-
7/28/2019 Phuong Phap Giai He Phuong Trinh
1/22
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
=
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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 + + + + = +
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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+ + + + = + + + =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
+ =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
= + + = + +
+ + = + + + = + + +
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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 > .
Bi 6. Gii h phng trnh
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 + .
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
= ++
= ++
Bi 7. Gii h phng trnh
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 :
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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.
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
=
+ =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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)
Bi 4. Gii h phng trnh
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
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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 =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
13/22
IV. Phng php t n ph.
Bi 1. Gii h phng trnh2 2
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
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
+ + = + + =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
= = = = + = =
= = = = + = + =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
= = + = + + =+ + + = + + =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
+ + =
+ =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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.
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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 =
Bi 15. Gii h phng trnh2 2
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)
Bi 16. Gii h phng trnh
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
= + +
+
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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= =
-
7/28/2019 Phuong Phap Giai He Phuong Trinh
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
