Sáng tạo và giải phương trình, bất phương trình, hệ phương trình, bất đẳng thức - Tài liệu ôn thi Đại học môn Toán: Phần 2

^ ^'^•^ dieu ki?n
P = x.y

b) Dat

S ( s 2 - 3 P ) = 19

> 4P h§ phucmg trinh da cho tro thanh:
SP = -8S

= -8S

=l
S ^ - 3 ( 2 - 8 S ) = 1 9 ^ | S 3 + 2 4 S - 2 5 = O ' ^ 1 P = -6
SP

S(8 + P) = 2

S

V i dv 2: G i i i cac h$ phuong trinh sau:
a)

b)

D^t

fS = a + b
P = ab

h? da cho tro tharJi:

dieu kif n

1+

d)

xV;

= 9

2 ( a ^ + b 3 ) = 3(a2b + b2a)
a+b=6

2xy
x+y

2

x^y + x^l + y + y^ j + y - 1 1 = 0

Giii:
a) Dgt Vx = a,7y = b dieu ki?n a,b > 0.

> 4P thi h^ da cho tro thanh.

[ 2 ( 3 6 - 3 P ) = 3P

phuong trinh tro thanh:

S=6

[S = 6

trinh thanh:

P =8

S=6

Suy ra a,b la 2 nghi^m cua phuong trinh:
x 2 - 6 X + 8 = 0 o X i = 2;X2=4=>

a = 2=>x = 8

fa = 4=>x = 64

b = 4=i>y = 64

[b = 2 = > y = 8

Vay h f da cho c6 hai cap nghi?m (x;y) = (8;64),(64;8)
xy >0
x,y>-l

. Dat

^
P = x.y

dieu ki^n

Va^+b^+N/2ab = 8V2

>4P h f phuong trinh da

^|ia + b)" - 4ab(a + b)^ + 2a^b^ + yflab = 8V2

IS = a + b
19^ > 4P
D^t -I „
,
dieu ki?n thi h? da cho tro thanh.
P = ab
S,P>0
V 2 5 6 - 6 4 P - 6 P 2 +N/2P = 8N/2

o S = P = 4«.a = b = 2 o x = y = 4

Ngoai ra ta cung c6 the giai ngan gpn hon nhu sau:
^ 2 ( x 2 + y 2 ) + 2 7 ; ^ = 16

S->/P=3

S>3;P = ( S - 3 f

S + 2 + 2VS + P + l = 1 6

2^S + ( S - 3 f + 1 = 1 4 - S

4(52 + 8S +10) = 196 - 28S +

. Ta viet Igi h^ phuong

a +b=4

S=4

cho tro thanh:

3£S<14;P = ( S - 3 f

^

x^y(l + y) + x V ( 2 + y) + x y ^ - 3 0 = 0

a +b= 4

2 f s ^ - 3 S P l = 3SP

d) Dieu ki^n:

= 5

(x + y ) 1 + xy

V^y h? da cho c6 hai cap nghi^m (x;y) = ( - 2 ; 3 ) , ( 3 ; - 2 )
c) Dat a = \/x, b = ^

c)

x^ + y

7x + y = x^ - y

Suy ra x,y la hai nghi^m ciia phuong trinh:
x 2 - X - 6 = 0c>Xj=3;X2=-2

7 x 2 + y 2 + ^ / ^ = 8N/2

3[S^ + 30S - 52 = 0

x + y + 27xy =16
<=>^2^x^ +y^^ = x + y^{x-yf
rs = 6
P = 9=>x = y = 3

=0ox =yo2N/x=4ox =4

Vay h? CO mpt cap nghi^m duy nhat (x;y) = (4;4)
b) Dieu k i f n: x + y > 0.
Bien doi phuong trinh (1):

Vay h? da cho c6 nghif m (x; y) = (3; 3 ) .
x+y

^

'

x+v
*

D§t x + y = S,xy = P ta CO phuong trinh: 5^+

'J

/

2P-1 = 0
s

o S 3 + 2 P - 2 S P - S = 0 o S ( S 2 - l ) - 2 P ( S - l ) = 0 » ( S - l ) ( S 2 + S - 2 P ) = 0.


Cty TNHH

Vi

> 4P,S > 0 suy ra

+y

X

x=

y = 0, y = 3

<=>x + y + l = l - x ^ - y ^ < = > x ^ + y ^ + x + y = 0 ( k h o n g

Vay he c6 n g h i ^ m : ( x ; y ) = ( l ; 2 ) , ( 2 ; l ) .

^

x + y + —+ — = 5
x y

1'

X + —

[

1
+ y+ - = 5
I yJ

1

y + -

X + —

y

1

x+—
Dat

2

<=>

-2P=9

S =5

oS

= 5,P-6

«

=9

x + - = 2;y + - = 3
X
y

d)

x = l;y =

tuong duong vol :

3±S]

M p t h? p h u o n g t r i n h 2 an x , y dup-c gpi la d o i x u n g loai 2 neu t r o n g h?
= S
=P

X + —
X

phuong

t r i n h ta d o ! v a i t r o x , y c h o nhau t h i p h u o n g

Tinh chat.: N e u (x,);yQ) la 1 nghi^m cua h$ t h i ( y o ; x o ) c i j n g la n g h i ^ m

+

Phuong phap giai:
T r u ve v o i ve hai p h u o n g t r i n h cua h ^ ta du^c m p t p h u o n g t r i n h c6 dang

3±75

(x-y)[f(x;y)] = 0 «

3±V5
x =—-—;y =1

'x-y =0
f(x;y) = 0

. Ta cijng CO the d u n g p h u o n g phap ham

so'de t i m quan he x = y
V i di^ 1: G i a i cat h ^ phuong trinh sau:

(3±^y5

a)

xy (x + y ) ( x + y + x y ) = 30

(x-l)(y2+6) = y(x2+l

x^ +v/x = 2 y

b)

( y - l ) ( x 2 + 6 ) = x(y2+l)

x y ( x + y ) + x + y + xy = 11
x-' + 3 X - 1 + V 2 x + 1 = y

Dat x y ( x + y ) = a;xy + X + y = b . Ta thu du(?c h f :

d)

y^ + 3y - 1 + ^ 2 y + 1 = x

x + \ / x 2 - 2 x + 2 =3^-^+1

xy(x + y) = 5

ab = 30

a = 5;b = 6

xy +

a + b=l l

a = 6;b = 5 '

xy(x + y) = 6
xy +

X

X

a) Dieu k i ^ n : x,y > 0 . T r u hai p h u o n g t r i n h ciia h$ cho nhau ta t h u dugc:
x2 + V ; ^ _ ( y 2 + ^ J . 2 ( y - x )

+ y=5

« (N/^ - Vy ) [ ( ^ + 7y )(^ + y ) + 1 + 2 ( 7 ^ + 7 y ) ] = 0

+ y=3

xy(x + y) = 6

X

xy +

xy = 3

X

+ y=5

x + y =2

x = 2;y = l
(L)

x = l;y=2

+ 2=3"-^+!

Giai:

+y- 6

xy = 2
THI:

t r i n h t r o thanh

+

<=>

x + — = 3;y + — = 2
X
y

Vay h ^ da cho c6 n g h i ^ m : (x; y ) = 1;

5 ± V 2 T 5q:>/2T^

p h u o n g t r i n h kia.

1

tro thanh:
S

2

II) He DOI XLfNG LOAI 2

da cho t u o n g d u o n g :
(

;y =
-

-;y = — ^ —

Vay h# da cho c6 n g h i ^ m ( x ; y ) = ( l ; 0 ) , ( - 2 ; 3 ) .
c) Dieu ki?n: \y ^0.

:
2

thoa

i

man dieu ki^n).

X

Vift

+ S - 2P > 0 . D o d o S = 1

V o i X + y = 1 thay vao (2) ta duXetX + y +1 =

M f V D W H Khattg

/fx

Vi ( v ; ; " + ^ ) ( x + y ) + i + 2 ( V ^ + 7 y ) > o
nen p h u o n g t r i n h da cho t u o n g d u o n g v o i : x = y .


Tdi li(u

on III, ,t,u ho,

s<i;rx t,u>

v.) gtdi

Fl, bat PI, hfPi,

bdl i J i -?Wglf

T r u hai p h u o n g t r i n h cvia h f cho nhau ta t h u dugrc:
x=0

x 3 + 3 x - l + V2x + 1 - ( y 3 + 3 y - l + 72y + l ) = y - x

H a y x ^ - 2 x + > / x = 0 < » x ^ + > / x = 2 x o V x | > / x - l | | x + >/x-lj = 0 <=> x = l
X =-

Vgiy

b)

CO

da cho

3 c l p n g h i ? m : (x;y) = ( 0 ; 0 ) , ( l ; l ) ,

o

(3-N/5

3-N/5^

< : : > ( x - y ) x^ + x y + y^ + 4(x - y ) +

yx^ + 6 y - x ^ - 6 = xy^ + x

x =y

x = y=2
x =y=3

M|it khac k h i cpng hai p h u o n g t r i n h ciia h$ da cho ta dupe:
x 2 + y 2 - 5 x - 5 x + 12 = 0 o ( 2 x - 5 f + ( 2 y - 5 f = 2 . D a t a = 2 x - 5 , b = 2 y - 5
fa + b = 0
Taco:

a^+h^=2

fa + b )f ^- -22aa bb = 2
(a

(a + 4 ) ( b + 4) = 15

ab + 4(a + b ) = - l

|lab = - l
a + b = -8
ab = 31

^

.
^
, fa + b = 0
T r u o n g h p p 1: •{ ,
<=> ( x ; y ) = (3;2),(2;3)
ab = - 1
T r u o n g h p p 2:

fa + b = - 8
[ab = 31

= 0O

^
vo nghi^m.

V g y n g h i l m cua h? d a cho la: ( x ; y ) = (2;2),(3;3),(2;3),(3;2)

= 0<::>X X ^ + l

1 +1

+

V2X + 1 + 1

x>-—;y^-—
2
2

De y r ^ n g X = y = - - ^ k h o n g p h a i la n g h i ? m .
Ta xet t r u o n g h p p x + y 5>t -1

X= y

' '

=0ox =0

,

""V'i-i-

d) D|it

a = x - l
b =y - l

'

Ta

h? m o i

CO

7 X < ! -r

'.-I •»

a + Va2+l=3^
b + Vb2+l=3'

Xet h a m so f ( t ) = t + V t ^ + l + 3' ta c6
f'(t) = l +

. ^
+ 3 ' l n 3 = ^^^J^=t^ + 3 * l n 3 > 0 suy ra f ( t ) d o n g b i e n .
Vt^+l
Vt^+l

Suy ra a = b

)

Ta can giai p h u o n g t r i n h : a + Va^ + 1 = 3*

ia

Lay loga theo co so' e ca hai ve ta c6:
/

r-r

In a + Va
V

\

r—

+ 1 =aln3<=>ln a + V a ^ + 1

J

V

(

, j rl

\

I

Xet h a m so f(a) = l n a + V a ^ + l

J

-aln3 =0

^

'
1

-aln3;f'(a)=

nghjch bien tren R
M a t khac ta c6: f(0) = 0 => a = 0 la n g h i ^ m d u y nhat.

Ket l u ^ n : P h u o n g t r i n h c6 m p t n g h i ^ m x = y = 1

c) D i e u k i f n :

= 0

T o m l ? i h ^ p h u o n g t r i n h c6 n g h i ^ m d u y nhat: x = y = 0 ^,

x + y-2xy +7=0

N e u x + y - 2 x y + 7 = 0 < » ( l - 2 x ) ( l - 2 y ) = 15.

N/2X + 1 + ^2y +1

V2X +

2 x y ( y - x ) + 7 ( x - y ) + ( x - y ) ( x + y ) = 0 o ( x - y ) ( x + y - 2 x y + 7) = 0

+

2

?

+xy+ y +4 + -

x(x2+l) + -

T r u ve theo v e hai p h u o n g t r i n h a i a h? ta duQc:

N e u x = y thay vao h ^ ta c6: x - 5x + 6 = 0 o

X

V2x + l+72y + l

K h i x = y xet p h u o n g t r i n h : x ^ + 2 x - l +V2x + 1 = 0 o x ' ' + 2 x + V2x + l - l = 0

xy^ + 6x - y^ - 6 = yx^ + y

+

0

<»(x-y)

2(x-y)

-ln3<0

n e n f(a)
,- ^


He c6

Y^u

T6

DANG CAP DANG CAP

De y rSng ne'u nhan cheo 2 phuong trinh ciia hf ta c6:
6(x^ + y'') = (8x + 2y)(x^ + 3y') day la phuong trinh dJing cap bac 3: Tu do

+ La nhung he chua cac phuong trinh dang cap
+ Hoac cac phuong trinh ciia hf khi nhan hoac chia cho nhau thi tao ra
phuong trinh dang cap.

- 8 x = t^x-' +2tx

Ta thuong gap dang h^ nay 6 cac hinh thuc nhu:
^

ax^ + bxy + cy^ = d
ex^+gxy + hy^ = k

^

^
-^ r •

ta CO 16i giai nhu sau:
Vi X = 0 khong la nghiem cua h^ nen ta dat y = tx . Khi do h? thanh:

- :x

^

i

<2-3 = 3(t2x2+l)

( l - t ^ ) = 2t + 8

^_^3

t+

l-3t^

3

<=>

l - 3 t ^ =6

^,

jax^+ bxy + cy^ =dx + ey

4 .-t-^ ,

t a

t = - i

4

ax + bxy + cy = d

'

gx'^ + hx^y + kxy^ + ly^ = mx + ny
Mot so' h^ phuong trinh tinh dang cap dugc giau trong cac bieu thuc chua
can doi hoi nguoi giai can tinh y de phat hi^n:
Phuong phap chung de giai h^ dang nay la: Tu cac phuong trinh cua h? ta
nhan hoac chia cho nhau de tao ra phuong trinh dang cap bac n :

t = -=>
3

y 5^ 0 ta dat x = ty thi thu dugc phuong trinh: ajt" + a^t""''.... + a„ = 0

+ Giai phuong trinh tim t sau do the vao h^ ban dau de tim x,y
Chii y: (Ta ciing c6 the dat y = tx )

'(i-st^j^e

y=±r
4N/78

t = --=>
4

x = ±3

x

x= ±

alx"+a,x"-^y^... + a „ y " = 0
Tu do ta xet hai truang hgp:
+ y = 0 thay vao de tim x

y-+-

13
13

Suy ra he phuong trinh c6 cac cap nghiem:
(x;y)=(3,l);(-3,-l);

^4^78

a)

b)

x 2 - 3 = 3 I..2
y^+1

THI: ,

xy x^ +y^ j + 2 = (x + y)^

TH2:

= ^x + 2y

x^+3y^=6

13 ' 13 '

VTS'

13 '

13

•

xy = 1
x2+y2=2

5 x ^ ' - 4 x v ' + 3v•^-2(x + y) = 0
[x = l , [ x = - l
^
'
<=> <^
va <
xy = 1
ly = ^
ly = -''
5x^y-4xy^+ 3y-''-2(x + y) = 0
x2+y2=2

a) Tabiendoih^: I'^'+

4^78

xy|x^ + y^j + 2 = x^ +y^ + 2xy o |x^ + y^ j(xy -1) - 2(xy -1) = 0
( x y - l ) ( x ^ +y^ - 2 ) = 0

5x^y - 4xy^ + 3y^ - 2 ( x + y) = 0
2
(x,y6K)

VTSI/

Phuong trinh (2) ciia he c6 dang:

Vi d\ 1: Giai cac h | phuong trinh sau:
x ^ - 8 x = y^ + 2y

3

< » 3 ( l - t ^ ) = (t + 4 ) ( l - 3 t ^ j « 1 2 t ^ - t - l = O o

gx + hxy + ky = Ix + my

+

>-U.

3^^Y

jsx^y -4xy^ + 3y^ = 2(x + y) (*)

<=> <

x2+y2.2

NC'u ta thay x"^ + y^ = 2 vao phuong trinh (*) thi thu dugc phuong trinh
d5ngca'pb|c3; 5 x ^ y - 4 x y ^ + 3 y ^ = | x ^ + y ^ j ( x + y)

Jt^r; .


Tdi li?u on thi dai

hQC

mi t>J -histuySiTrw^^^_

sang tao vd giai PT, hat PT, hf fl,

lang vtfT

Tu do ta CO loi giai nhu sau:
Ta thay y = 0 khong la nghi^m ciia h?.
Xet y ^ 0 d§t X = ty thay vao h? ta c6:

Chia hai phuoiig trinh aia h? ta Avtqc:

Dlit Vy = ^ y = t^x^ thay vao (1) ta du(?c:

i ,\ .
St^y^ - 4ty^ + 3y^

t2y2+y2 =2

= 2 (ty + y)

t=l

x-y
1 »
x = —y

x = l fx = - l
y = l

[y = - l

Vi dv 2: Giii cic phucmg trinh sau:

2N/2

X =•

2^/2

y=- s/5

+3+2y-3 =0
2(2y3 + x^) + 3y (x + i f + 6x(x +1) + 2 = 0
1 2 x _ x + 7y
b) 3x 3y 2x2+y
2(2x + 7y) = V2x + 6 - y

Vi dv 3: Gidi cac hf phuong trinh sau:
3x3-y3=^
x^^y + l - 2 x y - 2 x = l
a) \ + y
b) x^ - 3x - 3xy = 6
x2+y2=l

Giii:

Giii:

^Q

Thay vao phuong trinh (1) ta duQc: Vx^ - x + 2 = x + 4 o x = — ^ => y = ,
9 * 18
14._5_
V$y h^ CO mpt c^p nghi^m: (x; y) =

I 9'is;

b) De thay phuong trinh (1) ciia hf la phuong trinh dang cap ciia x va yjy

"t^^

V|iy nghifm ciia hf (x;y) = ^ / l 7 - 3 13-3N/I7^

7x2+2y

a) Dieu ki^n: x^ + 2y + 3 ^ 0.

Phuong trinh (2) tuoTig duong:
2(2y3 + x^) + 3y(x +1)^ + 6x2 + 6x + 2 = 0 o 2(x + i f +
^. ^
Day la phuong trinh diing cap giua y va x + 1 .
+ Xet y = 0 h^ v6 nghi^m
+ Xet y 7t 0. D§t x +1 = ty ta thu dug-c phuong trinh: 2t^ + St^ + 4 = 0
Suy ra t = -2 <=> X +1 = -2y

=

Riit gpn bien x ta dua ve phuong trinh an t:
-r,,rs;,u
( t - 2 f (t2 + t + lj = 0 o t = 2<»7y = 2 x > 0 .
' '
Thay vao (2) ta du^c:
4x2+8x = V2x + 6 «.4x2+10x + — = 2x + 6 + V2x + 6 + 4
4
V2x + 6 + i
2j
.
Giai ra ,ta dug-c
x = >/l7-3 => y = 13-3N/I7
—•

5^!zilll = l l i o t 3 - 4 t 2 + 5 t - 2 . 0

t^+l

+

a) Ta CO the viet lai h? thanh: 3x3-y3)(x + y) = l (1)
x2+y2=l
Ta thay ve trai ciia phuong trinh (1) la bac 4. De tao ra phucmg trinh dang
cap ta se thay ve phai thanh (x^ + y^ )2.
Nhu vay ta c6:
3x3-y3

^^^^y

x2+y2 0 2x''+3x3y-2x2y2-xy3-2y''=0

'

x=y
o (x - y)(x + 2y)(2x2 + xy + y^) = 0 <=> x = -2y
2x2 + xy + y2 = 0
Neu 2x2 + xy + y2 = 0 <=> —x2 +

/

Neu X = y ta c6 2x2 = 1 o x = ±

N2

X+y

2

= O o x = y = 0 khongthoaman.

V e t r a i ciia cac p h u o n g t r i n h t r o n g h? la p h u o n g t r i n h d i n g cap bac 3 doi
+

Neu x = - 2 y < » 5 y ^ = l » y = +
T o m lai

•u

—

vol x,yjy

p h u o n g t r i n h c6 cac cap nghi?m:
'2N/5

(x;y) =

5

2 ' 2

-^/5^
'

b) Dieu ki?n y > - 1 . Ta viet lai h ^ thanh:

f-275
5

5

.De thay y > 0 . Ta dat x = t ^

^ ( 2 t + t^) = 3

4£

t2+2

<=>

<=>

thi t h u d u g c h?:

3
,2
.
.
.
= - o 2 t ^ - 3 t + l = 0<=>

' 5

x2 7 y 7 T - 2 x ( y + l ) = l

5

x'^''-3x(y + l ) = 6

+

Ne'u t = 1 t h i

+

l t hthi
i xx = —l ^ y < : > y = 4x<=>x"'= — o x
Ne'u t = —
2
2

X

= ^y o

X =

t= l
2

1 => y = 1

Ta thay cac p h u o n g t r i n h ciia h$ deu la p h u o n g t r i n h d a n g cap bac 3 d o i v o l
T o m lai he c6 cac nghiem: ( x ; y ) = ( l ; l ) .

1

=

^ ^ ^ ^ ^

4

De thay y = - 1 k h o n g phai la nghiem ciia he p h u o n g t r i n h .
b) Dieu k i ^ n : x^y + 2y > 0 <» y > 0 .

Xet y > - 1 . Dat x = t ^ y + 1 thay vao h? ta c6:

T u p h u o n g t r i n h t h u nhat ta c6: xy = - x ^ - x - 3 thay vao p h u o n g t r i n h t h u
t^ - 2 t = 1
y^i)

t^-3t

» t ^ - 3 t - 6 ( t 2 - 2 t ) = 0<»

= 6

+

Ne'u

+

Neu t = 3 « 2 7 ^ ( y + l f -9^{y+

t= 0

hai ta t h u dugc:

t= 3

(x + l ) 2 + 3 ( y + l ) - 2 x 2 - 2 x - 6 - 2 ^ y ( x 2 + 2 ) = 0
«> x^ + 2 - 3y + 2^y{x^

t = 0 t h i x = 0 . K h o n g thoa man he
=

\

6 »

y

= 3^ - 1

x=

Dat J y = t./(x^ +2)

ta t h u duoc: 3t^ - 2t - 1 - 0 <=>

r * " •*
1

duoc: X = - 1 => y - 3
T o m lai h ^ p h u o n g t r i n h c6 m o t cap nghiem (x; y) = ( l ; - 3 )

+xy+x+3=0

V i du 5: G i a i cac

(x + 1)^ + 3 ( y + l ) + 2| x y - J x 2 y + 2y

X

a)

a) Dieu kien: y > 0 . P h u o n g trinh (2) ciia h ^ c6 dang:
o

"y = - l

x~

2x

8v + •3

+ y
/x-\x^
4

V3v

2xy + x"' = 3
xy +

X'

b)

y
2

x^y - 3x - 1 = 3 x ^ ( y r ^ - l)-"^
•Jsx^ - 3xy + 4y^ + ^ x y = 4y

Giai:

2xy + x-' = 3

T r u o n g h o p y - - 1 k h o n g thoa m a n dieu k i ^ n
T r u o n g hop 2xy + x^ = 3 ta c6 h?:

phuong trinh sau

= 0

Giai:

2xy(y + 1) + x^(y + 1) = 3{y + 1)

+2

'

t = 1 ta c6: y = x^ + 2 thay vao p h u o n g t r i n h t h u nhat cua h? ta t h u

Khi

X

2xy^ +(x-'' + 2 x - 3 ) y + x^ = 3
X

va Vx^

I —

V i dy 4: G i a i cac h? phuong trinh sau
a)

"

= 0

Day la p h u o n g t r i n h d i i n g cap bac 2 doi v o i ^

^

Vay h ^ CO 1 cap n g h i e m d u y nhat ( x ; y ) =

xy +

+2)

'

a) D i e u ki?n: y ; t O , x + y ? ^ 0 , — + — > 0 .
3y
4
P h u o n g t r i n h (2) t u o n g d u o n g :

,t.

,if ..


4x + 3y
Sy""

V 12y

6

8y l 6

,. x^
, 4x + 3y
Day la phuong trinh dang cap dol voi — va
oy
6

6 J
"

: • t

hayi^>0,i^>0.
8y

4x + 3y

By

X =

6y

Vay h? C O nghi?m (x; y ) =

r24

4^

. 7

7

ra phuong trinh c6 nghi?m khi va chi khi t = 1 o x = 1
Tom lai h^ phuong trinh c6 nghi^m ( x ; y ) - ( l ; l )
C h u y: Ta cung c6 the tim quan h? x,y d y a vao phuong trinh thu hai ciia
h^ theo each:
,(-8;12).

Phuong trinh c6 d^ing:

V8x^-3xy^4y^-3y^7;:^-y = 0 o

[x,y>0
x^l

8x + 5y
^ / 8 x 2 - 3 x y + 4y2 + 3 y

cap nghi^m nay khong thoa man h?.
T a chia phuong trinh thu hai cua h? cho

-3-+4+
y

-=4.
y

, (^"yK^^^^y)
78x2-3xy + 4y2+3y

^(pOy^p
V'^y + y

x=y

x , y . T a thay neu y = 0 thi tit phuong trinh thu hai cua h^ ta suy ra x = 0,

x

ta thay t = 1 th6a man phuong trinh.

N h u vay ham so' f(t) dong bie'n tren [ l ; +oo) suy ra f(t) > f(1) = 3 . T u do suy

De y rang phuong trinh thu hai cua h? la phuong trinh dang cap doi voi

y>0.

=3

Xet t > l . T a c 6 f ( t ) = f3t^ + 6t]fN/F + V T ^ f + - ^ ^ . ^ ^ ^ — j + 3t^ - 1 ] > 0
\2
Vt.Vt-1
^
'

4 ,
,
y= -^(L)
- y + y - 1 6 y = 16<=>
y = 12=>x = - 8 ( T M )

Xet

•

phuong trinh tro

o ( t ^ + 3t^ - l ) ( V t + yft^f

Xet ham so' f(t) - (t^ + 3t^ - lj(>/t + r / T ^

T H 2 : X = - - y thay vao (1) ta c6:

y^O

,111,.,

Chia bat phuong trinh cho x^ > 0 ta thu du^c phuong
I
\3

thanh: t^ + 31^ - 1 = 3(Vt - y f t ^ f

28
168,-,
y=
=>x =
(L)
| y 2 + y 2 - 1 6 y = 16<=> ^ 3 7
37 ^ '
4
24
y = —=>x = —

<=>

•'.

Phuong trinh thu nhat cua h^ tro thanh: x^ - 3x - 1 = 3 V ^ ( V l - x -1)^ .

trinh: 1 — - — - = 3 . - - l - - 7 =
. D^t - = t = > t ^ l
Vx
Vxj
X
x^
x-*

2 •

xy >0

o t =l

i M t ; - ',1

=y .

= 2ab o a = b

T H l : X = 6y thay vac (1) ta c6:

x
X

T a xet 0 < X < 1.

6

b) Dieu ki^n:

K h i t = 1 =>

[(t - l)(2t^ + 2t2 +1 + 3) = 0I

Dieu ki?n: 0 < x < 1. T a thay x = 0 khong th6a man phuong trinh.

6

e a t — = a, •^^^-!^ = b s u y r a a^ +
• 8y
6
'
X

[St'* - 4t2 + St - 1 2 = 0

t<4

t<4
|2t'* -1^ + 2t - 3 = 0

. x^
, 4x + 3y
,
cung dau
T a thay phuong trinh c6 nghi?m khi va chi khi — v a
oy
o

t^4

n ,
; —
ft ^4
V8t^-3t2+4 =4-to<^
,
St" - 3t2 + 4 = t^ - St +16

D|t

y ta thu dug-c:

thu duQfC phuong trinh

''•.••.in

I

y
V^y+y

= 0(3) • V i x , y > 0 nen ta suy ra x = y .


TTFTr

PHLTONG P H A P BIEN O O l TL/ONG DLTONG
Bie'n dot titvng ditong la phteang phdp gidi he dua tren nhimg ky thudt ca ban

Dat t = Vx + 1 + 7 4 - x > 0 => \lx + l.y/4-x
ta c6: t +

t2-5

= 5ci> t'^ + 2 t - 1 5 = 0<=>

= - y ^ • Thay vao p h u o n g t r i n h
t = -5
t =3

•J'.

nhir. The, bie'n doi cdc phuang trinh vedqng tich,cgng trk cac phuang trinh trong
he de tqo ra phuang trinh he qua c6 dang dqc biet...
*

>

x =0

K h i t = 3 => N/X + 1 . V 4 - X = 2 • » - X ^ + 3x = 0 O

x =3

Ta xet cac v i d u sau:
V i d u 1: G i a i cac
V ^
a)

p h u o n g t r i n h sau

T o m lai he c6 n g h i e m ( x ; y ) = ( 0 ; 0 ) .

+ V 4 - 2 y + x/5 + 2 y - ( x - l ) 2 ^5
3 x ^ + ( x - y ) 2 = 6 x 3 y + y2

N h a n x e t : D i e u kien t > 0 chua phai la dieu k i f n chat ciia bien t
(j)

That vay ta c6: t = Vx + 1 + V 4 - x => t^ = 5 + 27(x + l ) ( 4 - x ) => t^ > 5

x3-12x =y^-6y2+16
b)

M a t khac theo bat d^ng thuc Co si ta c6

+ y^ + xy - 4x - 6y + 9 = 0

2V(x + l ) ( 4 - x ) < 5 = ^ t 2 < 1 0 « t € [ N / 5 ; v ' l O

2xy - X + 2y = 3
c)

b) He viet lai d u o i dang

\^ +4y^ = 3 x + 6y^ - 4

x-''-12x = ( y - 2 ) 3 - 1 2 ( y - 2 )

X + x ( y - 4 ) + (y-3r

y^-7x-6-^y(x-6)=l

Dat t = y - 2 . Ta c(S he :

d)
sJ2(x-yf

=0

+ 6x - 2y + 4 - 7y = Vx+T

x^^-12x = t - ' ' - ] 2 t

Giai:

<=>

(x-tKx^+t^+xt-i2)=o

n

x ^ + t ^ + x t - 2 ( x + t) + l = 0

(2*)

x^ + x ( t - 2 ) + ( t - l ) ^ = 0

x>-l

T u (*) suy ra

a). D i e u k i ^ n s y < 2

x^+t^+ xt-12 = 0

.

(3*)

x=t

5 + 2y>(x-])^
Vol x = t thay vao ( 2 * ) ta c6 p h u o n g t r i n h 3x^ - 4x + 1 = 0
Xuat phat t u p h u o n g t r i n h (2) ta c6:

V

T u day suy ra 2 n g h i e m cua h ^ la ( x ; y ) = ( l ; 3 ) . '1
3'3

3x''-6x-V + { x - y ) 2 - y 2 = 0
<^ 3x^^(x - 2y) + x(x - 2y) = 0 » x(x - 2y)(3x2 +1) = 0 e>

x=0
X = 2y

Vol (3*) ke't h o p v o l ( 2 * ) ta c6 h ^

V o i x = 0 thay v a o ( l ) t a c 6 : 1+ 7 4 - 2 y + 7 4 + 2y = 5 c ^ 7 4 - 2 y + 7 4 + 2y = 4

(x + tr - x t - 1 2 = 0

Theo ba't dang thuc Cauchy-Schwarz ta c6

(x + t)^ - x t - 2 ( x + t) + l = 0 = 0

X+t = —

xt =

121

( V N ) . D o (x + t f < 4 x t

7 4 - 2 y + 74 + 2 y ) ^ < 2(4 - 2y + 4 + 2y) = 16 <=> 7 4 - 2 y + 74 + 2y < 4
Vay he p h u o n g t r i n h da cho c6 2 nghiem: ( x ; y ) = ( l ; 3 ) ,

Da'u = xay ra k h i : 4 - 2v = 4 + 2v <=> V = 0

(x + l ) ( 2 y - l ) = 2

V o i : X = 2y . Thay vao p h u o n g t r i n h tren ta d u g c
=5<=>

7^

[3'3)

H ? CO n g h i e m : ( 0 ; 0 )

NATTT + V 4 - X + ^5 + \-{x-lf

'I

+ V 4 - X + 7(x + l ) ( 4 - x ) = 5 (*)

) D u a he p h u o n g t r i n h ve dang:

12

"

-

^3 +. ^ ( 2 y -1)3 = 3{x +1)2 + | ( 2 y - 1 ) - 5
(X +1)-^


D$t:

a = x + l; b = 2y-l.

<:i>2(x + l - y ) 2 + ( V ^ - V y ) ^ = 0 « | ' ' / ^ ^ ^ < » x + l = y
[ Vx +1 = ^ y
Thay vao phuong trinh (2) ta c6:

Khi do ta thu dug-c h§ phuong trinh:
ab = 2

ab = 2
^y,,
2a^+b^=6a^+3b-10

,3+ib3=3a2+|b-5'

2'

2

^

^

Dlt a = ^ y ( y - 7 ) ta c6 phuong trinh:

,

:!

Tir h? phuong trinh ban dau ta nham dugc nghi?m la x = y = 1 nen ta se c6

a>-l
a^-1
a =0
<
o
a = -l
a^-a2-2a = 0
a=2

h# nay c6 nghi?m khi: a = 2; b = 1
[(a-2)b = 2 ( l - b )
Dod6tasephantichheved,ng:|^^_^^,^^^^^^^^_^^,^^^^^^
^

Va^ +1 = a +1

2(l-b)

Vi ta luon c6: b ^ 0 nen tu phuong trinh tren ta rut ra a - 2 = — - —

Voi a = 0:

y = 0=>x = - l
y = 7=>x = 6

The xuong phuong tririh duoi ta dugc:
y=
Vol a = - l = > y 2 - 7 y + l = 0<:> ^

i ^ ^ ^ ( a +1) = (b - l)Hh + 2) o (b - if [4(a +1) - b2(b + 2)] = 0

4(a + l) = b2(b + 2)

Voi a = 2 r : > y 2 - 7 y - 8 = 0 0

b+2

• The len phuong trinh tren taco:

b = -2 => a = -1 o X = -2; y = - i ( ^ = b^(b.2).
b
b^ = 4 (Khong TM)
V^y h? da cho c6 2 nghi^m la: (x;y) = (1;1),
x>-l
. Ta Viet lai h? phuong trinh thanh:
y >0

(x;y) = (-l;0),(6;7).

<=> ^ 2 ( x - y ) ^ + 6 x - 2 y + 4 = ^y + Vx + 1 . Binh phuong 2 ve ta thu dugc:

o2l{x

+ lf

-2y(x + l ) + y 2 l + (x + l + y) = 27y(x + l )

—

2

y=-l
(L)
y = 8=>x = 7

Vi
a)

c)

2: Giai cac

5-3N/5 7 - 3 V 5 '

5 + 375 7 + 375

,(7; 8)

phuong trinh sau

,2
x^-(2y + 2 ) x - 3 y 2 = 0

x^ - 2 x y + 2y^ +2y = 0

x^ + 2xy2 - (y + 3)x - 2y^ - 6y2 +1 = 0

x^-x2y + 2y2+2y-2x =0

x^ + x y + 9y = y^x + y^x^ + 9x
d)

x(y3-x^) = 7

xy^ - 3x^y - 4yx^ - y + 3x^ = 0
3x^y-y2+3xy + l = 0

Giii:

^ 2 ( x - y ) 2 + 6 x - 2 y + 4 - ^ = Vx+T

2x^ - 4 x y + 2y^ + 6x - 2y + 4 = x + y +1 + 2^y(x +1)

2

5-3S

H | phuong trinh da cho c6 nghi?m la :

Vol 4(a + l) = b^(b + 2).Talaic6: ab = 2 o b ( a + l) = b + 2 o a + l = - ^

d) Dieu ki^n:

=>x =

7 + 375
5 + 3>/5
y = —-—=>x =
^

b=l

Voi: b = 1 => a = 2 , suy ra: x = y = 1; .

7-3S

a) Cach 1: Lay phuong trinh thu hai tru phuong trinh thu nhat theo ve ta
du(?c: 2xy^ - (y + 3)x - 2y^ - 6y^ +1 + (2y + 2)x + 3y2 = 0
O 2xy2 + xy - 2y3 - 3y^ +1 - X = 0 <» X (2y^ + y _ 1J _ 2y3 - 3y^ +1 = 0

« ( y + l ) ( 2 y - l ) ( x - y - l ) = 0.
+ Neu y = -1 thay vao phuong trinh (1) ta c6: x^ = 3 o x = ±73


+

1
2
3±2-\/3
Neu y = - thay vao phuong trinh (1) ta c6: 4x - 12x - 3 = 0 <=> x =

^3
- X

+

- 2x^ - 3(x -1)2 = 0 o -4x2 + 6x - 3 = 0 . V6 nghi^m.
Ketluan: ( x ; y ) = ( ^ ^ ; l ) , ( - ^ / 3 ; l ) ,
»

3 - 2 V 2 1^

2 '2

l-t3
t

3 + 2N/2
2

'2

-t^

= 7.

(5).

D|itt = V ^ ( t > 0 ) .

(5)

CO d^ng

= 7 « t ^ - ( 3 - t ^ ) 3 + 7t = 0.

Xet ham so £(t) = t^ - (3 - 1 ^ )3 + 7 t (t > 0). Ta c6

nhu

sau

•
f'(t) =

+ 9t^ (3 - 1 ^ )2 + 7 > 0.

Cach 2: Phuong trinh thu hai phan richdu(?c: {2y^ + x ) ( x - y - 3 ) + l = 0

V^y phuong trinh f(t) = Oco toi da mpt nghi^m. Mat khac ta c6 f ( l ) = 0 nen

Phuong trinh t h u nhat phan tich du(?c: (x - y)^ - 2{x + 2y2) = 0

suy ra t = 1 la nghi^m duy nhat ciia phuong trinh f(t) = 0. T u do ta du
D^ta = x - y , b = x . 2 y

7

,

X = 1, y = 2. Vgy h? phuong trinh c6 mpt nghi?m duy nhat (x, y) = (1; 2).

, . . [a^ - 2 b = 0
t a c o h , : | ^ ^ ^ ^

d)

dugc viet lai n h u sau:

b) Lay phuong trinh t h u hai trir phuong trinh thu nha't, ta dug-c:
x ^ - x 2 - x 2 y + 2 x y - 2 x = 0, hay ( x ^ - x ^ - 2 x ) - y ( x 2 - 2 x ) = 0.
Do x^ - x^ - 2x = (x + l)(x2 - 2x) nen t u tren, ta c6 (x^ - 2x)(x + 1 - y) = 0.
+

- X"^

Neu y = X - 1 thay vao phuong trinh (1) ta c6:

Neu x = 0:

2:

( x y - y ) ( y - 3 x 2 ) = 4x2y

3x2y - y2 + 3xy + 1 = 0

3 x 2 - y 2 + 3 x y + l = 0>/-:

Xet voi y = 0 thay vao ta thay khong la nghi^m ciia h ^ .
Vol y ^ 0 ta bien doi h? thanh :

> = 0
y = -2

1
X

y=0
+

Neu

+

Neu y = x + l thay vao phuong trinh (1) ta thu dugc: l + 2 y 2 + 2 y = 0v6

X =

(xy2-y) + (3x2-3x3yj = 4 x 2 y ^

y

nghi^m.

D|it:

Ketluan:
H$ phuong trinh c6 cac c^p nghi^m la: (x;y) = (0;0),(0;-2),(2;0),
c) Truoc tien ta d i bien doi phuong trinh (1) trong h? ta dug-c
x2(x2 - y2) + xy(x2 - y^) -9(x - y) = 0 , o ( x - y)[x(x + y ^ - 9

'4

1
X

( y - 3 x 2 ) = 4x2

y)

3 x 2 - y + 3x + - i = 0

y= 3

3x'' - y +

X =

a= x

fab = 4x2
y
K h i do h? tro thanh h? :
a + b = 4x
b = y-3x2

,

•J

-

t - 4 x t + 4x2 o ( t - 2 x ) 2 = 0 o t = 2 x o
=0.

1
—

1
—

x(x + y ) 2 = 9

(3)

y=

[x(y3-x3) = 7

(2)

2x = - 1 - 3 x 2 3 x ^ + 2 x 2 + 1 = 0

Tu phuong trinh (3) ta suy ra dugc x, y > 0. Cung Kr (3) bSng phep rut an ta
thu duQ-c y = - = - x. Thay vao phuong trinh (2) ta thu A\xqc phuong trinh
Vx

-4x

Theo Viets thi ta c6 2 so a va b la nghi^m ciia phuong trinh :

R6 rang vol x - y = 0 thi h ^ v6 nghi^m khi do ta dua h? phuong trinh ban
dau ve h ^ phuong trinh

4x^

X

y=

X

Vhy H CO 1 nghi?m (x; y ) = ( - 1 ; l )

yv =
— x
V _
2x

^

y

.2
2x = y - 3x^

rx=-i

1

^

o
2x = - 1 - 3 x 2
X


My» nyi-

—.--

-o

- •

,

.

—•.^xtw^i'

V i d\ 3: Giai cac h f phuang trinh sau

x^ + 1 6 x - 1 5 > 0
- Zx^y - 15x = 6y(2x - 5 - 4y)

a)

c)

b)

- y^ + 9y = x(9 + y - y ^

x^ 2x
+— =
8y
3

4

6 x ^ - 3 x ^ y + 2xy + 4 = y^+4x + 6x^

_ ;j5jj^2 ^

36 =

x + 16-

X

15

= 0 o

x=y
x + y3-9 = 0

O X - — = - 1 8 < » x 2 + 1 8 x - 1 5 = 0<=>
X

Ta chi can giai truong hgp x = y . The vao phuong trinh ban dau ta
dugc. ^ 1 + x + y J T ^ = 2. D l t a = ^ l + x;b = V T o c (b > O) thi
.^"^'^"^
=^a^+(2-af = 2 o a 3 + a 2 - 4 a + 2 = 0 o { a - l ) f a 2 + 2 a - 2 ) = 0
a^+b2=2
^
'
V
/V
/
Tir do suy ra nghi^m cua phvrong trinh ban dau

;

x = 0;x = - 1 1 + 6N/3;X = - 1 1 - e V s

b) Phuang trinh t h u nhat ciia h? <=> (2y - x)[\^ - 12y - is) = 0 o
y=-

• +

2(x2-15)
36x^
x^-lS

thay vao phuang trinh t h u hai cua h | ta duQc:

12
—

4x^
= ,

3

Vx2-15

x2

. ^

X = -9-4N/6

_9-4V^;^^±1276]

N g h i f m ciia h f da cho la: (x; y) =
TH

- 9 + 476

X =

2:

x = 2y

x^

2x

4x

3

Thay

2x^ x^
-+
4
V 3x

vao

phuang

trinh

thii

hai

ciia

h$

ta

7
llx^
<::>X = 0 (loai) (do dieu ki§n
o—x =.
4
6
V 12
X

KL: Nghi^m ciia h$ da cho la: (x;y) =

x2-15

x^-15
12

) Dieu ki§n

x>2
y>3

Phuang trinh (2) ciia h^ tuang duong vai:
( 2 x - 2 - y ) ( 3 x 2 + y - 2 ) = 0<:>

y = 2x-2
y = 2-3x2

Voi y = 2x - 2 the vao phuang trinh (1) ta dugc:

24

(1)«7X-6N/2X-4-4V6X-15-4 =0

x2+16x-15) + (x2+16x-15) = 0
•-12,
^'"Vx^-lS

x2+16x-15>0

x2+16x-15^0

6

36

x2-15

f
V

2y = x

2x

<=>x = 5

/

Vay h f da cho c6 3 nghi^m la x = y = 0;x = y = - l l + 6V3; x = y = -11 - 6\/3

3x^

x = -3

+ N e u t = -18

Do do x + y'^ - 9 < - 1 < 0 nen x + y'' - 9 = 0 v6 nghi^m.

THI: y =

x=5

+ Neu t = 2<=>x- — = 2 < : > x 2 - 2 x - 1 5 = 0<»

Vi y < 1 va ^ 1 + x + ^ 1 - y = 2 nen ^ 1 + x < 2 o x < 7 .

x^ -15

j

t=2

Dat X - — = t = > t 2 + 1 6 t - 3 6 = 0<=>
x
t = -18

Giki:

x^ - y * +9y = x|9 + y - y ^ j < = > ( x - y ) ^ x + y^ - 9

_

_j g j p ^

Vi x = 0 khong phai la nghi^m. Ta chia hai ve phuang trinh cho x^ ta c6:

[2xy + y - y ^ = 2

a) Tir phuong trinh (2) ciia h§ ta c6:

_ j5

Xet phuang trinh (*) 36x2 ^

2

x ^ y - 8 y ^ + 3x^y = - 4

3x-6V2x-4=473y-9-2y

36x2 ^

= x^ +16X-15

(3)

Den day su dung bat dSng thuc Co si ta c6:
f6N/2x-4=3.2V2(x-2)<3x
^ /^ r
—
^ ^
,
1—
=i>6V2x-4+4V6x-15^7x-4
4V6x - 1 5 = 2.273(2x - 5) < 2(2x - 2)
Dau " = " xay ra khi chi khi x = 4

,

Tir (3) suy ra x = 4 la nghi^m duy rthat. V^iy h? c6 nghi?m (x;y) = (4; 6)

c6:

y^O)


j:tyTNHHMlV

-

V o i y = 2 - 3 x ^ <.2

v 6 n g h i ^ m do d i e u k i ? n y > 3

V^y h§ da cho chi c6 1 n g h i ^ m ( x ; y ) = (4;6)
d) The p h u o n g t r i n h 2 vao p h u o n g t r i n h 1 a i a h# ta
-

Ta c6:

>,
duQC

phuong trinh :

x^y - 8 y * + Sx^y = - 2 ( 2 x y + y - y^) <=> (x^ - 8 y ^ + 3x^ ) y = (-4x - 2 + 2 y ) y

suy ra: •

x^+4+x

vx^+4-x

=4; y y ^ + 4 + y

DWHKhangVift

y y ^ + 4 - y = 4 nen ta

•y/x^+4+x = - ^ y ^ + 4 + y

<=> X = y .

Vy^ + 4 - y = Vx^ + 4 - X

; ;

V i y = 0 k h o n g la n g h i ^ m a i a h?. Chia ca h a i ve cho y ta d u g c p h u o n g t r i n h

Thay vao p h u o n g t r i n h t h u h a i a i a h ^ ta c6:

x^ - 8y^ + Bx^ = - 4 x - 2 + 2 y o

x 2 - 8 x + 10 = (x + 2 ) V 2 x - l < » x 2 - 8 x + 1 0 - ^ ( x 2 + 4 x + 4 ) ( 2 x - l ) = 0

x^ + Sx^ + 4x = 8y^ + 2 y - 2

D a t : z = x + l = > x = z - l . K h i do ta c6 p h u o n g t r i n h :
+ z = 8y^ + 2 y o

(z - 2y){z^

+ 4y^ + 2zy) = 0 do (z^ + 4y^ + 2zy > 0

<=>z = 2 y = > x + l = 2y=J>x = 2 y - l

x2 + 4x + 4 - 6(2x - 1 ) - ^(x^ + 4 x + 4 ) ( 2 x - l ) = 0 . Chia

x^ + 4x + 4 > 0 . Ta

The vao p h u o n g t r i n h 2 a i a h | ta dug^c p h u o n g t r i n h :
y =l

3 y ^ - y - 2 = 0<»'

-2
V=—
/
3

=>x = l

DMt t = J —

-7
=>x= —
3

H§ phuong trinh da cho c6 hai nghi^m (x;y) = (1;1);
V i dy 4: GiAi cac h$ phuong trinh sau

-7

-2

l 3 ' 3 j

Ix^ + 4 x + 4

2x-l

2x-l

a)

> 0 t h u d u g c p h u o n g t r i n h : t^ - 1 - 6 = 0 <=>

Ket luan:

n
2y-yx^+2y + l

THl: ^

2x2-x3y = 2x2y2-7xy + 6

= ( x - y ) <=>

TH2: ^

a) P h u o n g t r i n h d a u a i a h f d u p e viet l a i n h u sau:

V
o

x +Vx^

/

yjx^ +2y + l

+4

.^yjy^

+ 4 - y = 4

J

=3y->

+2y + l = x + y

6xy = 9y^

x = l ; y = l(TM)
- 2 y - l o

xy = y^ + 3 y - 3

415
1 7 , ^ „•
x = — ; y = -(TM)

+ 2 y + l = x + y . B i n h p h u o n g hai ve p h u o n g t r i n h :

x + y >0

=log2 4

+i-y

V

'

^^S:)!!'

3y>x

x'' + 2y + 1 = 9y^ - 6xy + x^

Giai:

+ l o g 2 yjy

=>t = 3

+ 2 y + l = 3 y - x . B i n h p h u o n g h a i ve p h u o n g t r i n h ta dug^c:

f3y.:

x^ + 2xy + 6 y - (7 + 2 y ) x ^ = - 9

+4

t = -2

P h u o n g t r i n h (1) t u o n g d u o n g :

f

2x + ( 3 - 2 x y ) y 2 = 3

x + Vx

t= 3

p h u o n g t r i n h c6 cac n g h i ^ m la: ( x ; y ) = ( l ; l ) , ( l 3 ; 1 3 )

y(y-x) = 3-3y

logj

- 6 = 0.

x ^ + 2 y + l + x ^ + 2 y + l = x ^ - 2 x y + y^

3y^ + 1 + 2 y ( x + 1 ) = 4 y ^ x 2 + 2 y + l

d)

cho

x = l
Giai t - 3 « ^ ^ - ^ ^ i ^ = 9 < : ^ x 2 - 1 4 x + 13 = 0 o
2x-l
x = 13

xy - 4 ( x + y ) + 1 0 = (x + 2 ) . ^ 2 y - l

c)

trinh

b) D i e u k i ? n : x^ + 2 y + 1 > 0 .

l o g j f x + Vx^ + 4 +log2f>/y^ + 4 - y = 1

b)

, x^ + 4 x + 4

CO

phuong

x^ + 2 y + 1 = x^ + 2 x y + y^

x + y>0
2xy = - y ^ + 2y + 1
xy = y^ + 3y - 3
415,17

V | y h ? c 6 n g h i # m (x;y) = ( l ; l ) .

51 ' 3

x = l;y = l
o

x = —;y =
21^

—(L)
3


CtyTNHH.

c) Tu phuong trinh (1) ta thay: 2x(l - y^) = sjl - y^).
THI: y = l thay vao (2) ta c6:

3
+ xy + -3 = y^

-7x+ 6 = 0 <=> x = l;x = 3;x =-2.

TH2: Ket h
f2x + 2xy + 2xy2 =3 + 3y

,

.

(•)

, -

• .ov

d)

c)

- 3J = 0.

+ Neu: xy = 2 thay vao (*) ta c6:
2x + 4 + 4y = 3 + 3y

2x + 3-x^ + y ( 3 - x 2 ) = 3 + 3y=>y = - ^ - l =>2x ^

= 3-x^ <::>x = l;y = l

V^y he CO nghi?m (x;y) = (l;l),(3;l),(-2;l).
x 4 _ 7 x 2 + 9 - 2 y ( x 2 - x - 3 ) = 0 < : : > ( x 2 - x - 3 ) ( x 2 + x - 3 ) - 2 y ( x 2 - x - 3 ) = 0.
1-N/I3

79 + V I 3

l + ^/l3

79-yJl3'
36

=>y=-

c = ^/5=>y = l -

a)

36

,
[6y-3x + 3xy-12 = 0
o) H? tuong duong: <
[4x-* + 24x2 + 45x = -y^ + 6y - 20

36

Tru hai phuong trinh tren cho nhau ta duc^c:
4x^ + 24x2 + 48x + 32 - - y ^ + 3xy + 12y

N/5;1-

o 4(x + 2)^ + 4y^ = 3y^ + 3xy + 12y

p h u o n g trinh sau

xy - X - y = 1
4x^ - 12x2 + 9x = _y3 + 6y + 7

'

i;^

b)

ã :

ô ( x + y - l ) ( 2 x - 2 - y f =0

Vay h^ CO nghifm (x; y) =

. fl->/l3 79 + >/l3l f l + 7T3 79-7l3

5: G i i i cac

«

o 4 ( x + y - l ) ( x - l ) ^ - ( x - l ) y + y2 = 3 y ( y 2 + x y - y - l + l) -

Voi y = 2x - 2 thay vao (1) ta du(?c: 2x^ - 5x +1 = 0 <=>

Vay h^ CO nghi^m

Vi

4x^-12x2+9x = - y ^ + 6 y + 7"

X =

: = -yfs => y = 1 +

,

3xy-3x-3y = 3

Voi y = 1 - X thay vao (1) ta du(?c: x^ - x + 2 = 0 (v6 nghi^m).

TH2: 2y^ = x^ + X - 3 thay vao (2) ta c6:

(x;y) =

a) H? tuong duong:

o 4 ( x + y - l ) ( x - l f - ( x - l ) y + y2 =3y2(x + y - i )

d) Phuong trinh (1) tuong duong:

+ X - 3 x2-x3=10o

=_l

Giai:

o 4 ( x + y - l ) ( x - l ) ^ - ( x - l ) y + y2 =3y(y2+x + l)

+ Neu 2xy = 3-x^ thay vao (*) ta c6:

x=

- 4 y 2 . ^

o 4(x -1)^ + 4y^ = 3y^ + 3xy + 3y

Phuong trinh nay v6 nghi^m nen h? v6 nghi^m.

THI: x 2 - x - 3 = 0c>

3

Tru hai phuong trinh cho nhau ta dugc: 4(x -1)^ = -y'^ + 3xy + 3y
=> y(l + y) =-4 .

X=

+ X=

x + y-1

(xy + 2 f + - l - = 2y + i

[2x^-x^y = 2x^y^-7xy + 6
Phuong trinh (3) tuong duong voi: (xy - 2)^2xy +

x2 + y2

angViet

<»4(x + y + 2) (x + 2)^-(x + 2)y + y2 =3y(y2+x + 4)

xy - X + 2y = 4
4x^ + 24x2 +

= _y3 + 6y - 20

The X = xy + 2y - 4 vao VP ta dug^c:

5->/l7

4

5 + Vi7 "
x=-


Tai li$u on thi dai hoc sdng Uio va giai PT, bat PI, he PI. bai Pi

Cty TNHH MTV D W H miang Vift

\^^„i,c,t intn^]^'l

TH2: Ket h(?p vol (1) ta c6 h# moi: • ''^
xy + y +1
[x +y^+x = 3
Giai bang each:

4(x + y + 2)r(x + 2)^-(x + 2)y + y2 =3y(y2+2y + x y - 4 + 4) = 3y2(x + y + 2)
(x + y + 2)(^4(x + 2^ -4(x + 2)y + y2 = 0.

P T ( l ) - P T ( 2 ) « 3 y 2 + x y + x - y - 4 = 0<»(y + l)(x + 3 y - 4 ) = 0.

Voi y = - X - 2 thay vao (1) ta duVoi y = 2x + 2 thay vao (1) ta duQ-c: 2x^ -7x + 4 = 0<::>

X =

X =

V l 7 - 7 l + ^/l7

• Vay hf CO nghi^m (x; y) =

Vay nghi^m ciia h§

x/l7-7
4
N/T7+7'

(x;y) =

—

,yT7+7 1-N/17'

= 0«>xy = - - 2ci>y =

^4x^ + (4x - 9)(x - y) + ^

a)

1

vx

-1

1-1=
^x 2

M

2

b) <

c) •

2^^

*vX

(x + 1) y + ^xy + x ( l - x )

^x-y

d).

2(x2+y2)_3V2x-l=ll

.

Giai:

3

•6\

) Dieu kien: x, y > 0. Ta viet lai phuong trinh (1) ciia h? thanh:
V 4 x 2 + ( 4 x - 9 ) ( x - y ) - 2 y + 7 ^ - y = 0 (»)

= - 3 (v6 ly)

De thay x = y = 0 khong phai la nghi^m ciia h#. Ta xet x^ + y^ ?t 0.

Vay nghi|m ciia hf (x; y) =
d) Dieu ki?n: x + y 5^ 1 . Phuong trinh (2) tuong duong:
x^ -4y^){x + y - l ) + 2xy = -(x + y - l ) .
Phan tich nhan tu ta duc?c: (x + 2y - l)(x^ - 2y^ - xy + y +1 = 0 .
T H l : x + 2y - 1 = 0 thay vao (1) de dang tim dug-c:
V f - l - 2 > / l 4 3 + N/I41 f 2 V i 4 - l S-N/TI'

p. • '

=4

7x^+16(y-x) + y = 2 7 ^

Nhan lien hc?p (*) ta c6:

/

\

X-'3VX + 3 = 3 7 y - 5 - y

J

T H l : t = - = > x = 2=>y = - - .
TH2: 6t* 2-12t^ +2t^ +4t + 43 = 0 o 6

•

^ x y - ( x - y ) ( ^ - 2 ) + x / ^ = y + Vy

o (2t-l)(6t''-12t^+2t2+4t + 3 = 0.

1

'_10_17'|
,(i;i),(l;-l),{-2;-i)
, ll'lO

== 3y

47(x + 2)(y + 2x)=3(x + 3)

Thay vao (1) ta dugc:
^3

2^Ju-l 3-y[U

-l-27l4 3 + N/14'

Vi d\ 6) Giai h? phuong trinh voi nghifm la so thyc:

c) Dieu kif n: x ^ 0.
Phuong trinh (2) tuong duong: y + 2 - -

/

^ ( 4 x - 9 ) ( x - y ) - 4 y ^ ^ yOj-y) . Q

V4x^+(4x-9)(x-y)+2y

o(x-y)

8x + 4 y - 9
,/4x2+(4x-9)(x-y)+2y

Vxy+y

^/xy + y

^. ^
.r '

= 0 . De y rSng: Ttr phuong

9(x + 3)^
trinh thu hai ciia h§ ta c6: 8x + 4y =
'— suv ra:
4(x + 2)
^

gr- '••

8 X + 4 y - 9 = 4(x + 2) - 9 -4(x
^J^^4r
^ 0 « 8 x ^+ 4 y - 9 > 0. Nen ta CO ^x = y
+ 2)

Thay vao phuong trinh thu hai ciia hf ta thu dugc:

-

'


x=l
473x(x + 2) = 3(x + 3) <=>

/ ,

27

16-x

7= + 1 =

> 0 . Do do X = y thay vao phuone trinh (1)

thu dup-c: 2x = 3(Vx + 3 + V x - S J
Tom lai

''

'

'

c6 nghi?m duy nhat: ( x ; y ) = ( l ; l )

b) Dieu kifn: x , y > 0. Ta viet lai phuong trinh (1) cua h ? thanh:
^ x y - ( x - y ) ( ^ - 2 ) - y + >^-7y
man h^. T a xet x^ + y^

=0 (*). De thay x = y = 0 khong thoa

<^x''-9x3+9x2+324 = 0 c : > ( x - 6 f (x2+3x + 9) = 0 o x = 6
V|y h0 CO nghi?m x = y = 6.
x^y

0

d) Dieu ki$n:

NhMenh^pntaco: - J - V ) ^ - ! ^ - ^ ) ' . ^ ^ =0
^xy + ( x - y ) ( ^ - 2 ) + y
Vx+^y

1 7
x>-;x^ - x - y > 0

Phuong trinh dau ciia h? duq>c viet l^ii n h u sau:

=0

.(x-y)

x-y-1
Tir phuong trinh thu hai ciia h^ ta c6:
2

4
2
/—
4
-5
( x - l ) (x + 2)
+ x2-x-2--^^
^>0
x + 1- + X - X . y^ + J^^^- 2 = —
x+1
x+1
suy ra X = y thay vao phuong trinh thu hai cua h^ ta c6:

+

x^ - y^ - x - y
,
^
=0

'•J I ' , '

i('^-y)'+^+i
yjx^ - x-y

«(x-y-l)
Si^-yf

V x 2 x- x+ -yy +

+^x-y +l

+y
=0

'x = l
(x + l ) ( 3 x - x ^ ) = 4 o

Mat khac tu phuong trinh (1) ciia h f ta c6:

l±Vi7

X =•

Ket h(?p dieu ki$n ta c6: ( x ; y ) = ( l ; l ) /

' l + ^/l7
4

1 + N/I7
'

Neu

y<0=>3/;r7
^
> 0.
^N-y

v6 ly do x ^ i . N h u v|iy h ? c6 nghi?m

4

c) Dieu ki^n: x > 0,y > 5 . T a viet Igii phuong trinh (1) cua h^ thanh:
^ x y - ( x - y ) ( 7 x y - 2 ) - y + > / x - ^ = 0 (*). De thay x = y = 0 khong thoa
man h^. Ta xet x^ + y^ ;t 0 . Nhan lien h^p (*) ta c6:

X + •

khi y > 0 . Do do
^ ( x - y f + ^ x - y +1

->0

^]x^-x-y+y

Vay X - y -1 = 0 thay vao phuong trinh (2) ta c6: 4x2 - 4x - 9 - 3 7 2 x ^ = 0
o ( 2 x - l ) 2 - 3 V 2 x - l -10 = 0

/

X

<=>(x-yj

,

16-x
=

[Vx^ + i 6 ( y - x ) + 7 ^

y
+-T=^

=0

Vxy + y

T u phuong trinh (1) ta c6: y - 5 - ^ y - 5 + x + 3 - 3Vx + 3 + 2 = 0 . T a coi day
la phuong trinh b$c 2 an yJy-5

. Dieu ki?n de phuong trinh c6 nghi^m la:

A = 9 - 4 (x + 3)-3Vx + 3 + 2 l ^ 0 o V x + 3 ^ ^ " ^ ^ ^ < 1 6 . Tir do suy ra

Dat V 2 x - 1 = t > 0 ta CO
t ^ - 3 t - 1 0 = 0 o ( t - 2 ) ( t 3 + 2 t 2 + 4 t + 5) = 0<:>t = 2 o x = |
Vay h? CO mpt nghi?m la ( x ; y ) =

'5

3^

2'2

i I'll'

V i dv 7) Giai

+2y^ +2x + 8y + 6 = 0

a)

<=> -x^ - 3 x y - 8 x + 4y^ +13y + 9 = 0<:> x^ + (3y + 8)x - (4y2 + 13y + 9J = 0

phuong trinh v6i nghi^m la so thyc:

+ xy + y + 4x + l = 0

Ta C O A = (3y + 8)^ + 4(4y^ + 13y + 9) = 25y^ + lOOy +100 = (5y + lO)^

2x^ + 2xy + y - 5 = 0

b)

y2 + xy + 5x - 7 = 0

Giai:
*

Tu do tinh du^c:

x = u + a thay vao phuong trinh (1) cua h? ta c6:
y=v+b

Cachl:Dat

(u + a)2+2(v + b)2+2(u + a) + 8(v + b) + 6 = 0
o

x=
x=

3y + 8-(5y + 10)

^

2
3y + 8 + (5y + 10)

Tu do ta

a=-l

b + 2=:0

b = -2

CO

cac h dat an phu nhu sau: Dgt

u2+2v2=3
u^ +uv = 2

2x*^ + 2xy + y - 5 - ^y^ + xy + 5x -

X

= u-1

y = v-2

2x^ + (y - 5)x - y^ + y +12 = 0

Nhan xet: Khi gap cac he phuong trinh dang:

thay vao h$ ta c6:

a j X ^ + ajxy + a3y^ + a4X + agy +

=0

b j X ^ + b2xy + b3y^ + b 4 X + b j y + b^ = 0
+ Ta dat x = u + a,y = v + b sau do tim dieu ki?n de phuang trinh khong c6 so'
u =

V

j = S^u^ + uvj <=> u^ + 3uv - 4v^ = 0 <=> u = -4v

Cach 2:Ta cong phuang trinh (1) vai k Ian phuang trinh (2).
+2y^+2x + 8y + 6 + k x^+xy + y + 4x + l

hang bac 1 hoac khong c6 so' hang tu do.
+ Hoac ta cpng phuang trinh (1) voi k Ian phuong trinh (2) sau do chpn k sao
cho C O the bieu dien duQfC x theo y . De c6 dugic quan h^ nay ta can dya

Cong vifc con lai la kha don gian.

=0

<=>(l + k)x^ +(2 + 4k + ky)x + 2y^ +8y + ky + k + 6 = 0
Ta

7j = 0 o

.s ^

x=-y+2

day la h$ dSng cap.

T u h? ta suy ra 2(u^ + 2v^

n,u

b) Lay phuang trinh (1) tru phuang trinh (2) ta thu dugc:

Ta mong muon khong c6 so h^ng b^c nhat trong phuang trinh nen dieu
a+l= 0

= 4y + 9

Phan vi?c con lai la kha don gian.

v * >;

+2v^+2(a + l)u + 4v(b + 2) + a^+2b2+2a + 8b + 6 = 0.

ki?n la:

y. -1

= -V

CO

A = (2 + 4k + ky)2-4(k + l)(2y2+8y + ky + k + 6)
= (k^ - 8k - 8)y2 + (4k2 - 32k - 32)y + Uk^ - 12k - 20 .
Ta mong muon A c6 d^ng (Ay -hB)^ o A = 0 c6 nghi^m kep:

vao tinh chat. Phuang trinh ax^ + bx + c bieu dien du^c thanh dang:
(Ax + B)^ci>A = 0
Doi voi cac

Ta C O the van dung cac huang giai
+ Bie'n doi h^ de tao thanh cac hSng dang thiic
+ Nhan cac phuang trinh voi mpt bieu thiic d^i so' sau do cpng cac phuang
Vi dy 8) Giai h^ phuang trinh vai nghi^m la so' thyc:
a)

i

o (4k2 - 32k - 32)^ - 4(k2 - 8k - 8)(l2k2 - 12k - 2o) = 0 o k = - |
Tu do ta C O each giai nhu sau:
Lay 2 Ian phuang trinh (1) tru 3 Ian phuang trinh (2) cua h? ta c6:

d^i so bac 3:

b) .

x^ + 3xy^ = -49
x^ -8xy + y^ =8y-17x
x3-y3=35
2x^ +3y^ = 4 x - 9 y

c)

d)

x^ + 3x^y = 6xy - 3x -49
x^ - 6 x y + y^ =10y- 25x-9
x^ + y^ = (x - y)(xy - 1)
x^ - x^ + y +1 = xy(x - y - 1 )


Trir hai phuong trinh cho nhau ta c6: y = - 1 thay vao thi h? v6 nghif m

Giii:
a) Phan tich: Ta viet lai

Nh?n thay x = - 1 thi

+3xy^ +49 = 0

nhu sau:

.

KL: Nghi^m cua h# la: (x;y) =

y2+8(x + l ) y + x2+17x = 0

tro thanh:

-3y2+48 = 0
y2-16 = 0

<=>y = ±4

iity

1 3 + 3N/5^
2'
4

1 3-32/5'
2'
4

PHLTONG P H A P D A T A N P H U

Tir do ta CO loi giai nhu sau:

D|t an phu la vi?c chpn cac bieu thuc f(x,y);g(x,y) trong h? phuong trinh

Lay phuong trinh (1) cpng voi 3 Ian phuong trinh (2) ciia h? ta c6:

de d$t thanh cac an phy moi lam don g i ^ cau true cua phuong trinh, h^
phuong trinh. Qua do tao thanh cac h? phuong trinh m o i don gidn hon, hay
quy ve cac d^ng h ^ quen thupc nhu doi xung, dla\ cap...

x^ + 3xy^ + 49 + 3(x2 - 8xy + y^ - 8y + 17x) = 0
o ( x + l)r(x + l ) 2 + 3 ( y - 4 ) 2 ] = 0
T u do ta de dang tim dugc cac nghi^m cua h?: {x;y) = ( - l ; 4 ) , ( - l ; - 4 )
b) Lam tuong ty nhu cau a
Lay phuong trinh (1) cpng voi 3 Ian phuong trinh (2) thi thu du(?c:
(x +1) (x +1)^ + 3(y - 5)^ = 0 . T u do de dang tim dugc cac nghi^m cua h^.
c) Lay phuong trinh (1) t r u 3 Ian phuong trinh (2) ta thu dugc:

De t^o ra an phy ngudi giai can xu ly linh ho^it cac phuong trinh trong h?
thong qua cac ky thuat: Nhom nhan t u chung, chia cac phuong trinh theo
nhung so'hang c6 sin, nhom dya vao cac hang dSng thuc, doi bien theo dac
thii phuong trinh...
Ta quan sat cac VI dy sau:
V i dy 1: GiAi cac h^ phuong trinh sau
a) <

( x - 2 ) 3 = ( y + 3)^<::>x = y + 5
Thay vao phuong trinh (2) ta c6:
2(y + 5)2+3y2 ^4(y + 5 ) - 9 y o 5 y 2 + 2 5 y + 30 = 0 o

y = -3
y = -2

Vay h? phuong trinh c6 cac nghi^m la: (x;y) = (2;-3),(3;-2)

d) Lay 2 Ian phuong trinh (2) tru d i phuong trinh (1) ta thu du(?c:
( x - l ) r y 2 - ( x + 3)y + x 2 - x - 2
Truong hg^p 1: x
Truong hgp 2:

=0

2 x 2 - 2 x y - y 2 =2

b) <

2x^-3x2-3xy2-y3+l = 0

x*-4x^ + y ^ - 6 y + 9 = 0
x^y + x^ + 2 y - 2 2 = 0

Giii:
a) Ta viet lai h? phuong trinh thanh:
3 x 2 - ( x + y r =2

3x2-(x + y)2=2

[3x3+3x2y-(x + y ) 3 - 3 x 2 = - l

[3x^{x + y)-(\ y f = - 1

.2
D§t a = 3x ,b = x + y ta thu du^c h$ phuong trinh:

1 h# v6 nghi?m

o

a-b^

\"
=2

[ab-b^-a = - l "

y2 - ( x + 3)y + x ^ - x - 2 = 0

T u phuong trinh (1) suy ra a = b^ + 2 vao phuong trinh t h u hai aia h? ta

x^ +y^ = ( x - y ) ( x y - l )

thuduQc: ( b 2 + 2 ) b - b ^ - ( b 2 + 2) = - l o b 2 - 2 b + l = 0<=>b = l = > a = 3

Lay 2 Ian phuong trinh (2) t r u d i phuong trinh (1) ta thu dug-c:
(2x + l ) r y 2 - ( x - l ) y + x 2 - x + 2 = 0
^

NT-

+ Neu

1

X =

— => V =

Khi

3±3>/5

+ Neu y ^ - ( 2x - l ) ^y + x ^ -4x + 2 = 0 tacoh^:

y^ - ( x - l ) y + x^ - x + 2 = 0
y 2 - ( x + 3)y + x 2 - x - 2 = 0

a= 3

x2=l

y=0

b=l '

x+y=1

x=- l

Tom l ^ i h? phuong trinh c6 2 c$p nghi?m: (x; y ) = ( l ; O),(-1; 2) -

, ,


-cry

b) Ta viet l?i

Dat x^+v^
y + ~1 = a; X + y +1 = b . Ta c6:

(x^-2f.(y-3f=4

phuong trinh thanh:

x^y + x^ + 2 y - 2 2 = 0
D | t a = x^ - 2; b = y - 3 . Ta CO h? phuong trinh sau:
a2 + b 2 = 4

(a + 2)(b + 3) + a + 2 + 2(b + 3) = 22
(a + b r + 8 ( a + b)-20 = 0
ab + 4(a + b) = 8

a2+b2=4

j(a + b ) 2 - 2 a b = 4

[ab + 4(a + b) = 8
a + bi=2
ab = 0
a + b = -10.(L)
ab = 48

[ab + 4(a + b) = 8

<=> <

ab = 25
x2+y2=5(y
<=>a = b = 5<=>
a + b = 10
x+y=4

V|iy

a)

phuang trinh sau

x2 + y 2 j ( x + y + l) = 25(y + l)

+ xy + 2y^ + x - 8 y = 9

L _ + ^=:0
(x-y)2 8
b)
2 y - i - . ^ =0
^ x-y 4

a) De y rang khi y = -1 thi hf v6 nghiem

( x 2 + y 2 ] ( x + y + l ) = 25(y + l)

Xet y 7i - 1 . Ta vie't lai h? thanh: i
[ x 2 + y 2 + x ( y + l) + (y + l) =10(y + l)

Chia
x 2 +hai
y 2 phuang trinh cua hf cho y +1 ta thu
i ^ -du^c:
^ ( x + y + l) = 25
(x + y + l) = 25
V+1 ^ ^ '
y+1 •
o x2+y2
+ (x + y + l ) = 10
x 2 + y 2 + x ( y + l) + (y + l)^ =10(y + l)
y +1

x = 3;y = l •
x = - - - =11

CO nghiem (x; y) = (3; l). ' 23 ' 11^
2

2(x + y f - y - x + y-x

(y-x)
^-x + -y - x j
Dat x + y = a; y - X +
a + b=: —
4
2a2-b2=-^

x^ +y^ +6xy

Giai:

+ l)

b) Dieu ki^n: x ?t y.
da cho tuong duong:

a = 2,b = 0
Xet a + b = 2 a = 0,b = 2 .
ab = 0
x = ±^/2
+ Neu: a = 0 , b - 2 y = 5
x = ±2
+ Neu a = 2,b = 0=>.
y =3
Tom lai hf c6 cac cap nghiem: (x;y) = (%/2;5J,^-\/2;5J,(2;3),(-2;3)
Vi d\ 2: Giai cac

iNHHmiVDWHKftangVifr

y-x

y-x +y-x
= b; b > 2 h | thanh:
5
y +x= —
^
4
y - X = -2
y +x= 4

1
y-x=--

5
a =—
4
b= -5

V^y hf CO nghifm (x;y) = ^7

3Vl3

8'8 8 ' 8
Vi dvi 3: Giai cac h^ phuang trinh sau
a)

xVl7-4x2 +y7l9-9y2

25 „

=3

V17-4x2 + ^ 1 9 - 9 y 2 = 1 0 - 2 x - 3 y

13

3
X = —;y = —
8^
8

^7 ^3

3)

x^ + X

y^-4y2+y + l = 0

xy+ x 2 y 2 + l - ( 4 - x ^ ) y 3 =0

Giai:
1
Dieuki^n: - : ^ < x < : ^ ; - ^ < y
2
2
3
3
Dey x V l 7 - 4 x 2 lien quan den 2x va V l 7 - 4 x ^ y ^ l 9 - 9 y 2 lien quan deh
3y va
- 9y2 . Va tong binh phuang cua chiing la nh&ng hang so.


D l t 2x + \ / l 7 - 4 x ^ = a;3x + y-y/l^ - 9y^ = b . H | da cho tuong duong:
a + b = 10
b2 - 1 9

a ^-17
4

6

V i dv 4: G i i i cdc h# p h u o n g trinh s a u

a = 5;b = 5
a = 3;b = 7'

= 3

5x''-(x2-l)^y2-llx2

THl:

TH2:

6

6x2

(X

1'

(lo9i).

5x2

-9y2

5
+ 2

X

b) Ta viet l?i h# nhu sau:

->

(
X

6

2'

Dat X —
X

I
X 2 +.

-

= a,

y

x2.4

=2

x + -^ = 2

y

y')

I

y

J

=4

y

a+b=4
^ ^ = b ta CO h? moi
"^a =b=2
ab = 4
1
x+—
.
y,
x+- = 2
y

^-2

x . i =2
y
<r>x = y = l

Vay h^ CO mpt c|ip nghi^m duy nha't x = y = 1

y -y =o

\

2

/•

X

-

/

X -

y2-l =0

a = l,y„ _=o2

X —

1

=

l±Vl7

y = l

1

X =

[y = 2

y

xy+ 1

1^

- y=0

X =

X

x2+4 = 2

y

duoi d^ng:

- i = i
x ~2
y =l

^

x
,
3
,
+ — + l + x-^ =4
xy + 1

X -

= a . H? thanh: 5 a 2 - a 2 y 2 - i = o

y+1
y

X

/

-

Chia hai vecho a2 va d^t y + — = X, — = Y g i a i ra ta du(?c

xy + x2y2 + i + x^y^ =4y^

trinh thii 2 cho y^ ta dug-c:

X

5

6a2 - a y 2

6

^x^ + x y2 + y + l = 4y 2

x^ +x

Dgt

•

1^ y 2 - l l = 0

Ta thay y = 0 khong thoa man h^.Chia phuong trinh dau cho y^, phuong

Viet l ^ i

6
\

' l 5 + N/I31 f 1 5->/l3'
2'

2

f

y2-y-12 = 0

X

V9yh?c6nghi?m (x;y) =

xy

Chia hai ve phuong trinh cho x^ ta c6:

5±Vl3

-4x2

3y +

5x + y-

a) Nhan thay x = 0 khong la nghi^m ciia h ^ .

x=2

y=—7—
2x +

=4

x+y

Giii

"" = 2

-9y2

3y +

x -y

b)

=-5

1

2x + 7l7- -4x2

5y

6x'* - ( x ^ - x)y2 - ( y +12)x^ = - 6

1±

Vs
1>C

y = 2

V^y h? CO nghi^m (x;y) =
b). Dieu ki^n: \,y ^

Phuong trinh (2) tuong duong:
D § t i ^ = a,ii±y^.b.
X

X

x2.

y+

y

+ 5x - ^

X

=5

2

2

o i i U L + 5.iL-lil = 5
X

X


r
Vay

(x^-y) = x

1 5
1
—+—=4
1 . 5R
a b
c^a=-,b = - «
b + 5a = 5
^

thanh:

CO nghi?m (x;y) =

-1

I

]

2' .

3
o
x =__,y =3
1
x=l,y=-

(x + y 2 ) = 5y

1^

fa

b) Phuang trinh (2) tuong duong:
(2x - y2 ) ( y - 9x2 ) = i8x2y2 ^ g^l^l ^ jg^3 ^ y 3 ^ 2xy

.

9x2y2+18x^+y3
„
^
ISx^ y^
>'= 5 .
^ = 2<::>9xy +
+— + 2 = 4
c,
X
xy
o9x

3^

H

2x^
y

U'2j

2x

+

= 4 < » 9x + ^

J

2x
Dat a = 9x + ^ ;b = y + . H ? thanh:
x>

V i d\ 5: G i a i cac h$ phuong trinh sau
{xy + 3 ) % ( x + y ) = 8
a)

X

y

_

x^+1

y^+1

b)

1

2x

a+2b=4
= 18

-1

ab = 2

GiAi

9x + ^ = 4
<=>a = 2;b = l o

2x

(1) o x^y^ + 6xy + 9 + x^ + 2xy + y^ = 8 o x^y^ + x^ + y^ +1 = - S x y
o ( x 2 + l ) ( y ^ + l ) = -8xy.

(4x-9x2j^ +2x = 4x-9x^

X = —=>

Phuang trinh (1) khi do la: ^ ^ ^ - ^ " ^ =

9

9x + y = 4x

1 •
y=—
^ 3

9'3

V i d y 6: G i a i cac h ? phuong trinh sau

•

a) .

XN/X^ +

6 + yVx^ + 3 = 7 x y

b) •

xVx^ + 3 + y-y/y2 + 6 =x^ +y^ + 2

D a t - ^ = a ; ^ =- b . H ? da cho tuang duang vai:
x^+l
y2+l

2x2y + y ^ = 2 x * + x^
(x + 2 ) 7 y + l = ( x + l ) '

Gi4i
Giai he:.

r

1
a = —

x^+1

b=i

y _ i
2,1
4
y +

2

;

4

1
a =—
4

b=-i
-

1

11

y^ +2x = y

x = 0(L)

Vay h§ CO nghi^m (x; y ) =

Nhan thay x = 0, y = 0 khong la nghi^m cua h?.

,

y = 4x-9x^

a) Trien khai phuang trinh (1)

a +b = —
4

2x
= 4.--y +yJ

2

<=>

H f phuang trinh tuang duang voi :

:
x =- l

/

y = 2±V3

y
y^+1

V^y h ? CO nghifm (x;y) = (-1;2 -

_

y=-i

N

J

x=2+±V3
4

y

x J y ^ + 6 + y + y Vx^ + 3
X

.

Vx^+3-x

V

+y

= 9xy

+ X

x'^ + 3

+ X

=9

/

=2
<

y +6 + y

Vx^+3-x

)

+yf^y2+6-

1
2

T =9
\/y^+6-y

x^+3-x

|x V x 2 + 3 - x j +

y[^7y2+6-y = 2

),(-1;2 + >/3),(2 - VS; -l),(2 + V S ; - l ) ,
" O K I (1), cf-fln?

;

=2


rat H f « on mt aat nQesangratrva^ttj

D|t

X

n,

mi

m,

J i,mri vi

-Nguyen

irung^*^;^

Xet h a m so f ( x ) = x^ + x t a c6 f (x) = Sx^ + 1 > 0 suy

Vx^ + 3 - x l = a;y i/y^ + 6 - y = b .

d i $ u tang.
H ? thanh:

b

a

x^

T u do suy ra f ( V 2 - x ) = f ( 7 2 y - l ) < » ^ 2 y - l = V 2 - x < » x = 3-2y

^=3-^=3

vao

2

= 1

+3-X

1V

3/5-2y+27y + 2 = 5 < » D | [ t

ta c6:

x= l

a = l;b = 2
a+2b=5

= 1

a^+2b2 =9
X

TH2:

4

=

<=>s

y = 2

1

3
Vay n g h i ^ m ciia h? ( x ; y ) =

_2_

'4 tVIs'

(x;y) = (-l;2).

•X

ta c6 h?

y =2

a=

-3-^/65 .
23 + ^/65
;b =
:
4
'
8

a =

>/65-3 ^ 23-^/65
-;b = 8

o

Vay h# CO n g h i | m

15

Vl5j

P H U O N G PHAP HAM

^23N/65-185 233 - 23N/65
16

233 + 23V65
1
32
233-23V65 ,
y=
1
>
^
32
y =
^

23V65+185 233 + 23N/65'

32

16

32

b) D i e u k i f n : y > - 1 .
V i X = 0 k h o n g la n g h i f m cua h ^ nen chia p h u o n g t r i n h (1) cho x^ v i x = 0

S6

D i e m m a u chot k h i g i i i h f b a n g p h u o n g p h a p h a m so l a d u a m p t p h u o n g
t r i n h ciia h? ve d?ng:

suy ra h a m so f(x) d o n d i | u tang.

f [ u ( x ; y ) ] = £ [ v ( x ; y ) ] t r o n g d o h a m so dac t r u n g

+
/

y

N3

ta c6: 2 y
x
V

f(t) d o n d i ? u tang, hoac d o n d i ? u g i a m t u do suy ra u ( x ; y) = v(x; y)
+

a = ^5-2y;b = ^ y + 2

thay

phucmg t r i n h sau:

THl:

Vx^+3-x

don

-

a = ^;b = l
c>

a+b = l

ra h a m so f(x)

X

= 2x + x ^ X e t h a m s o f(t) = t ^ + 2 t ta c6 f'(t) = 3 t 2 + 2 > 0

Theo bai ra ta c6: f y

= f(x)

« i

= x o y = x^

De phat h i ^ n ra f [ u ( x ; y ) ] = f [ v ( x ; y ) ] ngoai vi^c t h a n h thao cac k y nang
Thay vao (2) ta dupe:

bien d o i h a n g d a n g thiic, n h o m n h a n t u c h u n g d o i k h i ta can chia cho m p t
bieu t h u c g ( x ; y ) ho?c the m p t b i e u thuc t u p h u o n g t r i n h t h u nhat vao

(x + 2 ) N £ 2 7 l = ( x + l ) ' ^ ( x + 2 f ( x 2 + l ) = (x + l ) '

p h u o n g t r i n h con lai de t^o ra p h u o n g t r i n h c6 cau true h a m so'.

: = -V3,y

Ta xet cac v i dy sau;

V^y h?

Vi dv 1: G i i i cac h$ p h u c m g t r i n h sau
a)

2x^y + y ^ = 2 x ' ^ + x ^

(3-x)>/2^-2y72y-l=0
i , ;
,
^( + 2 + 2^/y + 2 = 5

b)

(x + 2 ) V ^

= (x + l f

GiAi

CO

nghi^m (x;y) =

2

( 2 - x ) V r ^ + V 2 ^ = (2y-l)72y-l + 7 2 y - l o f ( V 2 x - l ) = f(72y-l).

3

(thoa man).

jjj, j j, v

2x3 _ 4^2 + 3x _ 1 ^ 2x3 ^2 - y ) ^ 3 - 2 y

x5+xy''=yl°+y^
b)

V4x + 5 + 7 y ^ + 8 = 6

yf^

= ^U-Xyl3-2y

+1

Giai

a) D i e u k i ^ n : x < 2 , y > - .
Phucmg t r i r J i (1) t u o n g d u o n g :

=

(±^/3;s).

V i dv 2: G i A i cac h ^ p h u o n g t n n h sau
a)

3

c = N/3,y =

a) D i e u k i ^ n : x > — .
4

F, ;

Ta thay y = 0 k h o n g la n g h i ? m ciia h?. chia hai ve ciia (1) cho y^ ta dupe:


u i i »n» mm

T r u theo ve'hai phuang trinh tren ta duQC
X

y

+ i = y 5 + y . Xet ham so f(t) = t ^ + t ta c6 f'(t) = 5t^+1 >0 suy
y

ham so f(x) dan di^u t^ng. Theo bai ra ta c6: f

X

= f(y) <z> — = y<=>x = y
,

74x + 5 + Vx + 8 = 6 o
2V4X + 5

X =

• > 0. Suy ra f(x) dan di^u tang.

Ta CO f ( l ) = 0 => x = 1 la nghi?m duy nha't. T u do tinh dugrc y = ±1
Vay h f da cho

CO

nghi^m ( x ; y ) =

^3

u + Vu^ + 1 = 3 " o I n f u + V u ^ T l

i-' •

(l;±l).

= uln3;

.4

u

1+
f'(u) =

u + Vu^ +1
Mat khac f(0) - 0

u = 0 la nghi^m duy nha't ciia phuang trinh.
.

b) D^t z = 75 - 2y => z = - y - thay vao phuang trinh (1) ta c6:

/

1-1

4x

X

Xet ham so f ( x ) = x^ + x t a c6 f ( x ) = 3x^+1 >0suy

di^utang. Taco f ( j 3 - 2 y ) = f [ ^ l - - J

.(i&'^i'yV

- l n 3 < OVu => f(u) la ham so nghjch bie'n.

- 3 = /

T u do suy r a u = v = 0<=>x = y = l

*0:

W-2-^4-4- = ( 4 - 2 y ) 7 ^
1-i

+ 3' l n 3 > OVt suy ra ham so f(t) dong bieh tren R. Ta c6
Vt + 1
-

Xet ham so f ( u ) = ln(u + Vu^ +1) - u In 3 ta c6

b) Dieu ki?n: x > -2; y < | .Ta thay khi x = 0 thi h# khong c6 nghi§m.
Chia phuang trinh (1) cho

f'(t) = 1 +

f(u) = f(v) <» u = V .Thay vao phuang trinh dau ciia h? (*) ta c6:

1 => y = ± 1 . Xet ham so f(x) = V4x + 5 + Vx + 8 - 6

2Vx + 8

u + Vu^ + 1 + 3 " = v + V v 2 + l + 3 ' ' . Xet ham so f(t) = t + Vt^ + 1 + 3 ' . Ta c6

J )

Thay vao (2) ta du(?c:

Taco f'(x) =

ra

yJS-ly

ra ham so f(x) dan

+ X

= z 3-

5 - z ,2>

o Sx'' + 2x = z^ + z .

Xet f (t) = t^ +1 ^ f ' ( t ) = 3t2 + 1 > 0 suy ra ham f (t) luon dong bieh.

^1--.
Tir do suy ra f(z) = f ( 2 x ) o z = 2 x o 7 5 - 2 y

Thay vao (2) ta dupe: x + 2 - sJ\5-\ 1.

=2xo y=-

111

phuang trinh (2) ciia h# ta dupe: g(x) = 4x2 +

98

Vay h§ CO n g h i f m (x; y) =
V i dv 3: Giai cac
a)

voi x e

-1

y + ^ y 2 - 2 y + 2=3''"^+l

b)

4x2 + l ) x + ( y - 3 ) 7 5 - 2 y = 0
4x2 + y 2 + 2 V 3 - 4 x = 7

Giiii
a) Dat u = x - l ; v = y - l h? thanh:

U + N/U27I=3^

v + > / ^ = 3""

g'(x) = 8 x - 8 x
/

^

Tacog-

'5-4x2

\2

+ 2V3-4x-7 = 0

.De thay x = 0 ho^c x = - deu khong la nghi?m

phuang trinh sau

x +Vx2-2x + 2=3y"Ul

the vao

x>0

Ta thay ve'trai la ham dan di^u tang nen phuang trinh c6 nghi^m duy nha't
x = 7=>y=:

^' ( '

(5

73-4x

= 4x 4 x 2 - 3

V3-4x

< 0 voi

X

\

= 0 = > x = - ; y = 2 1a nghifm duy nha't ciia h?. I 1.

f •

e

^

3^


Thay vao phuong trirUi (2) ta dupe

Vi dv 4: Giii cac h? phuong trinh sau
4 + 9.3x'^-2y _ 4+9"

a)

4" + 4 = 4x + 4^2y - 2x + 4

b)

x + Vx^-2x + 5 = l + 2^y2+2y
(1)
(2)

X+

(y + l ) ' + y ^ / 7 T l = x + ^
: + V x 2 - 2 x + 5 = l + 272x-4y + 2 •
Giii

4 + 3'

Xet ham so
f(.) = 4

/ 1 ^'

In

=4

\
v7y

4+3

2-z

.Z+2

'1^ t
T

3

In

r3^
17J

< » (X -1) + V(x -1)2 + 4 = 2y + y]i2yf

+ 4

Xet ham so f(t) = t + Vt2+4;f'(t) = 1+

*
it'+4

Thay vao (3) ta du^c:
i l | - o f ( . . 2 ) = f(2z).

(y + ^jy^+lf

,t€R taco

= 4 o y + ^y2+l=2o

5
3
Thu lai thay x = - ;y = - thoa man.
2
4

< OVt G R . Do do f(t) nghjch bien tren R,

Theo phuong trinh ta c6 f (t + 2) = f (2t) o t + 2 = 2t o t = 2, tuc la

V|y h? phuong trinh c6 nghi^m (x;y) =
Vi dy 5: Giai cac

x 2 - 2 y = 2.Suyra ( l ) o 2 y = x 2 - 2

a)

Thay vao phuong trinh (2) ta c6:

trong do s = x - 1 , Loga co so' e ca hai ve ta thu du^c:

b)

s + Vs^+1 = 4 ' < » l n s +•is
Vs^+1 = sln4

's^+l

s + Vs^+l

M^t khac f (0) = 0

-ln4 = -

^5 3^
2'4

2^2x + y + 5 + 373x + 2y + l l = x^ + 6x +13
x ^ - y ^ + 2 ( y - x ) = 61n

(1)
(2)

y + Vy2+9
x+
Giai

1

r - In 3 < 0Vs => f (s) la ham so nghjch bien.
VsVl

s = 0 la nghi?m duy nhat cua phuong trinh.

x<5
y <4

Dieu ki^n:
2x + y + 5 > 0
3x + 2y + 11^0
Bien doi phuong trinh (1) ta c6:
3(5 - x) + 2]

'•'-2

= [3(4 - y) + 2 ] 7 4 ^ ( 3 )

Xet ham so: f(t) = (31^+2)t voi t>0 taco: f'(t) = 9t2+2>0

b) Dieu ki?n: x - 2y +1 > 0
Phuong trinh (1) tuong duong voi 2x - 4y + 2 = 2y^ +1 + 2y^y^ +1

, « y = l=>x = |

y^ = x ^ - 2 x + l

Tuc la X - 1 = 0. Suy ra nghi^m duy nhat cua h^ phuong trinh da cho la
(x;y) =

y ' + l = (2-y)

(17 - 3 X ) N / 5 ^ + (3y -14)^4-y = 0

Xet ham so f(s) - ln(s + Vs^+1) - sln4 ta c6

f'(s)-

y<2

phuong trinh sau

4'<+4 = 4x + 4 7 x 2 - 2 - 2 x + 4 < » 4 ' ' " ^ = x - l + . J ( x - l f + l « 4 ' = s + >/s2+l,

1+

> 0, Vt e R

Dodo f ( x - l ) = f(2y)«>x = 2y + l

a) Dieuki^n: y - x + 2>0.D$t z = x 2 - 2 y
P T ( I ) O 4 + 3^+2 =(4 + 9^)7

7(x -1)^ + 4 = 1 + 2f y + ^y2 +1

(3)

Do do f(t) la ham so dong bien tren R.
Tir phuong trinh (3) ta suy ra:


1 ui nnu un ini ««»wpc yung luo vu giui yi,

QUI

fi, i-i,oui rrr- Nfuyju

— J I - . X T X I

Tning^ferr

IVITV

UVVH

KHafig'VifT

Dieu kif n xac djnh ciia phuang trinh (4) la: x > --^ *

Phuong trinh nay c6: f (-2)f (o) < 0 ;f (o)f (l) < 0 ;f (l)f (2) < 0 .Vay phuang
trinh c6 3 nghiem thupc doan (-2;2) nen ta dat x = 2cost voi xerO;n
Thay vao ta c6:
•'' '
8cos^t-4cos^ t-4cost + l = 0=>sint|8cos''t-4cos^ t-4cost + l j = 0

(4) c^x^ + x + 2(x + 2->y3x + 4) + 3(x + 3-V5x + 9) = 0

«sin4t = sin(-3t)«t = ^ ; t = ^ ; t = - ^

f (Ts^) = f ( > / 4 ^ ) = ^ / 4 ^ « y = X -1
Thay vao (2) ta c6:

+ 6x +13 = 2V3x + 4 + sVSxT?

(4)

^ '

2(x2+x)
Sfx^ + x)
A
.=J= + ^- ^ = = 0
o \ X + X + 2 + N/3X + 4 X + 3 +V5X + 9
2
3
=0
o x +X
; + 2 + N/3X + 4 X + 3 + V5X + 9 J
x^ + x = 0

1 + X + 2 +V3x + 4
(*) x ^ + x =

0

X + 3 +V5x + 9

a)

=0

2— = = +
3
4
- = = > 0 do dieu kif n x > —X + 2 +73x^+4 X + 3 + N/5X + 9
3
Ketluan: (x;y) = (0;-l),(-l;-2)

Taco 1 +

b) Phuong trinh dau tien tuong duong voi:
x^-2x + 6lnfx + \/x^ +9l = y^ -2y + 6lnfy + ^/y^ +9
\
V

Theo bat dSng thiic Cauchy ta c6:
27

1

# 7 9 ^ 9 27

^ 9

7

b)

V

^x v

2cos:^;2cos^
7
7

x3(4y2+l) +2(x2+l)V;: = 6

xV

2+

2 7 4 y 2 + l l = x + V x 2 + l (2)

x(x + y) + 7 ^ = 72y f

+1

V

x^y - 5 x 2 + 7(x + y) - 4 = ^^^y _ ^ ^ ^
Giai
Dieu ki^n: x > 0.
<;
Ta thay x = 0 khong la nghiem ciia h§. Chia hai ve ciia (2) cho x^ ta dupe:
2y + 2 y 7 4 y 2 + l = i + l

Xethams6:f(t) = t 3 - 2 t + 61n t + V t ^ + 9 ; t e R .

2

7

37t .
371
Vay h$ CO2cos—;2cos—
3 nghiem la: 2cos—;2cos—
7

7
7
7
(x;y) =
Vi 6: Giai cac h? phuang trinh sau

x = 0=>y = -1
X = -1 => y = -2

Taco f'(t) = 3 t ^ - 2 + , ^ = 3 t ^ .

7

Er^.

X xVx^

bCethamso f(t) = t + t V t ^ taco f'(t) = l + V t ^ +

3j
29
3

Nen t^ + . ^ -2- > 0
3
Vay ham so f(t) dong bieh tren R nen f(x) = f(y) o x = y.
Thay vao phuong trinh thii hai ta c6: x'^ - x^ - 2x +1 = 0.

1)

>O.Nhuvay

nay don di^u tang. Vay tu do suy ra f (2y) = f - « 2y = - thay vao
ta c6: x^ - + 1 - 2 ( x 2 + l ) V ; ^ = 6 « x 3 + x + 2 ( x 2 + l ) 7 ^ = 6

<ethamso f(x) = x^+x + 2 ( x 2 + l ) V ^ - 6 vdi x > 0 taco
t'(x) = 3x2 +1 + 4x>A( + - ^ = - 1 > 0.
Vx


I di u?u un ini uui nyt

sung

T a xet cac v i
V ^ y h? CO n g h i ^ m ( x ; y ) =

Vi

b) D i e u k i ? n : y S O , x + y > 0 . N h ^ n thay y = 0 t h i h? v 6 n g h i e m . Ta xet k h i
y >0

1 : G i i i cac h f p h u o n g t r i n h sau

a) •

T\x p h u c m g t r i n h (1) ta svr d y n g p h u o n g phap lien h o p :
-(x-y)

R6 rang x + 2 y = x + y + y > 0 ;

-1

V2y + 7xT7

sau:

b) •

xy + x + y = x2 - 2 y 2

(1)

x72y-yVx-l =2x-2y

(2)

' .

''' '\'

+ l ) - 2 y 2 - y = 0.

Ta

2x2 + y2 _ 3xy + 3x - 2y + 1 = 0
4x2 - y2

+ X +

4=

+ y + ^ x + 4y
Giii

< 0 , t u d o suy ra x = y

Xet p h u o n g t r i n h (1) ciia h f ta c6:
xy + x + y = x 2 - 2 y 2 « x 2 - x ( y

coi

day

la

T h a y vao (2) ta d u ^ c : x^ - 5x^ + 14x - 4 = 6\/x^ - x + 1 .

t r i n h bac 2 cua X t h i ta c6: A = ( y +1)2 + 8y2 + 4y = (3y +1)2 . T u do suy ra

Bien d o i p h u o n g t r i n h da cho h i o n g d u o n g :

; ^ _ y + l-(3y + l ) _
2
_ y + l + (3y + l )

x^ + 3 x 2 + 6 x + 4 = 8x2 _ 8 X + 8 + 3 N / 8 X 2 - 8 X + 8

X =

o ( x + l f + 3 ( x + l ) - 8 x 2 - 8 x + 8 + 3\/8x2-8x + 8 .
Xet h a m f ( t ) = t^ + 3t ta c6 f "(t) = St^ + 3 > 0 . T u d o ta c6:
f ( x + l ) = f f ^ 8 x 2 - 8 x + 8 l « x + l = \/8x2-8x + 8 o x = l ; y - l .
v.
/
V | y h? CO n g h i ? m ( x ; y ) = ( l ; l ) .

^

.,

= 2y + l

[x>l
T r u a n g h g p 1: x = - y . T u p h u o n g t r i n h (2) ciia h? ta c6 d i e u ki?n:
len: <
y >0
suy ra p h u o n g t r i n h v6 n g h i f m
T r u a n g h o p 2: x = 2y + 1 thay vao p h u o n g t r i n h t h u hai ta c6:
(2y + l ) V 2 y - y V2y = 2y + 2 o

KHI T R O N G H f C 6 CHCTA PHUONG TRINH BAC 2
T H E O AN X, HOAC y

o

(y + l ) ( 7 2 y - 2) = 0 «

y 7 2 7 + 7 2 y = 2(y +1)

y = 2 => X = 5

Vay h? CO m p t cap n g h i ^ m : ( x ; y ) = (5;2)
) Xet p h u o n g t r i n h (1) ciia h? ta c6:
2x2 + y 2 _ 3 x y + 3 x - 2 y + l = 0 o 2 x 2 + x ( 3 - 3 y ) + y 2 - 2 y + l = 0.

K h i t r o n g h? phucmg t r i n h c6 chua p h u o n g t r i n h bac hai theo an x hoac y
ta CO the n g h i deh cac h u o n g x u l y n h u sau:
*

C o i day la p h u o n g t r i n h bac 2 ciia x ta c6:
A = (3-3y)2-8(y2-2y + l) = y2-2y + l = (y-i)2

N e u A c h i n , ta giai x theo y r o i the vao p h u o n g t r i n h con l a i cua h f de
giai tiep

*

N e u A k h o n g c h i n ta t h u o n g x u l y theo each:

+

C p n g hoac t r u cac p h u o n g t r i n h cua h? de tao dug-c p h u o n g t r i n h b^c h a i c6
A chan hoac tao t h a n h cac hang dang t h i i c

+

phuong

D i i n g d i e u k i ^ n A > 0 de t i m m i e n gia t r j cua bien x , y . Sau d o d i i n g h a m

so de d a n h gia p h u o n g t r i n h con l ^ i tren m i e n gia t r i x, y v u a t i m d u ^ c :

x_3y-3-(y-l)^y-l
Suy ra

. X

X

T r u o n g h(?p 1: y = x + 1 thay vao p h u o n g t r i n h (2) ta t h u du(?c:
3x2 - x + 3 = >/3x + l +V5x + 4
o 3 x 2 - 3 x + (x + l - V 3 x + l ) + (x + 2 - V 5 x + 4) = 0

.,t