Phuong Phap Giai He Phuong Trinh

7/28/2019 Phuong Phap Giai He Phuong Trinh

1/22


2/22

Bi 1 : Mt s dng h phng trnh c bit.

1) H bc nht hai n, ba n.

a)2 4 0

2 5 0

x y

x y

+ = + =

b)2 3 7 0

2 4 0

x y

x y

+ = + =

c)1 02 2 0

2 3 4 0

x y zx y z

x y z

+ = + = + + =

d)1 02 2 0

2 3 4 0

x y zx y z

x y z

+ + = + = + + =

2) H gm mt phng trnh bc nht v phng trnh bc cao. PP chung : S dng phng php th.- H 2 phng trnh.- H 3 phng trnh.

3) H i xng loi 1. PP chung : t n ph ( );a x y b xy= + =

4) H i xng loi 2.

PP chung : Tr tng v hai phng trnh cho nhau ta c : ( ). ( ; ) 0x y f x y =

5) H phng trnh ng cp bc hai.PP chung : C 2 cch gii - t n ph .y t x=

- Chia c hai v cho 2y , v tx

ty

=


3/22

Bi 2 : Mt s phng php gii h phng trnh

I. Phng php th.* C s phng php. Ta rt mt n (hay mt biu thc) t mt phng trnh trong h v th vo

phng trnh cn li.* Nhn dng. Phng php ny thng hay s dng khi trong h c mt phng trnh l bc nht i

vi mt n no .

Bi 1 . Gii h phng trnh2 2

2 3 5 (1)3 2 4 (2)x yx y y

+ =

+ =Li gii.

T (1) ta c5 3

2

yx

= th vo (2) ta c

2

25 33 2 4 02

yy y

+ =

2 2 2 593(25 30 9 ) 4 8 16 23 82 59 0 1,23

y y y y y y y y + + + = = =

Vy tp nghim ca h phng trnh l ( )31 59

1;1 ; ;23 23

Bi 2 Gii h phng trnh sau :2 2

2 1 0

2 3 2 2 0

x y

x y x y

=

+ + =

Bi 3 Gii h :3 2

2

3 (6 ) 2 0

3

x y x xy

x x y

+ =

+ = - PT (2) l bc nht vi y nn T (2) 23y x x= + thay vo PT (1).- Nghim (0; 3); ( 2;9)

Bi 4 a)Gii h :3 2

2

3 (5 ) 2 2 04

x y x xy xx x y

+ = + =

- PT (2) l bc nht vi y nn T (2) 24y x x= + thay vo PT (1).

b) Gii h :3 2 2 2

2 2

3 (6 ) 2 0

3

x y x xy

x x y

+ + + =

+ =

Bi 6 (Th T2012) Gii h :2 2 1 4

2 2( ) 2 7 2

x y xy y

y x y x y

+ + + =

+ = + +.

- T (1)2 2

1 4x y y xy+ = thay vo (2). Nghim (1; 2); ( 2;5)

Bi 7. Gii h phng trnh4 3 2 2

2

2 2 9 (1)

2 6 6 (2)

x x y x y x

x xy x

+ + = +

+ = +Phn tch. Phng trnh (2) l bc nht i vi y nn ta dng php th.

Li gii.TH 1 : x = 0 khng tha mn (2)

TH 2 :26 6

0, (2)2

x xx y

x

+ = th vo (1) ta c

22 2

4 3 26 6 6 62 2 92 2x x x xx x x xx x + + + + = +


4/22

2 24 2 2 3

0(6 6 )(6 6 ) 2 9 ( 4) 0

44

xx xx x x x x x x

x

=+ + + + = + + = =

Do 0x nn h phng trnh c nghim duy nht17

4;4

Ch .: H phng trnh ny c th th theo phng php sau:

- H ( )

22

22

22 2

2

6 62 9 2 92

6 66 6

22

x xx xy x x

x xx xy x x

x xy

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

+ = - Phng php th thng l cng on cui cng khi ta s dng cc phng php khc

Bi 8(D 2009 ) Gii h : 22

( 1) 3 0

5( ) 1 0

x x y

x yx

+ + =

+ + =. T (1) th

31x y

x+ = v thay vo PT (2).

Bi 9 Gii h :2 2 2( ) 7( 2 ) 2 10

x y x yy y x x

+ + + = =

HD : Th (1) vo PT (2) v rt gn ta c : 2 2 4 2 3 0 ( 1)( 2 3) 0x xy x y x x y+ + + + = + + + =


5/22

II. Phng php cng i s.

* C s phng php. Kt hp 2 phng trnh trong h bng cc php ton: cng, tr, nhn, chia ta thuc phng trnh h qu m vic gii phng trnh ny l kh thi hoc c li cho cc bc sau.

* Nhn dng. Phng php ny thng dng cho cc h i xng loi II, h phng trnh c v tring cp bc k.

Bi 1 Gii h phng trnh2

2

5 4 0

5 4 0

x y

y x

+ =

+ =

Bi 2. Gii h phng trnh

2 23

2

2 23

2

yy

x

xx

y

+=

+=

Li gii.- K: 0xy

- H

2 2

2 23 2 (1)3 2 (2)x y yy x x

= + = +. Tr v hai phng trnh ta c

2 2 2 20

3 3 3 ( ) ( )( ) 03 0

x yx y xy y x xy x y x y x y

xy x y

= = + + = + + =

- TH 1. 0x y y x = = th vo (1) ta c 3 23 2 0 1x x x = =

- TH 2. 3 0xy x y+ + = . T2

2

23 0

yy y

x

+= > ,

2

2

23 0

xx x

y

+= >

3 0xy x y + + > . Do TH 2 khng xy ra.

- Vy h phng trnh c nghim duy nht (1 ; 1)

Bi 2 Gii h phng trnh

1 12 2 (1)

1 12 2 (2)

yx

xy

+ =

+ =

Li gii.

- K:1 1

,2 2

x y .

- Tr v hai pt ta c1 1 1 1

2 2 0y xx y

+ =

( )

1 12 2

0 01 1 1 1

2 2 2 2

y x y x y xy x

xy xy x yxy

y x y x

+ = + =+

+ +

- TH 1. 0y x y x = = th vo (1) ta c1 1

2 2

xx

+ =


6/22

- t1

, 0t tx

= > ta c

2

2 2 2

2 0 22 2 1 1

2 4 4 2 1 0

t tt t t x

t t t t t

= = =

= + + = v 1y =

- TH 2. ( )1 1

01 1

2 2

xy x y

xy y x

+ =+

+

. TH ny v nghim do K.

Vy h c nghim duy nht (1; 1)

Bi 5 Gii h phng trnh:2 2

2 2

2 2

3

x xy y

x xy y

+ =

+ + =

Bi 3. Gii h phng trnh2 2

2 2

3 5 4 38

5 9 3 15

x xy y

x xy y

+ =

=Phn tch. y l h phng trnh c v tri ng cp bc hai nn ta s cn bng s hng t do v

thc hin php tr v. Li gii.

- H2 245 75 60 570 2 2145 417 54 02 2190 342 114 570

x xy yx xy y

x xy y

+ = + + =

=

- Gii phng trnh ny ta c1 145

,3 18

y x y x= = th vo mt trong hai phng trnh ca h ta

thu c kt qu (3;1); ( 3; 1) * Ch - Cch gii trn c th p dng cho pt c v tri ng cp bc cao hn.- Cch gii trn chng t rng h phng trnh ny hon ton gii c bng cch t , 0y tx x=

hoc t , 0x ty y= .

Bi 4. Tm cc gi tr m h2 2

2 2

3 2 11

2 3 17

x xy y

x xy y m

+ + =

+ + = +c nghim.

- Phn tch. c kt qu nhanh hn ta s t ngay , 0y tx x= Li gii.

- TH 1.

22

22

11110 17

3 17 3

yyx m

yy m

= = = +== +

Vy h c nghim17

0 11 163

mx m

+= = =

- TH 2. 0x , t y tx= . H2 2 2 2

2 2 2 2

3 2 11

2 3 17

x tx t x

x tx t x m

+ + = + + = +

22 2 2

2 2

22

11(3 2 ) 11 3 2

11(1 2 3 ) 17 (1 2 3 ). 173 2

xt t x t t

t t x m t t mt t

= + + = + +

+ + = + + + = + + +


7/22

2

2

2

11

3 2

( 16) 2( 6) 3 40 0 (*)

xt t

m t m t m

= + + + + + + =

- Ta c2

110,

3 2t

t t>

+ +nn h c nghim pt (*) c nghim. iu ny xy ra khi v ch khi

16m = hoc 216, ' ( 6) ( 16)(3 40) 0m m m m = + +

5 363 5 363m +- Kt lun. 5 363 5 363m +

Bi 5. Tm cc gi tr ca m h

2 2

2 2

5 2 3

2 21

x xy y

mx xy y

m

+

+ +

(I) c nghim.

Li gii.

- Nhn 2 v ca bpt th hai vi -3 ta c

2 2

2 2

5 2 3

16 6 3 3 1

x xy y

x xy y m

+

- Cng v hai bpt cng chiu ta c 2 2 21 1

4 4 ( 2 )1 1

x xy y x ym m

+

- iu kin cn h bpt c nghim l1

0 11

mm

> >

- iu kin . Vi 1m > . Xt h pt2 2

2 2

5 2 3

2 2 1

x xy y

x xy y

+ =

+ + =(II)

- Gi s 0 0( ; )x y

l nghim ca h (II). Khi 2 22 2 0 0 0 00 0 0 0

2 22 2

0 0 0 00 0 0 0

5 2 35 2 3

2 22 2 11

x x y yx x y ym

x x y yx x y ym

+ + = + + + + =

- Vy mi nghim ca h (II) u l nghim ca h (I)

(II)2 25 2 3 2 24 4 0 2 0 2

2 26 6 3 3

x xy yx xy y x y x y

x xy y

+ = = + = =

=

- Thay 2x y= vo pt th 2 ca h (II) ta c2 2 2 2 1 28 4 1 5 1

5 5y y y y y x + = = = =m

- H (II) c nghim, do h (I) cng c nghim. Vy 1m > .


13 1 2

17 1 4 2

xx y

yx y

+ =+

=+

- Phn tch. Cc biu thc trong ngoc c dng a + b v a b nn ta chia hai v pt th nht cho

3x v chia hai v pt th hai cho 7y .Li gii.

- K: 0, 0, 0x y x y + .


8/22

- D thy 0x = hoc 0y = khng tha mn h pt. Vy 0, 0x y> >

- H

2 4 2 1 2 21 22 1 (1)1

3 7 3 73

1 4 2 2 2 4 2 1 2 2 11

7 3 7 3 7

x y x y x yx

x y x y x yy x y x y

= + + =+ =+

= = =+ + +

- Nhn theo v hai pt trong h ta c 1 2 2 1 2 2 13 7 3 7 x yx y x y

+ =+

2 2

61 8 1

7 38 24 0 43 7

7

y xy xy x

x y x y y x

= = = + =

- TH 1. 6y x= th vo pt (1) ta c1 2 11 4 7 22 8 7

121 73 21

x yx x

+ ++ = = =

- TH 2.

4

7y x= khng xy ra do 0, 0x y> > .

- Vy h pt c nghim duy nht ( )11 4 7 22 8 7

; ;21 7

x y+ +

=

.

- Ch . H phng trnh c dng2

2

a b m m n a

a b n m n b

+ = + = = =

. Trong trng hp ny, dng th

nht c v phi cha cn thc nn ta chuyn v dng th hai sau nhn v mt cn thc.

- Tng qut ta c h sau:

a nm

px qybx

c nm

px qydy

= ++

= ++


2 2 2 2 2( ) (3 1)2 2 2 2 2( ) (4 1)2 2 2 2 2( ) (5 1)

x y z x x y z

y z x y y z x

z x y z z x y

+ = + +

+ = + +

+ = + +

- Phn tch. Nu chia hai v ca mi phng trnh cho 2 2 2x y z th ta c h mi n gin hn.

- TH 1. 0xyz= . Nu 0x = th h 2 20

0 ,

yy z z t t

= = = hoc

0

,

z

y t t

= =

- Tng t vi 0y = v 0z = ta thu c cc nghim l (0;0; ), (0; ;0), ( ;0;0),t t t t - TH 2. 0xyz . Chia hai v ca mi pt trong h cho 2 2 2x y z ta c

21 1 1 1

3 (1)2

21 1 1 1

4 (2)2

21 1 1 1

5 (3)2

z y x x

x z y y

y x z z

+ = + +

+ = + +

+ = + +

. Cng v 3 phng trnh ca h ta c :


9/22

2 221 1 1 1 1 1 1 1 1 1 1 1

122 2 2z y x z y x x y z x y z

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

1 1 1

4(4)21 1 1 1 1 1

12 01 1 1

3 (5)

x y z

x y z x y z

x y z

+ + =

+ + + + =

+ + =

- T (4) v (1) ta c2

2

1 1 1 9 94 3 13

13x

x x x x = + + = =

- T (4) v (2) ta c3

4y = . T (4) v (3) ta c

9

11z =

- Tng t, t (5), (1), (2), (3) ta c5 5

, 1,6 4

x y z= = = .

- Vy h c tp nghim l

S =9 3 9 5 5

( ;0;0); (0; ;0); (0;0; ); ; ; ; ; 1; ,13 4 11 6 4t t t t

- Nhn xt. Qua v d trn ta thy: t mt h phng trnh n gin, bng cch i bin s ( trn

l php thay nghch o) ta thu c mt h phc tp. Vy i vi mt h phc tp ta s ngh nphp t n ph h tr nn n gin.


10/22

III. Phng php bin i thnh tch.

* C s phng php. Phn tch mt trong hai phng trnh ca h thnh tch cc nhn t. i khi cnkt hp hai phng trnh thnh phng trnh h qu ri mi a v dng tch.

Bi 1 (Khi D 2012) Gii h 3 2 2 22 0 (1)

2 2 0 (2)

xy x

x x y x y xy y

+ =

+ + =- Bin i phng trnh (2) thnh tch.- Hoc coi phng trnh (2) l bc hai vi n x hoc y.

- H cho2

2 0

(2 1)( ) 0

xy x

x y x y

+ =

+ =. H c 3 nghim

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

2x y

=

Bi 2. (D 2008) Gii h phng trnh

2 22 (1)

2 1 2 2 (2)

xy x y x y

x y y x x y

+ + =

= - Phn tch. R rng, vic gii phng trnh (2) hay kt hp (1) vi (2) khng thu c kt qu kh

quan nn chng ta tp trung gii (1).Li gii.

K: 1, 0x y (1) 2 2( ) ( ) ( )( 1 ) 0y x y x y x y x y y x y + + + = + + + =TH 1. 0x y+ = (loi do 1, 0x y )TH 2. 2 1 0 2 1y x x y+ = = + th vo pt (2) ta c(2 1) 2 2 4 2 2 ( 1) 2 2( 1)y y y y y y y y y+ = + + = +

1 0 1

22 2

y y

yy

+ = = ==

. Do 0 2y y = . Vy h c nghim ( ; ) (5;2)x y =

- Ch . Do c th phn tch c thnh tch ca hai nhn t bc nht i y (hay x) nn c th giipt (1) bng cch coi (1) l pt bc hai n y (hoc x).

Bi 3. (A 2003) Gii h phng trnh3

1 1(1)

2 1 (2)

x yx y

y x

= = +

- Phn tch.T cu trc ca pt (1) ta thy c th a (1) v dng tch.Li gii.

K: 0xy . (1)1 1 1

0 0 ( ) 1 0x y

x y x y x yx y xy xy

+ = + = + =

TH 1. x y= th vo (2) ta c 3 2 1 0 1x x x + = = hoc 1 52

x = (t/m)

TH 2.1 1

1 0 yxy x

+ = = th vo (2) ta c 4 2 2 21 1 3

2 0 ( ) ( ) 02 2 2

x x x x+ + = + + + = .

PT ny v nghim.

Vy tp nghim ca h l S =1 5 1 5 1 5 1 5

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

+ +

Bi 3. (Thi th GL) Gii h phng trnh

1 1(1)3 3

( 4 )(2 4) 36 (2)

x y

x y

x y x y

=

+ =


11/22

Li gii.

2 2

2 2

3 3 3 3

3 3

1 1 ( )( )( )

1

x yy x y xy x

x y x y y xy xx y x y

x y

= + + = = + + =

TH 1. x y= th vo pt th hai ta c 26

4 12 02

xx x

x

= + = =

TH 2.2 2

3 31 0

y xy xxy

x y

+ += < .

(2) 2 2 2 22 4 9 4 16 36 2( 1) 4( 2) 9 18x y xy x y x y xy + + = + + =

Trng hp ny khng xy ra do 2 20 2( 1) 4( 2) 9 0xy x y xy< + + >Vy tp nghim ca h phng trnh l S = { }(2;2); ( 6; 6)


2 2

2

816 (1)

(2)

xyx y

x y

x y x y

+ + = +

+ = - Phn tch.R rng, vic gii phng trnh (2) hay kt hp (1) vi (2) khng thu c kt qu kh

quan nn chng ta tp trung gii (1)Li gii.

K: 0x y+ > . (1) 2 2( )( ) 8 16( )x y x y xy x y + + + = +2

( ) 2 ( ) 8 16( )x y xy x y xy x y + + + = + 2( ) ( ) 16 2 ( 4) 0x y x y xy x y + + + =

[ ]( 4) ( )( 4) 2 0x y x y x y xy + + + + =

TH 1. 4 0x y+ = th vo (2) ta c 23 7

6 02 2

x yx x

x y= =+ = = =

TH 2. 2 2( )( 4) 2 0 4( ) 0x y x y xy x y x y+ + + = + + + = v nghim do KVy tp nghim ca h l S = { }( 3;7); (2;2)

Bi 5(Th T 2013) Gii h phng trnh( )( 2)

2( 1)( ) 4

xy x y xy x y y

x y xy x x

+ + = + + + + =

- iu kin :

; 0

( )( 2) 0

x y

xy x y xy

+

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

( )( 2)0

( )( 2)

x y y xy x y

x yxy x y xy y

+ + =

++ +

2 1( ) 0 (3)

( )( 2)

y xyx y

x yxy x y xy y

+ + = ++ +

0,25

- T PT (2) ta c

2 24 4

( 1) 1 2 21 1y xy x x x xx x

+ = + = + + + + + 2 1

0( )( 2)

y xy

x yxy x y xy y

+ + >

++ +

0,25


12/22

- PT (3) x y = , thay vo PT (2) ta c : 3 22 3 4 0x x x + =

1x = hoc1 17

2x

=

0,25

- Kt hp vi iu kin ta c 1x = ,1 17

2x

+=

- KL: Vy h cho c hai nghim (x; y) l :1 17 1 17

(1; 1); ;2 2

+ +

0,25

Bi 6 (A 2011 ) Gii h PT :2 2 3

2 2 2

5 4 3 2( ) 0 (1)

( ) 2 ( ) (2)

x y xy y x y

xy x y x y

+ + =

+ + = +

HD : Bin i PT (2) thnh tch ta c2 2

1

2

xy

x y

= + =

.

- TH1:1

yx

= thay vo PT (1).

- TH 2: PT(1) 2 2 2 23 ( ) 2 4 2( )y x y x y xy x y + + + ( 1)(2 4 ) 0xy x y =

Bi 7(Th GL 2012) Gii h : 3 3 4(4 )2 21 5(1 )

x y x y

y x = + = +

HD : T (2) 2 24 5y x= thay vo (1) ta c : 3 3 2 2( 5 )(4 )x y y x x y =


13/22

IV. Phng php t n ph.


1

7

x y xy

x y xy

+ + =

+ =Li gii.

y l h i xng loi I n gin nn ta gii theo cch ph bin.

H 2( ) 1( ) 3 7x y xyx y xy+ + = + =

tx y S

xy P

+ = =

( )2, 4x y S P ta c 21 1, 2

4, 33 7

S P S P

S PS P

+ = = = = = =

TH 1.1 1 1, 2

2 2 2, 1

S x y x y

P xy x y

= + = = = = = = =

TH 2.4 4 1, 3

3 3 3, 1

S x y x y

P xy x y

= + = = = = = = =

. Vy tp nghim ca h l

S = { }( 1;2); (2; 1); ( 1; 3); ( 3; 1) Ch .

- Nu h pt c nghim l ( ; )x y th do tnh i xng, h cng c nghim l ( ; )y x . Do vy, hc nghim duy nht th iu kin cn l x y= .

- Khng phi lc no h i xng loi I cng gii theo cch trn. i khi vic thay i cch nhnnhn s pht hin ra cch gii tt hn.

Bi tp tng t: (T 2010) Gii h phng trnh:2 2 1

3x xy yx y xy

+ + = =

Bi 2(D 2004 )Tm m h c nghim :1

1 3

x y

x x y y m

+ =+ =

Bi 4. Gii h phng trnh2 2 18

( 1)( 1) 72

x y x y

xy x y

+ + + =

+ + =Phn tch. y l h i xng loi I

- Hng 1. Biu din tng pt theo tng x y+ v tch xy

- Hng 2. Biu din tng pt theo 2x x+ v 2y y+ . R rng hng ny tt hn.Li gii.

H2 2

2 2

( ) ( ) 18

( )( ) 72

x x y y

x x y y

+ + + = + + =

. t

2

2

1,41

,4

x x a a

y y b b

+ = + =

ta c

18 6, 12

72 12, 6

a b a b

ab a b

+ = = = = = =

TH 1.2

2

6 6 2, 3

12 3, 412

a x x x x

b y yy y

= + = = =

= = = + = TH 2. i vai tr ca a v b ta c

3, 4

2, 3

x x

y y

= =

= = . Vy tp nghim ca h l


14/22

S = { }(2;3); (2; 4); ( 3;3); ( 3; 4); (3;2); ( 4; 2); (3; 3); ( 4; 3) Nhn xt. Bi ton trn c hnh thnh theo cch sau

- Xut pht t h phng trnh n gin18

72

a b

ab

+ = =

(I)

1) Thay 2 2,a x x b y y= + = + vo h (I) ta c h

(1)

2 2 18

( 1)( 1) 72

x y x y

xy x y

+ + + =

+ + = chnh l v d 2.

2) Thay 2 2,a x xy b y xy= + = vo h (I) ta c h

(2)2 2

2 2

18

( ) 72

x y

xy x y

+ =

=3) Thay 2 2 , 2a x x b x y= + = + vo h (I) ta c h

(3)2 4 18

( 2)(2 ) 72

x x y

x x x y

+ + =

+ + =

4) Thay1 1

,a x b yx y

= + = + vo h (I) ta c h

(4)2 2

( ) 18

( 1)( 1) 72

x y xy x y xy

x y xy

+ + + =

+ + =

5) Thay 2 22 ,a x xy b y xy= + = vo h (I) ta c h

(5)2 2 18

( 2 )( ) 72

x y xy

xy x y y x

+ + =

+ =

a. Nh vy, vi h xut (I), bng cch thay bin ta thu c rt nhiu h pt mi.

b. Thay h xut pht (I) bng h xut pht (II)2 2

7

21

a b

a b

+ =

=v lm tng t nh trn ta li

thu c cc h mi khc. Chng hn :6) Thay 2 2 ,a x y b xy= + = vo h (II) ta c h

(6)2 2

4 4 2 2

7

21

x y xy

x y x y

+ + =

+ + =

7) Thay

1 1

,a x b yx y= + = + vo h (II) ta c h

(7)2 2

2 2

1 17

1 121

x yx y

x yx y

+ + + = + =

8) Thay1

,x

a x by y

= + = vo h (II) ta c h

(8) 2 2 21 7

( 1) 21

xy x y

xy x y

+ + = + + =


15/22

9) Thay1

,a x y by

= + = vo h (II) ta c h

(9) 2 2 2( ) 1 9

( 2) 21 1

x y y y

x y y y

+ + =

+ =10)Thay 2 22 , 2a x x b y x= + = + vo h (II) ta c h

(10)

2 2

4 4 2 2

4 7

4 ( ) 21

x y x

x y x x y

+ + =

+ =...

Bi 5(D 2007 ) Tm m h c nghim :3 3

3 3

1 15

1 115 10

x yx y

x y mx y

+ + + = + + + =

.

t n ph

1

1

a xx

b y y

= +

= +

iu kin ; 2a b

Ta c h3 3

5

3 3 15 10

a b

a a b b m

+ =

+ = Bi 6 Gii h phng trnh :

a) (C 2010 )2 2

2 2 3 2

2 2

x y x y

x xy y

+ =

=b) (B 2002)

3

2

x y x y

x y x y

=

+ = + +

c)2 2 2

3 4 2 2 4 1

x y x y

x y

= +

+ =d)

Bi 7 (St hch khi 10 nm 2012)Gii h :

a)3 2

2

3 (6 ) 2 18 0

3

x y x xy

x x y

+ =

+ = b)

3 2

2

2 (6 ) 3 18 0

7

x

x

y x xy

x y

+ = + + = a) H

( 2)(3 ) 18 0

( 2) (3 ) 0

x x x y

x x x y

+ = + + =

t( 2)

3

a x x

b x y

= + =

Nghim 1; 3x =

b)H( 3)(2 ) 18 0

( 3) (2 ) 0

x x x y

x x x y

+ = + + =

t( 3)

2

a x x

b x y

= + =

Nghim

Bi 8(D 2009 ) Gii h phng trnh : 22

( 1) 3 0

5( ) 1 0

x x y

x yx

+ + = + + =

- K. 0x . H 22

11 3. 0

1( ) 5. 1 0

x yx

x yx

+ + = + + =

t1

,x y a bx

+ = = ta c h :

2 2 2 2

2, 1 11 3 0 3 11 1 3

, 2,5 1 0 (3 1) 5 1 0 2 2 2

a b x ya b a b

a b x ya b b b

= = = = + = =

= = = = + = + =


16/22

Bi 9(A 2008) Gii h phng trnh :

2 3 2

4 2

5

45

(1 2 )4

x y x y xy xy

x y xy x

+ + + + = + + + =

- H

2 2

2 2

5( ) ( 1)

4

5( )4

x y xy x y

x y xy

+ + + + = + + =

. t2x y a

xy b

+ =

=

ta c :

2

22

5 5( 1) 0,0

4 455 1 3

,44 2 2

a b a a ba a ab

b aa b a b

+ + = = = = = + = = =

- Vy tp nghim ca h pt l S = 3 33 5 25

1; ; ;2 4 16

Bi 10 Gii h phng trnh :2 2 2( ) 7

( 2 ) 2 10

x y x y

y y x x

+ + + =

=

- H2 2

2( ) 7

( 2 ) 2 10

x y x y

y y x x

+ + + =

=

2 2

2 2

( 1) ( 1) 9

( ) ( 1) 9

x y

y x x

+ + + =

+ =.

- t 1, 1a x b y b a y x= + = + = ta c h2 2

2 2

9

( ) 9

a b

b a a

+ =

=

2 2 2 2 2

( ) 2 0a b b a a a ab a + = = = hoc 2a b= - Vi 0 3 1, 2a b x y= = = = hoc 1, 4x y= =

- Vi2 3 62 5 9

5 5a b b b a= = = = m

6 3

1 , 15 5

x y = = + hoc6 3

1 , 15 5

x y= + =

Cch 2: Th (1) vo PT (2) v rt gn ta c : 2 2 4 2 3 0 ( 1)( 2 3) 0x xy x y x x y+ + + + = + + + =

Bi 11(A 2006) Gii h phng trnh :3

1 1 4

x y xy

x y

+ =

+ + + =- K: 1, 1, 0x y xy

- H3 3

2 2 ( 1)( 1) 16 2 1 14

x y xy x y xy

x y x y x y x y xy

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

- t ,x y a xy b+ = = . 2 22, 0, 4a b a b ta c h pt

22 2

3 3 3

3 26 105 02 1 14 2 4 11

a b a b a b

b ba a b b b b

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


17/22

3 3

6 3

b x

a y

= = = =

(tha mn k)

Bi 12(Th T2010) Gii h phng trnh:8

2 29 9 10

x y

x y

+ =

+ + + =. Bnh phng c 2 PT.

Bi 13(Th GL 2012) Gii h :

2 2

2 2

1 12 7

6 11

x y

x y

x y xy

+ + + =

+ = +

- PT (1) 2 21 1

( ) 2 ( ) 2 2 7x yx y

+ + + =

- PT (2)1 1

6 ( ) ( ) ( ) 6x y

x y x yxy x y

+ + = + + + + = Ta c

2 2

6

2 2 2 7

a b

a b

+ =

+ =

Bi 14(T 2011) Gii h :

( 7) 1 0

2 2 221 ( 1)

y x x

y x xy

+ + =

= + . Ln lt chia cho2

;y y v t n ph.

Bi 15(B 2009 ) Gii h :2 2 2

1 7

1 13

xy x y

x y xy y

+ + =

+ + =. Ln lt chia cho 2;y y v t n ph.

Bi 16(Th T2012) Gii h :2 2 1 4

2 2( ) 2 7 2

x y xy y

y x y x y

+ + + =

+ = + +Chia 2 v ca 2 PT cho y v t n ph.

Bi 17 Gii h phng trnh:

2 2 2 2(2 ) 5(4 ) 6(2 ) 0

1

2 32

x y x y x y

x yx y

+ + =

+ =


18/22

V. Phng php hm s.

* C s phng php. Nu ( )f x n iu trn khong ( ; )a b v , ( ; )x y a b th :( ) ( )f x f y x y= =

Bi 1 Gii cc HPT sau :

a)3 3

2 2

2

x x y y

x y

+ = +

+ =

b)5 5

2 2

2

x x y y

x y

+ = +

+ =

Bi 2 Gii h phng trnh :5 55 5 (1)

2 2 1 (2)

x x y y

x y

=

+ =

Bi 3. Gii h phng trnh3 33 3 (1)

2 2 1 (2)

x x y y

x y

=

+ =

Phn tch. Ta c th gii h trn bng phng php a v dng tch. Tuy nhin ta mun gii hny bng phng php s dng tnh n iu ca hm s. Hm s 3( ) 3f t t t= khng n iu trn

ton trc s, nhng nh c (2) ta gii hn c x v y trn on [ ]1;1 .Li gii.T (2) ta c [ ]2 21, 1 , 1;1x y x y

Hm s 3( ) 3f t t t= c 2'( ) 3 3 0, ( 1;1) ( )f t t t f t= < nghch bin trn on [ ]1;1 .

[ ], 1;1x y nn (1) ( ) ( )f x f y x y = = th vo pt (2) ta c 22

x y= = .

Vy tp nghim ca h l S =2 2 2 2

; ; ;2 2 2 2

Nhn xt. Trong TH ny ta hn ch min bin thin ca cc bin hm s n iu trn on .

Bi 4 Gii h phng trnh:

3 23 ( 3) (1)

2 2( 1) 2 5 0 (2)

x x y y

y y x y x

= + + + + + + + =

PT 3 3(1) 3 3x x y y + = +Xt hm 3( ) 3f t t t= + . HS ng bin. T (1) ( ) ( )f x f y x y = =Thay v (2) tip tc s dng PP hm s CM PT (2) c 1 nghim duy nht 1 1x y= = .

Bi 5 (A 2003) Gii h : 3

1 1(1)

2 1 (2)

x yx y

y x

=

= +

- Xt hm s2

1 1( ) ( 0) '( ) 1 0f t t t f t

t t= = + > nn hm s ng bin.

- T (1) ( ) ( )f x f y x y = =

- Thay vo (2) c nghim1 5

1;4

x

=

Bi 6(Th GL) Gii h phng trnh

1 13 3

( 4 )(2 4) 36 (2)

(1)x yx y

x y x y

=

+ =

.

- Xt hm s3 4

1 3( ) ( 0) '( ) 1 0f t t t f t

t t= = + > nn hm s ng bin.


19/22

- T (1) ( ) ( )f x f y x y = =- Thay vo (2) c nghim 2; 6x = . vy h c nghim (2; 2); ( 6; 6) .

Bi 7 (Thi HSG tnh Hi Dng 2012)3 33 ( 1) 9( 1) (1)

1 1 1 (2)

x x y y

x y

=

+ = - T iu kin v t phng trnh (2) c 1; 1 1x y

-

3 3

(1) 3 ( 1) 3 1x x y y = , xt hm s3

( ) 3f t t t= trn [1; )+- Hm s ng bin trn [1; )+ , ta c ( ) ( 1) 1f x f y x y= =

- Vi 1x y= thay vo (2) gii c 1; 2x x= = 1 2

,2 5

x x

y y

= = = =

Bi 8 (A 2012) Gii h phng trnh

3 2 3 2

2 2

3 9 22 3 9

1

2

x x x y y y

x y x y

+ = +

+ + =

- T phng trnh (2) 2 21 1

( ) ( ) 12 2

x y + + = nn3 1 1 3

1 ; 12 2 2 2

x v y

+

- 3 3(1) ( 1) 12( 1) ( 1) 12( 1)x x y y = + + nn xt 3( ) 12f t t t= trn3 3

[ ; ]2 2

- Ch ra f(t) nghch bin. C ( 1) ( 1) 1 1f x f y x y = + = +

- Nghim1 3 3 1

( ; ) ( ; ); ( ; )2 2 2 2

x y

=

Bi 9. (A 2010) Gii h phng trnh2(4 1) ( 3) 5 2 0 (1)

2 24 2 3 4 7 (2)

x x y y

x y x

+ + =

+ + =

Li gii.

- (1) 2(4 1)2 (2 6) 5 2 0x x y y + + =

( ) ( )2 32 3

(2 ) 1 (2 ) 5 2 1 5 2 (2 ) 2 5 2 5 2x x y y x x y y + = + + = +

(2 ) ( 5 2 )x f y = vi 3( )f t t t= + . 2'( ) 3 1 0, ( )f t t t t= + > B trn .

Vy25 4

(2 ) ( 5 2 ) 2 5 2 , 02

xf x f y x y y x

= = =

- Th vo pt (2) ta c

225 42

4 2 3 4 7 0 ( ) 02

x

x x g x

+ + = =

- Vi

225 4 32( ) 4 2 3 4 7, 0;2 4

xg x x x x

= + +

. CM hm g(x) nghch bin.

- Ta c nghim duy nht1

22

x y= =

Bi 10.(Thi th T 2011) Tm cc gi tr ca m h phng trnh sau c nghim3 3 2

2 2 2

3 3 2 0

1 3 2 0

x y y x

x x y y m

+ =

+ + =Li gii.

- iu kin. 1 1, 0 2x y


20/22

(1) 3 33 ( 1) 3( 1)x x y y = - Hm s 3( ) 3f t t t= nghch bin trn on [ 1;1]

[ ], 1 1;1x y nn ( ) ( 1) 1 1f x f y x y y x= = = +

Th vo pt (2) ta c 2 22 1 (3)x x m = H c nghim Pt (3) c nghim [ ]1;1x

Xt [ ]2 2 21( ) 2 1 , 1;1 , '( ) 2 1 1g x x x x g x x x = = + '( ) 0 0g x x= = . (0) 2, ( 1) 1g g= =

Pt (3) c nghim [ ]1;1 2 1 1 2x m m

Bi 11(Th T 2012) Gii h :( )

5 4 10 6 (1)

24 5 8 6 2

x xy y y

x y

+ = + + + + =

.

TH1 : Xt 0y = thay vo h thy khng tha mn.

TH2 : Xt 0y , chia 2 v ca (1) cho5

y ta c5 5

( ) (3)

x x

y yy y+ = +- Xt hm s 5 4( ) '( ) 5 1 0f t t t f t t= + = + > nn hm s ng bin.

- T2(3) ( ) ( )

x xf f y y x y

y y = = =

- Thay vo (2) ta c PT 4 5 8 6 1x x x+ + + = = . Vy h c nghim ( ; ) (1;1)x y =


2 2 ( )( 2)

2

x y y x xy

x y

= +

+ =Phn tch.Nu thay 2 22 x y= + vo phng trnh th nht th ta s c ht

Li gii.Thay 2 22 x y= + vo phng trnh th nht ta c

2 2 3 3 3 32 2 ( )( ) 2 2 2 2x y x y x yy x xy x y y x x y = + + = + = + (1)Xt hm s 3( ) 2 ,tf t t t= + c 2'( ) 2 ln 2 3 0,tf t t t= + > suy ra ( )f t ng bin trn .

(1) ( ) ( )f x f y x y = = th vo pt th hai ta c1x y= = . Vy tp nghim ca h l S = { }(1;1); ( 1; 1)


2

2

1 3

1 3

y

x

x x

y y

+ + =

+ + =Li gii.

Tr v hai pt ta c

( )2 2 2 21 1 3 3 1 3 1 3y x x yx x y y x x y y+ + + + = + + + = + + +

( ) ( )f x f y= vi 2( ) 1 3tf t t t= + + + .2

( ) 1 3 ln3 0,1

ttf tt

= + + > +

( )f t ng bin trn . Bi vy ( ) ( )f x f y x y= = th vo pt th nht ta c

( )2 21 3 1 3 1 (0) ( )x xx x x x g g x+ + = = + =

Vi ( )2( ) 3 1xg x x x= + . ( )2 2'( ) 3 ln3 1 3 11x x xg x x x

x

= + +

+


21/22

( )2 21

3 1 ln3 0,1

x x x xx

= + >

+ do 2 1 0x x+ > v 2 1 1x +

Suy ra ( )g x ng bin trn . Bi vy ( ) (0) 0g x g x= =Vy h phng trnh c nghim duy nht x = y = 0

Bi 17. Chng minh h

20072 1

20072 1

yxey

xye

x

=

=

c ng 2 nghim 0, 0x y> >

Li gii.

K:2

2

1 0 ( ; 1) (1; )

( ; 1) (1; )1 0

x x

yy

> +

+ > . Do

0

0

x

y

>

>nn

1

1

x

y

>

>

Tr v hai pt ta c 2 2 2 21 1 1 1x y x yx y x ye e e e

x y x y = =

Hay ( ) ( )f x f y= vi 2( ) , (1; )1t tf t e t

t= +

.

( )2 21

'( ) 0, (1; ) ( )1 1

tf t e t f tt t

= + > + ng bin trn

(1; )+ .

Bi vy ( ) ( )f x f y x y= = th vo pt th nht ta c

2 22007 2007 0 ( ) 0

1 1x xx xe e g x

x x= + = =

Vi2

( ) 2007, (1; )1

x xg x e xx

= + +

. Ta c

2

2 2 2 3 2

1 3 ( 1)'( ) ; ''( ) 0, (1; )

( 1) 1 ( 1) 1x x x xg x e g x e x

x x x x

= = + > +

Suy ra '( )g x ng bin trn (1; )+ . '( )g x lin tc trn (1; )+ v c

1lim '( ) , lim '( )

xxg x g x

+ += = + nn '( ) 0g x = c nghim duy nht 0 (1; )x + v

0 0 0'( ) 0 '( ) '( ) . '( ) 0 1g x g x g x x x g x x x> > > < < > (1) ln(1 ) ln(1 ) ( ) ( )x x y y f x f y + = + = vi ( ) ln(1 ) , ( 1; )f t t t t= + +

1'( ) 1 0 0 ( 1; ) ( )

1 1

tf t t f t

t t

= = = = +

+ +B trn ( 1;0) v NB trn (0; )+

TH 1. , ( 1;0)x y hoc , (0; )x y + th ( ) ( )f x f y x y= =

Th vo pt (2) ta c 0x y= = (khng tha mn)TH 2. ( 1;0), (0; )x y + hoc ngc li th 2 20 12 20 0xy x xy y< + >TH 3. 0xy = th h c nghim 0x y= = . Vy h c nghim duy nht 0x y= =


22/22

VI. Phng php s dng bt ng thc.

1) C s phng php : S dng BT chng minh VT VP hoc ngc li, du bng xy ra khix y=

2) Mt s BT quen thuc.

Bi 1 Gii h :

2 2 2 2

2 33

(1)2 3

2 1 14 2 (2)

x y x xy yx y

x y x x y y

+ + ++ = +

+ + =

- HD : T (1) VT VP, du bng khi x y= thay vo PT (2) ta c :2 332 1 14 2x x x x + =

Ta c :

2 2

2

223

2 1 0 2 1 02 1 0 1 2

2 1 014 2

x x x xx x x

x xx x

= =

Bi 2(Thi th T 2013) Gii h :2 22x 3x 4 2y 3y 4 18

2 2x y xy 7x 6y 14 0

+ + =

+ + + =

( )( ) ( , )x y

- (2) + + + =2 2( 7) 6 14 0x y x y y . 7

0 13

xx y 0,25

- (2) + + + =2 2( 6) 7 14 0y x y x x . 10

0 23

yy x 0,25

- Xt hm s 23

( ) 2 3 4, '( ) 4 - 3, '( ) 0 14

= + = = = > =2 ( ) (2) 6x f x f Kt hp vi 1y = = + + >2 2( ) (1) 3 ( ). ( ) (2 3 4)(2 3 4) 18f y f f x f y x x y y .

- TH 2. 2x = h tr thnh2

2

12 3 1 0 1,

24 4 0

2

y y y y

y yy

+ = = =

+ = =

v nghim

- Vy h cho v nghim.

0,25

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