^ ^'^•^ 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)
• :
« ( 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