ( ) ( )2;2 3;7; ,x y =



Gii h phng trnh 2 2 23 3 3

7

371

x y z

x y z

x y z

=+

+ = + =

Kt qu: ( ) ( ) ( ); 9;10;12 10;9;12,x y =



Gii h phng trnh

3 2 3

3 2 3

3 2 3

2 3 18

2 3 18

2 3 18

x y y

y y z z

z z x x

+ = +

+ = + + = +


70/70


Hng dn: t 3 2( ) 2 3 18f t t t = + v 3( )g t t t = + th

phng trnh c vit li:( ) ( )

( ) ( )

( ) ( )

f x g y

f y g z

f z g x

=

= =

Gi s ( )max , ,x y z = th ( ) ( )( ) ( )x y g x g yx z g x g z

Do hm sng bin( ) ( )

( ) ( )

x f y

x f z

Kt qu: H c nghim duy nht ( ) ( )2;2;2; ;x y z =



Gii h phng trnh

3 3

1 19

1 1 1 11 1 18

x y

x yx y

+ = + + + =

Hng dn: t3 3

1 1;a b

x y= =

Kt qu: ( ) 1;1 1

; , ;18 8

x y =

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