tuyển chọn 410 hệ phương trình và các phương pháp giải hệ phương trình
DESCRIPTION
Tài liệu rất hay và đầy đủ về các phương pháp giải hệ phương trình thông qua giải chi tiết 410 bài hệ.TRANSCRIPT
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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inhTu
n
Nguyn Minh TunSinh vin K62CLC - Khoa Ton Tin HSPHN
TUYN CHN 410 BI HPHNG TRNH I S
BI DNG HC SINH GII V LUYN THI I HC - CAONG
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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n
H Ni, ngy 9 thng 10 nm 2013
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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nMc lc
Li ni u 4
1 Mt s phng php v cc loi h c bn 5
1.1 Cc phng php chnh gii h phng trnh . . . . . . . . . . . . . . . . . . 51.2 Mt s loi h c bn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Tuyn tp nhng bi h c sc 7
2.1 Cu 1 n cu 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Cu 31 n cu 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Cu 61 n cu 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Cu 91 n cu 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Cu 121 n cu 150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6 Cu 151 n cu 180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7 Cu 181 n cu 210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.8 Cu 211 n cu 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.9 Cu 241 n cu 270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.10 Cu 271 n cu 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
2.11 Cu 301 n cu 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
2.12 Cu 331 n cu 360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
2.13 Cu 361 n cu 390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
2.14 Cu 391 n cu 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Ti liu tham kho 228
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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inhTu
nLi ni u
H phng trnh i s ni chung v h phng trnh i s hai n ni ring l mt phnquan trng ca phn i s ging dy THPT . N thng hay xut hin trong cc k thi hcsinh gii v k thi tuyn sinh i hc - Cao ng.
Tt nhin gii tt h phng trnh hai n khng phi n gin . Cn phi vn dng ttcc phng php, hnh thnh cc k nng trong qu trnh lm bi. Trong cc k thi i hc, cu
h thng l cu ly im 8 hoc 9.
y l mt ti liu tuyn tp nhng kh dy nn ti trnh by n di dng mt cun schc mc lc r rng cho bn c d tra cu. Cun sch l tuyn tp khong 400 cu h c sc,t n gin, bnh thng, kh, thm ch n nh v kinh in. c bit, y hon ton lh i s 2 n. Ti mun khai thc tht su mt kha cnh ca i s. Nu coi Bt ng thc3 bin l phn p nht ca Bt ng thc, mang trong mnh s uy nghi ca mt ng hong thH phng trnh i s 2 n li mang trong mnh v p gin d, trong sng ca c gi thnqu lm say m bit bao g si tnh.
Xin cm n cc bn, anh, ch, thy c trn cc din n ton, trn facebook ng gp vcung cp rt nhiu bi h hay. Trong cun sch ngoi vic a ra cc bi h ti cn lng thmmt s phng php rt tt gii. Ngoi ra ti cn gii thiu cho cc bn nhng phng phpc sc ca cc tc gi khc . Mong y s l mt ngun cung cp tt nhng bi h hay chogio vin v hc sinh.
Trong qu trnh bin son cun sch tt nhin khng trnh khi sai st.Th nht, kh nhiubi ton ti khng th nu r ngun gc v tc gi ca n. Th hai : mt s li ny sinh trongqu trnh bin son, c th do li nh my, cch lm cha chun, hoc trnh by cha p dokin thc v LATEX cn hn ch. Tc gi xin bn c lng th. Mong rng cun sch s honchnh v thm phn s. Mi kin ng gp v sa i xin gi v theo a ch sau y :
Nguyn Minh TunSinh Vin Lp K62CLC
Khoa Ton Tin Trng HSP H NiFacebook :https://www.facebook.com/popeye.nguyen.5
S in thoi : 01687773876Nick k2pi, BoxMath : Popeye
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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inhTu
nChng 1Mt s phng php v cc loi h cbn
1.1 Cc phng php chnh gii h phng trnhI. Rt x theo y hoc ngc li t mt phng trnh
II. Phng php th1. Th hng s t mt phng trnh vo phng trnh cn li2. Th mt biu thc t mt phng trnh vo phng trnh cn li3. S dng php th i vi c 2 phng trnh hoc th nhiu ln.
III. Phng php h s bt nh1. Cng tr 2 phng trnh cho nhau2. Nhn hng s vo cc phng trnh ri em cng tr cho nhau.3. Nhn cc biu thc ca bin vo cc phng trnh ri cng tr cho nhau
IV. Phng php t n ph
V. Phng php s dng tnh n iu ca hm s
VI. Phng php lng gic ha
VII. Phng php nhn chia cc phng trnh cho nhau
VIII. Phng php nh gi1. Bin i v tng cc i lng khng m2. nh gi s rng buc tri ngc ca n, ca biu thc, ca mt phng trnh3. nh gi da vo tam thc bc 24. S dng cc bt ng thc thng dng nh gi
IX. Phng php phc ha
X. Kt hp cc phng php trn
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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6 Mt s phng php v cc loi h c bn
1.2 Mt s loi h c bnA. H phng trnh bc nht 2 n
I. Dng ax + by= c (a2 + b2 = 0)
ax + by=c (a2
+ b2
= 0)II. Cch gii1. Th2. Cng i s3. Dng th4. Phng php nh thc cp 2
B. H phng trnh gm mt phng trnh bc nht v mt phng trnh bc hai
I. Dng ax2 + by2 + cxy+ dx + ey+ f= 0
ax + by=c
II. Cch gii:Th t phng trnh bc nht vo phng trnh bc hai
C. H phng trnh i xng loi II. Du hiui vai tr ca xvycho nhau th h cho khng iII. Cch gii:Thng ta s t n ph tng tch x + y = S, xy=P (S2 4P)
D. H phng trnh i xng loi II
I. Du hiui vai tr ca xvycho nhau th phng trnh ny bin thnh phng trnh kiaII. Cch gii:Thng ta s tr hai phng trnh cho nhau
E. H ng cpI. Du hiu
ng cp bc 2
ax2 + bxy+ cy2 =d
ax2 + bxy+ cy2 =d
ng cp bc 3ax3 + bx2y+ cxy2 + dy3 =e
ax3 + bx2y+ cxy2 +dy3 =e
II. Cch gii:Thng ta s t x= ty hoc y = tx
Ngoi ra cn mt loi h na ti tm gi n l bn ng cp, tc l hon ton c th av dng ng cp c .Loi h ny khng kh lm, nhng nhn nhn ra c n cn phikho lo sp xp cc hng t ca phng trnh li. Ti ly mt v d n gin cho bn c
Gii h :
x3 y3 = 8x + 2yx2 3y2 = 6
Vi h ny ta ch vic nhn cho v vi v s to thnh ng cp. V khi ta c quynchn la gia chia c 2 v cho y3 hoc t x= ty
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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nChng 2Tuyn tp nhng bi h c sc
2.1 Cu 1 n cu 30
Cu 1
(x y) (x2 + y2) = 13(x + y) (x2 y2) = 25
Gii
D dng nhn thy y l mt h ng cp bc 3, bnh thng ta c nhn cho ln ri chia 2v cho x3 hoc y3. Nhng hy xem mt cch gii tinh t sau y:Ly(2) (1)ta c : 2xy(x y) = 12 (3)Ly(1) (3)ta c : (x y)3 = 1 x= y + 1V sao c th c hng ny ? Xin tha l da vo hnh thc i xng ca h. Ngon lnh
ri. Thay vo phng trnh u ta c
(y+ 1)2 + y2 = 13
y = 2y = 3
Vy h cho c nghim (x; y) = (3; 2), (2;3)
Cu 2 x3 8x= y3 + 2y
x
2
3 = 3 (y2
+ 1)
Gii
nh sau : Phng trnh 1 gm bc ba v bc nht. Phng trnh 2 gm bc 2 v bc 0(hng s).R rng y l mt h dng na ng cp. Ta s vit li n a v ng cpH cho tng ng :
x3 y3 = 8x + 2yx2
3y2 = 6
Gi ta nhn cho hai v a n v dng ng cp
6 x3 y3 = (8x + 2y) x2 3y2 2x (3y x) (4y+ x) = 0
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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8 Tuyn tp nhng bi h c sc
TH1 : x= 0thay vo (2) v nghimTH2 : x= 3ythay vo (2) ta c:
6y2
= 6 y= 1, x= 3y= 1, x= 3TH3 : x= 4ythay vo (2) ta c:
13y2 = 6
y=
6
13, x= 4
6
13
y=
6
13, x= 4
6
13
Vy h cho c nghim :(x; y) = (3; 1), (3;1),4
6
13 ;6
13
,
46
13 ;6
13
Cu 3
x2 + y2 3x + 4y= 13x2 2y2 9x 8y = 3
Gii
khi nhn 3 vo PT(1) ri tr i PT(2) s ch cn y . Vy
3.P T(1) P T(2) y2 + 4y= 0
y= 0 x=
372
y= 4 x= 3
7
2
Vy h cho c nghim : (x; y) =
3 7
2 ; 0
,
37
2 ;4
Cu 4
x2 + xy+y2 = 19(x y)2x2 xy+y2 = 7 (x y)
Gii
Nhn xt v tri ang c dng bnh phng thiu, vy ta th thm bt a v dng bnhphng xem sao. Nn a v (x y)2 hay (x+ y)2. Hin nhin khi nhn sang v phi ta schn phng n u
H cho tng ng
(x y)2 + 3xy= 19(x y)2(x y)2 + xy= 7 (x y)
t x y = a vxy= b ta c h mi
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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2.1 Cu 1 n cu 30 9
b= 6a2
a2 + b= 7a
a= 0, b= 0a= 1, b= 6
x y = 0xy= 0
x y = 1xy= 6
x= 0, y= 0x= 3, y= 2x= 2, y = 3
Vy h cho c nghim :(x; y) = (0; 0) , (3;2)(2;3)
Cu 5
x3 + x3y3 + y3 = 17x +xy+ y= 5
GiiH i xng loi I ri. No problem!!!
H cho tng ng (x + y)3 3xy(x + y) + (xy)
3 = 17(x + y) +xy = 5
t x + y= avxy= bta c h mi
a3 3ab + b3 = 17a +b= 5
a= 2, b= 3a= 3, b= 2
x + y= 2xy= 3
x + y= 3xy= 2
x= 2, y= 1x= 1, y= 2
Vy h cho c nghim (x; y) = (1; 2), (2; 1)
Cu 6
x(x + 2)(2x + y) = 9x2 + 4x + y = 6
Giiy l loi h t n tng tch rt quen thuc
H cho tng ng
(x2 + 2x) (2x +y) = 9(x2 + 2x) + (2x + y) = 6
t x2 + 2x= av2x + y = b ta c h mi ab= 9a + b= 6
a= b = 3
x2 + 2x= 32x + y = 3
x= 1, y = 1x= 3, y= 9
Vy h cho c nghim (x; y) = (1; 1), (3;9)
Cu 7
x +y xy= 3x + 1 +
y+ 1 = 4
GiiKhng lm n g c c 2 phng trnh, trc gic u tin ca ta l bnh phng ph skh chu ca cn thc
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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10 Tuyn tp nhng bi h c sc
(2) x + y+ 2 + 2
xy+ x + y+ 1 = 16
M t (1) ta c x + y= 3 + xynn
(2) 3 + xy+ 2 + 2xy+ xy+ 4 = 16 xy= 3 xy= 9x +y = 6 x= y = 3Vy h cho c nghim (x; y) = (3; 3)
Cu 8
x + 5 +
y 2 = 7x 2 + y+ 5 = 7
Giii xng loi II. Khng cn g ni. Cho 2 phng trnh bng nhau ri bnh phng tungte ph s kh chu ca cn thciu kin : x, y 2T 2 phng trnh ta c
x + 5 +
y 2 = x 2 +
y 5
x + y+ 3 + 2
(x + 5)(y 2) =x + y+ 3 + 2
(x 2)(y+ 5)
(x + 5)(y
2) = (x
2)(y+ 5)
x= y
Thay li ta c
x + 5 +
x 2 = 7 x= 11
Vy h cho c nghim : (x; y) = (11;11)
Cu 9 x2 + y2 +
2xy= 8
2
x +y = 4
Gii
H cho c v l na i xng na ng cp, bc ca PT(2) ang nh hn PT(1) mtcht. Ch cn php bin i bnh phng (2) s va bin h tr thnh ng cp va ph bbt i cniu kin : x, y 0H cho
2(x2 + y2) + 2xy= 16x +y+ 2xy= 16 2 (x2 + y2) =x + y x= yThay li ta c : 2
x= 4 x= 4
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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2.1 Cu 1 n cu 30 11
Vy h cho c nghim (x; y) = (4; 4)
Cu 10
6x
2
3xy+x= 1 yx2 + y2 = 1
Gii
Mt cch trc gic khi nhn thy h cha tam thc bc 2 l th xem liu c phn tch cthnh nhn t hay khng ? Ta s th bng cch tnh theo mt n c chnh phng haykhng. Ngon lnh l PT(1) xp nh tin.Phng trnh u tng ng (3x 1)(2x y+ 1) = 0Vi x=
1
3 y= 2
2
3
Vi y= 2x + 1 x2 + (2x + 1)2 = 1
x= 0, y = 1
x= 45
, y=3
5
Vy h cho c nghim (x; y) =
1
3;2
2
3
, (0, 1),
4
5;3
5
Cu 11 x 2y xy= 0
x
1 +
4y
1 = 2
Gii
Phng trnh u l dng ng cp riiu kin x 1, y1
4T phng trnh u ta c :
x +
y
x 2y = 0 x= 4yThay vo (2) ta c
x 1 + x 1 = 2 x= 2
Vy h cho c nghim (x
;y
) =
2;
1
2
Cu 12
xy+ x +y = x2 2y2x
2y yx 1 = 2x 2y
Gii
iu kin : x
1, y
0
Phng trnh u tng ng
(x + y) (2y x + 1) = 0
x= yx= 2y+ 1
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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12 Tuyn tp nhng bi h c sc
Vi x= yloi v theo iu kin th x, yphi cng duVi x= 2y+ 1 th phng trnh 2 s tng ng
(2y+ 1)
2y y
2y = 2y+ 2
2y(y+ 1) = 2y+ 2 y= 2 x= 5
Vy h cho c nghim (x; y) = (5; 2)
Cu 13
x + 1 +
y+ 2 = 6x +y = 17
Gii
iu kin x, y 1H cho tng ng
x + 1 + y+ 2 = 6(x + 1) + (y+ 2) = 20t
x + 1 =a 0,y+ 2 =b 0. H cho tng ng
a + b= 6a2 +b2 = 20
a= 4, b= 2a= 2, b= 4
x= 15, y = 2x= 3, y= 14
Vy h cho c nghim (x; y) = (15; 2), (3;14)
Cu 14
y2 = (5x + 4)(4 x)y2 5x2 4xy+ 16x 8y+ 16 = 0
Gii
Phng trnh 2 tng ng
y2 + (5x + 4)(4 x) 4xy 8y= 0 2y2 4xy 8y = 0
y= 0y= 2x + 4
Vi y= 0th suy ra : (5x + 4) (4 x) = 0 x= 4
x= 45
Vi y= 2x + 4th suy ra (2x + 4)2 = (5x + 4)(4 x) x= 0Vy h cho c nghim (x; y) = (4; 0),
4
5; 0
, (0; 4)
Cu 15
x2 2xy+x +y = 0x4 4x2y+ 3x2 + y2 = 0
Gii
H cho tng ng
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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2.1 Cu 1 n cu 30 13
x2 + y=x(2y 1)(x2 + y)
2+ 3x2 (1 2y) = 0 x
2(2y 1)2 + 3x2(2y 1) = 0 x2(2y 1)(2y 4) = 0
x= 0, y = 0
y=1
2(L)
y= 2, x= 1 2Vy h cho c nghim (x; y) = (0; 0), (1; 2), (2; 2)
Cu 16 x + y+ xy(2x +y) = 5xyx + y+ xy(3x
y) = 4xy
Gii
P T(1) P T(2) xy(2y x) =xy
xy= 0x= 2y 1
Vi xy= 0 x + y= 0 x= y = 0Vi x= 2y 1
(2y 1) +y+ (2y 1)y(5y 2) = 5(2y 1)y y= 1, x= 1
y=
9
41
20 , x= 1 +
41
10
y=9 +
41
20 , x=
41 1
10
Vy h cho c nghim(x; y) = (0; 0), (1; 1),
1 +
41
10 ;
9 4120
,
41 1
10 ;
9 +
41
20
Cu 17 x2 xy+y2 = 32x3 9y3 = (x y)(2xy+ 3)
Gii
Nu ch xt tng phng trnh mt s khng lm n c g. Nhng 2 ngi ny b rngbuc vi nhau bi con s 3 b n. Php th chng ? ng vy, thay 3 xung di ta s ra mtphng trnh ng cp v kt qu p hn c mong iTh 3 t trn xung di ta c
2x3
9y3 = (x
y) x2 +xy+ y2 x3 = 8y3 x= 2y
(1) 3y2 = 3 y = 1, x= 2Vy h cho c nghim (x; y) = (2; 1), (2;1)
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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14 Tuyn tp nhng bi h c sc
Cu 18
x + y+
x y = 1 +
x2 y2x +
y = 1
Gii
iu kin :x y 0Phng trnh u tng ng
x +y 1 = x y x + y 1 x + y= 1
x y = 1
x=
1 yx=
1 +y
T
1 y+ y= 1y+ 1 +
y= 1
y= 0, x= 1y= 1, x= 0(L)
y= 0, x= 1
Vy h cho c nghim (x; y) = (1; 0)
Cu 19
2x y = 1 +x(y+ 1)x3 y2 = 7
Gii
iu kin : x(y+ 1) 0T (2) d thy x >0 y 1(1)
x
y+ 1 2x +y+ 1 = 0 x= y + 1 (y+ 1)3 y2 = 7 y = 1, x= 2Vy h cho c nghim (x; y) = (2; 1)
T cu 20 tr i ti xin gii thiu cho cc bn mt phng php rt mnh gii quyt gn p rt nhiu cc h phng trnh hu t. gi h s bt nh(trong y ti s gi n bng tn khc : UCT). S mt khong hn chc v d din t trn vn phng php ny
Trc ht im qua mt mo phn tch nhn t ca a thc hai bin rt nhanh bng mytnh Casio. Bi vit ca tc gi nthoangcute.
V d 1 : A= x2 + xy 2y2 + 3x + 36y 130Thc ra y l tam thc bc 2 th c th tnh phn tch cng c. Nhng th phn tchbng Casio xem .Nhn thy bc ca x v y u bng 2 nn ta chn ci no cng cCho y = 1000ta c A= x2 + 1003x 1964130 = (x + 1990) (x 987)Cho 1990 = 2y 10 v 987 = y 13A= (x + 2y
10)(x
y+ 13)
V d 2 : B= 6x2y 13xy2 + 2y3 18x2 + 10xy 3y2 + 87x 14y+ 15
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5/25/2018 Tuy n ch n 410 h ph ng tr nh v c c ph ng ph p gi i h ph ng tr nh
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2.1 Cu 1 n cu 30 15
Nhn thy bc ca x nh hn, cho ngay y = 1000B= 5982x2 12989913x + 1996986015 = 2991 (2x 333)(x 2005)Cho 2991 = 3y 9 ,333 =
y 13
, 2005 = 2y + 5
B= (3y 9)2x y
1
3
(x 2y 5) = (y 3)(6x y+ 1) (x 2y 5)
V d 3 : C=x3 3xy2 2y3 7x2 + 10xy+ 17y2 + 8x 40y+ 16Bc ca xvynh nhauCho y = 1000ta c C=x3 7x2 2989992x 1983039984Phn tch C= (x 1999)(x + 996)2Cho 1999 = 2y 1v996 =y 4C= (x 2y+ 1) (x + y 4)2
V d 4 : D= 2x2y2 + x3 + 2y3 + 4x2 +xy+ 6y2 + 3x + 4y+ 12Bc ca xvynh nhauCho y = 1000ta c D= (x + 2000004) (x2 + 1003)Cho 2000004 = 2y2 + 4v1003 =y + 3D= (x + 2y2 + 4) (x2 +y+ 3)
V d 5 : E=x3y+ 2x2y2 + 6x3 + 11x2y xy2 6x2 7xy y2 6x 5y+ 6Bc ca ynh hn
Cho x = 1000 ta c E = 1998999y2
+ 1010992995y + 5993994006 =2997 (667y+ 333333) (y+ 6)
o ha E=999 (2001y+ 999999) (y+ 6)Cho 999 =x 1, 2001 = 2y+ 1, 999999 =x2 1E= (x 1) (y+ 6) (x2 + 2xy+y 1)
V d 6 : F = 6x4y+ 12x3y2 + 5x3y 5x2y2 + 6xy3 +x3 + 7x2y+ 4xy2 3y3 2x2 8xy+3y2 2x + 3y 3Bc ca y nh hnCho x= 1000ta c F= 5997y3 + 11995004003y2 + 6005006992003y+ 997997997Phn tch F= (1999y+ 1001001) (3y2 + 5999000y+ 997)Cho 1999 = 2x 1, 1001001 =x2 + x + 1, 5999000 = 6x2 x, 997 =x 3F = (x2 + 2xy+ x y+ 1) (6x2y xy+ 3y2 +x 3)
Lm quen c ri ch ? Bt u no
Cu 20
x2 +y2 =1
5
4x2 + 3x5725
= y(3x + 1)
Gii
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16 Tuyn tp nhng bi h c sc
Li gii gn p nht ca bi trn l
25.P T(1) + 50.P T(2) (15x + 5y 7)(15x + 5y+ 17) = 0
n y d dng tm c nghim ca h : (x; y) = 25
;1
5 ,11
25
; 2
25
Cu 21
14x2 21y2 6x + 45y 14 = 035x2 + 28y2 + 41x 122y+ 56 = 0
Gii
Li gii gn p nht ca bi ny l
49.P T(1) 15.P T(2) (161x 483y+ 218)(x + 3y 7) = 0V n y cng d dng tm ra nghim (x; y) = (2;3), (1; 2)
Qua 2 v d trn ta t ra cu hi : V sao li th ? Ci nhm thnh nhn t th ti khngni bi t hn cc bn c n trn ri. V sao y l ti sao li ngh ra nhng hng skia nhn vo cc phng trnh, mt s tnh c may mn hay l c mt phng php. Xin tha chnh l mt v d ca UCT. UCT l mt cng c rt mnh c th qut sch gn nh tonb nhng bi h dng l hai tam thc. Cch tm nhng hng s nh th no. Ti xin trnhby ngay sau y. Bi vit ca tc gi nthoangcute.
Tng Qut:
a1x2 +b1y
2 + c1xy+ d1x + e1y+ f1 = 0a2x
2 +b2y2 + c2xy+ d2x + e2y+ f2 = 0
Gii
Hin nhin nhn xt y l h gm hai tam thc bc hai. M nhc n tam thc th khngth khng nhc ti mt i tng l . Mt tam thc phn tch c nhn t hay khngphi xem x hoc y ca n c chnh phng hay khng. Nu h loi ny m t ngay mtphng trnh ra k diu th chng ni lm g, th nhng c hai phng trnh u ra rtk cc th ta s lm nh no. Khi UCT s ln ting. Ta s chn hng s thch hp nhn vomt (hoc c hai phng trnh) p sao cho chnh phng.
Nh vy phi tm hng s k sao cho P T(1) +k.P T(2)c th phn tch thnh nhn tt a= a1+ ka2, b= b1+ kb2, c= c1+kc2, d= d1+kd2, e= e1+ke2, f=f1+ kf2S kl nghim ca phng trnh sau vi a = 0
cde+ 4abf=ae2 +bd2 +f c2
D vng c hn mt cng thc gii h phng trnh loi ny. Tc gi ca n kh xutsc !!!. Th kim chng li v d 21 nha= 14 + 35k, b=
21 + 28k, c= 0, d=
6 + 41k, e= 45
122k, f=
14 + 56k
S ks l nghim ca phng trnh
4(14+35k)(21+28k)(14+56k) = (14+35k)(45122k)2+(21+28k)(6+41k)2 k= 1549
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2.1 Cu 1 n cu 30 17
Nh vy l P T(1) 1549
.P T(2)hay 49.P T(1) 15.P T(2)Mt cht lu l khng phi h no cng y cc hng s. Nu khuyt thiu phn no thcho hng s l 0. Ok!!Xong dng ny ri. Hy lm bi tp vn dng. y l nhng bi h ti tng hp t nhiu
ngun.
1.
x2 + 8y2 6xy+ x 3y 624 = 021x2 24y2 30xy 83x + 49y+ 585 = 0
2.
x2 + y2 3x + 4y= 13x2 2y2 9x 8y= 3
3.
y2 = (4x + 4)(4 x)y2 5x2 4xy+ 16x 8y+ 16 = 0
4. xy 3x 2y= 16x2
+ y2
2x 4y= 335.
x2 + xy+ y2 = 3x2 + 2xy 7x 5y+ 9 = 0
6.
(2x + 1)2 + y2 + y= 2x + 3xy+x= 1
7.
x2 + 2y2 = 2y 2xy+ 13x2 + 2xy y2 = 2x y+ 5
8.
(x 1)2 + 6(x 1)y+ 4y2 = 20x2 + (2y+ 1)2 = 2
9. 2x2 + 4xy+ 2y2 + 3x + 3y 2 = 0
x2 + y2 + 4xy+ 2y= 0
10.
2x2 + 3xy= 3y 133y2 + 2xy= 2x + 11
11.
4x2 + 3y(x 1) = 73y2 + 4x(y 1) = 3
12.
x2 + 2 =x(y 1)y2 7 =y(x 1)
13.
x2 + 2xy+ 2y2 + 3x= 0xy+ y2 + 3y+ 1 = 0
Cu 22
x3 y3 = 352x2 + 3y2 = 4x 9y
Gii
Li gii ngn gn cho bi ton trn l
P T(1) 3.P T(2) (x 2)3 = (y+ 3)3 x= y + 5
Thay vo (2) ta d dng tm ra nghim(x; y) = (2;3), (3;2)Cu hi t ra y l s dng UCT nh th no ? Tt nhin y khng phi dng trn nari. Trc ht nh gi ci h ny
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18 Tuyn tp nhng bi h c sc
- Bc ca xvyl nh nhau- Cc bin x,yc lp vi nhau- Phng trnh mt c bc cao hn PT(2)Nhng nhn xt trn a ta n tng nhn hng s vo PT(2) P T(1) +a.P T(2)ac v dng hng ng thc A3 =B3P T(1) +a.P T(2) x3 + 2ax2 4ax y3 + 3ay2 + 9ay 35 = 0Cn tm asao cho v tri c dng (x + )3 (y+ )3 = 0
Cn bng ta c :
3 3 = 353= 2a32 = 4a
a= 3= 2= 3
VyP T(1) 3.P T(2) (x 2)3 = (y+ 3)3OK ?? Th mt v d tng t nh
Gii h: x3 + y3 = 91
4x2
+ 3y2
= 16x
+ 9y
Gi : P T(1) 3.P T(2) (x 4)3 = (y+ 3)3
Cu 23
x3 + y2 = (x y)(xy 1)x3 x2 + y+ 1 =xy(x y+ 1)
Gii
Hy cng ti phn tch bi ton ny. Tip tc s dng UCTnh gi h :-Bc ca x cao hn bc ca y-Cc bin x,y khng c lp vi nhau-Hai phng trnh c bc cao nht ca x v y nh nhauV bc x ang cao hn bc y v bc ca y ti 2 phng trnh nh nhau nn ta hy nhn tungri vit li 2 phng trnh theo n y. C th nh sau :
y2 (x + 1) y (x2 + 1) +x3 + x= 0y2x y (x2 +x 1) +x3 x2 + 1 = 0
By gi ta mong c rng khi thay x bng 1 s no vo h ny th s thu c 2 phngtrnh tng ng. Tc l khi cc h s ca 2 phng trnh s t l vi nhau . Vy :
x + 1
x =
x2 + 1
x2 + x 1= x3 + x
x3 x2 + 1 x= 1
Rt may mn ta tm c x = 1. Thay x = 1 li h ta c 2 (y2 y+ 1) = 0y2 y+ 1 = 0 2.P T(2) P T(1)s c nhn tx 1
C th l (x 1) (y2 (x + 3) y+x2 x 2) = 0TH1 :x= 1thay vo th v nghim
TH2: Kt hp thm vi PT(1) ta c h mi : y2 (x + 3) y+x2 x 2 = 0 (3)x3 +y2 x2y+ x + xy2 y = 0
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2.1 Cu 1 n cu 30 19
Nhn xt h ny c c im ging vi h ban u l bc y nh nhau. Vy ta li vit li htheo n y v hi vng n s li ng vi x no . Tht vy, l x =1
2. Tip tc thay n
vo h v ta s rt ra :
2P T(2) P T(1) (2x + 1) y2 (x 1) y+ x2 x + 2TH1 : x= 1
2 y=5 3
5
4TH2 : Kt hp vi (3) ta c
y2 (x 1) y+x2 x + 2 = 0y2 (x + 3) +x2 x 2 = 0
Vi h ny ta ch vic tr cho nhau s ra y= 1 x2 + 2 = 0(V nghim)Vy h cho c nghim :(x; y) =
1
2
;5 + 3
5
4 ,
1
2
;5 35
4
Cu 24
2 (x +y)(25 xy) = 4x2 + 17y2 + 105x2 + y2 + 2x 2y= 7
Gii
Hnh thc bi h c v kh ging vi cu 23
Mt cht nh gi v h ny- Cc bin x v y khng c lp vi nhau- Bc cao nht ca x 2 phng trnh nh nhau , y cng vyVi cc c im ny ta th vit h thnh 2 phng trnh theo n x v y v xem liu h cng vi x hoc y no khng. Cch lm vn nh cu 23. Vit theo x ta s khng tm c y,nhng vit theo y ta s tm c x = 2 khin h lun ng. Thay x = 2 vo h ta c
21y2 42y+ 21 = 0y2 2y+ 1 = 0 P T(1) 21P T(2) (x 2)
2y2 + 2xy+ 4y 17x 126 = 0
TH1 : x= 2 y= 1
TH2 :
2y
2
+ 2xy+ 4y 17x 126 = 0x2 + y2 + 2x 2y 7 = 0H ny c cch gii ri nh ??3.P T(2) P T(1) (x y+ 5)2 + 2x2 + x + 80 = 0(V nghim)Vy h cho c nghim : (x; y) = (2; 1)
Tip theo chng ta s n vi cu VMO 2004.
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20 Tuyn tp nhng bi h c sc
Cu 25
x3 + 3xy2 = 49x2 8xy+y2 = 8y 17x
Gii
Li gii ngn gn nht ca bi trn l :
P T(1) + 3.P T(2) (x + 1) (x + 1)2 + 3(y 4)2 = 0n y d dng tm ra nghim (x; y) = (1;4), (1;4)
Cu hi c t ra l bi ny tm hng s nh th no ? C rt nhiu cch gii thch nhngti xin trnh by cch gii thch ca ti :tuzki:Lm tng t theo nh hai cu 23 v 24 xem no. Vit li h cho thnh
3xy2 + x3 + 49 = 0y2 + 8(x + 1)y+ x2 17x= 0
Mt cch trc gic ta th vi x= 1. V sao ? V vi x= 1phng trnh 2 s khng cnphn y v c v 2 phng trnh s tng ng. Khi thay x= 1h cho tr thnh
3y2 + 48 = 0y2 16 = 0
Hai phng trnh ny tng ng. Tri thng ri !! Vy x =1chnh l 1 nghim cah v t h th hai ta suy ra ngay phi lm l P T(1) + 3.P T(2). Vic cn li ch l phntch nt thnh nhn t.
Tip theo y chng ta s n vi mt chm h d bn ca tng trn. Ti khng trnhby chi tit m ch gi v kt qu
Cu 26
y3 + 3xy2 = 28x2 6xy+y2 = 6x 10y
Gi : P T(1) + 3.P T(2) (y+ 1) (3(x 3)2 + (y+ 1)2) = 0Nghim ca h : (x; y) = (3;1), (3;1)
Cu 27
6x2y+ 2y3 + 35 = 05x2 + 5y2 + 2xy+ 5x + 13y= 0
Gi : P T(1) + 3.P T(2) (2y+ 5)
3
x +
1
2
2+
y+
5
2
2= 0
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2.1 Cu 1 n cu 30 21
Cu 28
x3 + 5xy2 = 352x2 5xy 5y2 + x + 10y 35 = 0
Gi : P T(1) + 2.P T(2) (x 2) (5(y 1)2
+ (x + 3)2
) = 0
Cu 29
x3 + 3xy2 = 6xy 3x 49x2 8xy+y2 = 10y 25x 9
Gi : P T(1) + 3.P T(2) (x + 1) ((x + 1)2 + 3(y 5)2) = 0
im qua cc cu t cu 23 n cu 29 ta thy dng nh nhng cu h ny kh c bit.Phi c bit th nhng h s kia mi t l v ta tm c x = hay y = l nghim cah. Th vi nhng bi h khng c c may mn nh kia th ta s lm nh no. Ti xin giithiu mt phng php UCT rt mnh. C th p dng rt tt gii nhiu bi h hu t (kc nhng v d trn). l phng php Tm quan h tuyn tnh gia x v y. V ta skhng ch nhn hng s vo mt phng trnh m thm ch nhn c mt hm f(x)hay g(y)vo n. Ti s a ra vi v d c th sau y :
Cu 30
3x2 + xy 9x y2 9y= 02x3 20x x2y 20y= 0
Gii
Bi ny nu th nh cu 23, 24, 25 u khng tm ra ni x hay y bng bao nhiu l nghim cah. Vy phi dng php dng quan h tuyn tnh gia x v y. Quan h ny c th xy dng
bng hai cch thng dng sau :- Tm ti thiu hai cp nghim ca h- S dng nh l v nghim ca phng trnh hu t
Trc ht ti xin pht biu li nh l v nghim ca phng trnh hu t :Xt a thc : P(x) =anx
n + an1xn1 + ....+a1x + a0a thc c nghim hu t
p
q p l c ca a0 cn q l c ca an
OK ri ch ? By gi ta hy th xy dng quan h theo cch u tin, l tm ti thiu haicp nghim ca h ( Casio ln ting :v )D thy h trn c cp nghim l (0;0v(2;1)Chn hai nghim ny ln lt ng vi ta 2 im, khi phng trnh ng thng quachng s l : x + 2y= 0 x= 2y
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22 Tuyn tp nhng bi h c sc
Nh vy quan h tuyn tnh y l x= 2y. Thay li vo h ta c 9y (y+ 1) = 020y (y+ 1) (y 1) = 0
Sau ta chn biu thc ph hp nht nhn vo 2 phng trnh. y s l 20 (y 1) .P T(1) + 9.P T(2)Nh vy
20 (y 1) .P T(1) + 9.P T(2) (x + 2y) 18x2 + 15xy 60x 10y2 80y = 0TH1 : x= 2ythay vo (1)TH2 : Kt hp thm vi PT(1) na thnh mt h gm hai tam thc bit cch giiNghim ca h :
(x; y) = (0; 0), (2;
1), (10; 15),
15 1452
; 11
145 ,
15 +
145
2 ; 11 +
145
S dng cch ny chng ta thy, mt h phng trnh hu t ch cn tm c mt cpnghim l ta xy dng c quan h tuyn tnh v gii quyt bi ton. y chnh l uim ca n. Bn c th vn dng n vo gii nhng v d t 23 n 29 xem. Ti th lmcu 25 nh : Cp nghim l(1;4), (1;4)nn quan h xy dng y l x = 1. Thay livo h v ta c hng chn h s nhn.
Tuy nhin cch ny s chu cht vi nhng bi h ch c mt cp nghim hoc nghim qul khng th d bng Casio c. y l nhc im ln nht ca n
No by gi hy th xy dng quan h bng nh l nh.
Vi h ny v phng trnh di ang c bc cao hn trn nn ta s nhn avo phng trnhtrn ri cng vi phng trnh di. V bc ca x ang cao hn nn ta vit li biu thc saukhi thu gn di dng mt phng trnh bin x. C th l
2x3 + (3a y) x2 + (ay 9a 20) x y (ay+ 9a + 20) = 0()Nghim ca (*) theo nh l s l mt trong cc gi tr
1,12
,y2
,y, ....Tt nhin khng th c nghim x= 1
2hayx = 1c. Hy th vi hai trng hp cn li.
* Vi x= y thay vo h ta c 3y2 18y= 0y3 40y= 0
Khi ta s phi ly(y2 40).P T(1) 3(y 6).P T(2). R rng l qu phc tp. Loi ci ny.* Vi x= ythay vo h ta c
y2 = 03y3 = 0
Khi ta s ly 3y.PT(1) +P T(2). Qu n gin ri. Khi biu thc s l
(x + y)
2x2 + 6xy 3y2 + 27y+ 20 = 0Cch s hai rt tt thay th cch 1 trong trng hp khng tm ni cp nghim. Tuy nhinyu im ca n l khng phi h no dng nh l cng tm c nghim. Ta phi bit kt
hp nhun nhuyn hai cch vi nhau. V hy th dng cch 2 lm cc cu t 23 n 29 xem.N s ra nghim l hng s.
Lm mt cu tng t na. Ti nu lun hng gii.
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2.2 Cu 31 n cu 60 23
2.2 Cu 31 n cu 60
Cu 31
x2y2 + 3x + 3y 3 = 0x2y 4xy 3y2 + 2y x + 1 = 0
Gii
P T(1) (y 1).P T(2) (x + y 1) 3y2 +xy 2y+ 2 = 0TH1 : x= 1 y. No problem !!!Th2 :
3y2 + xy 2y+ 2 = 0x2y 4xy 3y2 + 2y x + 1 = 0
y li l h c bit, ta tm c x= 3l nghim ca h. Thay vo v rt ra kt qu
PT(1) + PT(2) (x 3) (xy 1) = 0
Vy h cho c nghim (x; y) = (0; 1), (1; 0)
Bi vit v phng php UCT hay cn gi l h s bt nh kt thc y. Qua hn chccu ta thy : s dng phng php UCT nng cao (tm quan h tuyn tnh gia cc n) lmt phng php rt mnh v rt tt gii quyt nhanh gn cc h phng trnh hu t. Tuynhin nhc im ca n trong qu trnh lm l kh nhiu. Th nht : tnh ton qu tru b
v hi no. Hin nhin ri, dng quan h tuyn tnh kh, sau cn phi nhc cng phntch mt a thc hn n thnh nhn t. Th hai, nu s dng n mt cch thi qu s khinbn thn tr nn thc dng, my mc, khng chu my m suy ngh m c nhn thy l laou vo UCT, c khc g lao u vo khng ?
Mt cu hi t ra. Liu UCT c nn s dng trong cc k thi, kim tra hay khng ? Xintha, trong nhng VMO, cng lm tng ca h l dng UCT dng c bn, tc l nhnhng s thi. UCT dng c bn th ti khng ni lm g ch UCT dng nng cao th tt nhtkhng nn xi trong cc k thi. Th nht mt rt nhiu thi gian v sc lc. Th hai gy khkhn v c ch cho ngi chm, h hon ton c th gch b ton b mc d c th bn lmng. Vy nn : CNG NG LM RI MI DNG NH !! :D
y c l l bi vit ln nht m ti km vo trong cun sch. Trong nhng cu tip theoti s ci nhng bi vit nh hn vo. n xem nh. Nhng cu tip theo c th cn mt scu s dng phng php UCT. Vy nn nu thc mc c quay tr li t cu 20 m xem. Tmthi gc li , ta tip tc n vi nhng cu tip theo.
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24 Tuyn tp nhng bi h c sc
Cu 32
x5 + y5 = 1x9 + y9 =x4 + y4
Gii
Nhn thy r rng y l loi h bn ng cp. Ta nhn cho hai v vi nhau c
x9 + y9 = (x4 + y4)(x5 + y5) x4y4(x + y) = 0
TH1 : x= 0 y= 1TH2 : y= 0 x= 1TH3 : x= ythay vo (1) r rng v nghimVy h cho c nghim (x; y) = (1; 0), (0; 1)
Cu 33
x3 + 2xy2 = 12y8y2 +x2 = 12
Gii
Li thm mt h cng loi, nhn cho hai v cho nhau ta c
x3 + 2xy2 =y(8y2 +x2) x= 2y
Khi (2) s tng ng12y2 = 12 y= 1, x= 2
Vy h cho c nghim (x; y) = (2; 1), (2;1)
Cu 34
x2 +y2 +
2xy
x + y = 1
x + y = x2 y
Gii
iu kin : x + y >0R rng khng lm n c t phng trnh (2). Th bin i phng trnh (1) xem
(1) (x + y)2 1 + 2xyx +y
2xy= 0
(x + y+ 1)(x + y 1) 2xy(x + y 1)x + y
= 0
C nhn t chung ri. Vi x + y= 1thay vo (2) ta c
1 = (1 y)2 y y= 0, y = 3
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2.2 Cu 31 n cu 60 25
Gi ta xt trng hp cn li. l x + y+ 1 = 2xy
x +y
x +y+ 1 = 1 x2 y2 x2 + y2 + x + y= 0
R rng sai v t iu kin cho ngay x +y >0Vy h cho c nghim (x; y) = (1; 0), (2;3)
Cu 35
x3 y3 = 3(x y2) + 2x2 +
1 x2 3
2y y2 + 2 = 0
Gii
iu kin :1 x 1, 0 y 2Thng th bi ny ngi ta s lm nh sau. phng trnh (1) mt cht
(1) x3 3x= (y 1)3 3(y 1)
Xt f(t) =t3 3tvi1 t 1th f(t) = 3t2 3 0Suy ra f(t)n iu v t suy ra x= y 1thay vo (2)Cch ny n. Tuy nhin thay vo lm vn cha phi l nhanh. Hy xem mt cch khc rt mim m ti lm
(2) x2 +
1 x2 + 2 = 3
2y y2 f(x) =g(y)
Xt f(x)trn min [1;1]ta s tm c 3 f(x) 134Ta li c : g(y) = 3
y(2 y) 3 y+ 2 y
2 = 3
Vyf(x) g(y). Du bng xy ra khi y= 1x= 1, x= 0 Thay vo phng trnh u ch c cp(x; y) = (0; 1)l tha mn
Vy h cho c nghim (x; y) = (0; 1)
Cu 36
x3 3x= y3 3yx6 + y6 = 1
Gii
D thy phng trnh (1) cn xt hm ri, tuy nhin f(t) =t33tli khng n iu, cn phib thm iu kin. Ta s dng phng trnh (2) c iu kin. T (2) d thy 1 x, y 1.Vi iu kin r rng f(t)n iu gim v suy ra c x= yThay vo (2) ta c
2x6
= 1 x
= 1
62Vy h cho c nghim :(x; y) =
16
2;
16
2
,
1
6
2; 1
6
2
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26 Tuyn tp nhng bi h c sc
Cu 37
x3(2 + 3y) = 1x(y3 2) = 3
Gii
Nhn thy x= 0khng l nghim. H cho tng ng
3y+ 1 = 1
x33
x+ 2 =y3
y = 1x
Thay li (1) ta c
2x3 + 3x2 1 = 0
x= 1 y= 1x=
1
2y = 2
Vy h cho c nghim :(x; y) = (1;1),
1
2; 2
Cu 38
x2 + y2 + xy+ 1 = 4yy(x +y)2 = 2x2 + 7y+ 2
Gii
S dng UCT s thy y= 0l nghim ca h. Thay li v ta s c
2P T(1) +P T(2) y(x + y+ 5)(x +y 3) = 0 y= 0x= 5 y
x= 3 y
Vi y= 0thay li v nghimVi x= 5 ykhi phng trnh (1) s tng ng
(y+ 5)2 +y2
y2
5y+ 1 = 4y
V L
Tng t vi x= 3 ycng v nghimVy h cho v nghim
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2.2 Cu 31 n cu 60 27
Cu 39
x +y x y= y2
x2 y2 = 9
Gii
iu kin : y min{x}Ta khng nn t n tng hiu v vn cn st li
y
2s lm bi ton kh khn hn. Mt cch
trc gic ta bnh phng (1) ln. T (1) ta suy ra
2x 2
x2 y2 = y2
4
n y nhn thy
x2 y2 theo (2) bng 3. Vy suy ra
2x 6 = y2
4 y2
= 8x 24Thay vo (2) ta c
x2 8x + 15 = 0 x= 3 y= 0(T M)x= 5 y= 4(T M)
x= 5 y= 4(T M)Vy h cho c nghim (x; y) = (3; 0), (5; 4), (5;4)
Cu 40
x y+ 1 =52
y+ 2(x 3)x + 1 = 34
Gii
iu kin : x, y 1Khng tm c mi quan h c th no. Tm thi ta t n d nhnt
x + 1 =a
0,
y+ 1 =b
0. H cho tng ng
a2 1 b= 5
2
b2 1 + 2a(a2 4) = 34
Ta th b=7
2 a2 t (1) vo (2) v c :
72 a
22+ 2a(a
2
4) 1
4 = 0
a= 3 b= 112
(L)
a=
2
b=
1
2(L)
a= 1 b= 52
(T M)
a= 2 b= 12
(L)
x= 0
y = 34
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28 Tuyn tp nhng bi h c sc
Vy h cho c nghim : (x; y) =
0;34
Cu 41
(x2 + xy+ y2)
x2 + y2 = 185
(x2 xy+ y2)
x2 + y2 = 65
Gii
Thot nhn qua th thy y l mt h ng cp bc 3 r rng. Tuy nhin nu tinh ta emcng 2 phng trnh cho nhau s ch cn li x2 + y2Cng 2 phng trnh cho nhau ta c
2(x2 +y2)
x2 +y2 = 250 x2 + y2 = 5Khi thay li h ta c
(25 +xy).5 = 185(25 xy).5 = 65
xy= 12x2 + y2 = 25
x= 3, y = 4x= 4, y = 3x= 3, y= 4x= 4, y= 3
Vy h cho c nghim (x; y) = (3; 4), (4; 3), (3;4), (4;3)
Cu 42
y
x+
x
y =
7xy
+ 1
x
xy+y
xy= 78
Gii
iu kin : xy 0H cho tng ng
x + yxy
=7 +
xy
xyxy(x +y) = 78
t x + y= a,xy=b. H cho tng ng
a b= 7ab= 78
a= 13b= 6
a= 6b= 13 (L)
x +y = 13xy= 36
x= 9, y= 4x= 4, y= 9
Vy h cho c nghim (x; y) = (9; 4), (4; 9)
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2.2 Cu 31 n cu 60 29
Cu 43
x3 y3 = 9x2 + 2y2 x + 4y= 0
Gii
Dng UCTP T(1) 3.P T(3) (x 1)3 = (y+ 2)3 x= y + 3
n y d dng tm nghim (x; y) = (1;2), (2;1)
Cu 44
8x3y3 + 27 = 18y3
4x2y+ 6x= y2
Gii
y l mt h hay. Ta hy tm cch loi b 18y3 i. V y = 0khng l nghim nn (2) tngng
72x2y2 + 108xy= 18y3
n y tng r rng ri ch ? Th 18y3 t (1) xung v ta thu c
8x3y3
72x2y2
108xy+ 27 = 0
xy= 3
2
xy=
21
9
5
4
xy=21 + 9
5
4
Thay vo (1) ta s tm c y v x
y= 0(L)
y= 3
8(xy)3 + 27
18 = 3
2
5 3 x= 1
4
35
y= 3
8(xy)3 + 27
18 =
3
2 3 +
5x=
1
4 3 +
5Vy h cho c nghim :(x; y) =
1
4
35 ;3
2
5 3 ,1
4
3 +
5
;3
2
3 +
5
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30 Tuyn tp nhng bi h c sc
Cu 45
(x +y)
1 +
1
xy
= 5
(x2 + y2)
1 +
1
x2y2
= 9
Gii
iu kin : xy= 0Ta c nhn ra . H tng ng
x +y+1
x+
1
y = 5
x2 + y2 + 1
x2+
1
y2 = 9
x +
1
x
+
y+
1
y
= 5
x +1
x
2+
y+
1
y
2= 13
x +
1x
= 2, y+1y
= 3
x +1
x= 3, y+
1
y = 2
x= 1, y =
3 52
x=3 5
2 , y= 1
Vy h cho c nghim : (x; y) =
1;
3 52
,
35
2 ; 1
Cu 46
x2 + y2 + x + y = 18x(x + 1)y(y+ 1) = 72
Gii
Mt bi t n tng tch cng kh n gint x2 + x= a, y2 + y= b. Ta c
a + b= 18ab= 72
a= 12, b= 6
a= 6, b= 12
x2 + x= 6y2 +y = 12
x
2
+ x= 12y2 +y = 6
x= 2, x= 3y = 3, y = 4
x= 3, x= 4y = 2, y = 3
Vy h cho c c thy 8 nghim
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2.2 Cu 31 n cu 60 31
Cu 47
x3 + 4y= y3 + 16x1 +y2 = 5(1 + x2)
Gii
H cho tng ng x3 16x= y (y2 4)y2 4 = 5x2
Nh vy phng trnh (1) s l
x3 16x= 5x2y x= 0, y = 2
y=x2 16
5x
Trng hp 2 thay vo (2) s l(x2 16)2
25x2 4 = 5x2
x2 = 1
x2 = 6431
x= 1, y = 3x= 1, y= 3
Vy h cho c nghim (x; y) = (0; 2), (0;2), (1;3), (1;3)
Cu 48 x +y2 x2 = 12
y
x
y2 x2 = 12
Gii
iu kin : y2 x2 x
y2 x2 sinh ra t vic ta bnh phng (1). Vy th bm theo hng xem. T (1)
ta suy tax2 + y2 x2 + 2x
y2 x2 = (12 y)2
y2 + 24 = (12 y)2 y= 5
Thay vo (2) ta c x
25 x2 = 12 x= 3, x= 4i chiu li thy tha mnVy h cho c nghim (x; y) = (3; 5), (4; 5)
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32 Tuyn tp nhng bi h c sc
Cu 49
x4 4x2 + y2 6y+ 9 = 0x2y+x2 + 2y 22 = 0
Gii
nu t x2 = ath h cho bin thnh h tam thc bc 2 ta hon ton bit cchgii. C th y s l
P T(1) + 2.P T(2) (x2 + y)2 2(x2 + y) 35 = 0
TH1 : x2 +y = 7 x2 = 7 ythay (2) ta c
(7 y)y+ 7 y+ 2y 22 = 0
y = 3 x= 2y = 5 x= 2
TH2 : x2
+y = 5 x2
= 5 y. Hon ton tng t thay (2) s cho y v nghimVy h cho c nghim : (x; y) = (2; 3), (2;3), (2;5), (2;5)
Cu 50
x2 +y+ x3y+ xy+ y2x= 54
x4 +y2 + xy(1 + 2x) = 54
Giiy l cu Tuyn sinh khi A - 2008. Mt cch t nhin khi gp hnh thc ny l ta tin hnhnhm cc s hng liH cho tng ng
(x2 + y) +xy+ (x2 +y)xy= 5
4
(x2 + y)2 +xy = 54
n y hng i r rng. t x2 +y = a, xy= b ta c
a + b + ab= 54
a2 + b= 54
a= 0, b=
5
4
a= 12
, b= 32
x2 + y= 0xy= 5
4
x2 +y = 12
xy= 32
x= 3
5
4, y= 3
25
16
x= 1, y = 32
Vy h cho c nghim (x; y) =
3
5
4; 3
25
16
,
1;3
2
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2.2 Cu 31 n cu 60 33
Cu 51
x2 + 1 + y(y+x) = 4y(x2 + 1)(x + y 2) =y
Gii
H gn nh ch l cu chuyn ca x2 + 1v x+y. Tuy nhin ychen vo khin h tr nnkh chu. Hy dit yi . Cch tt nht l chia khi m y = 0khng phi l nghim cah. H cho tng ng
x2 + 1
y + x + y 2 = 2
x2 + 1
y (x + y 2) = 1
Hng i r rng. t x2 + 1
y =a, x +y 2 =b
H cho tr thnh a +b= 2ab= 1
a= 1b= 1
x2 + 1 =yx + y= 3
x= 1, y= 2x= 2, y = 5
Vy h cho c nghim (x; y) = (1; 2), (2;5)
Cu 52 y+ xy2 = 6x2
1 +x2y2 = 5x2
Gii
Loi h ny khng kh. tng ta s chia bin v phi tr thnh hng sNhn thy x= 0khng l nghim. H cho tng ng
y
x2+
y2
x = 6
1
x2+y2 = 5
y
x
1
x+y
= 6
1
x+ y
2
2 yx
= 5
t y
x=a,
1
x+ y=b. H tr thnh
ab= 6b2 2a= 5
a= 2b= 3
y = 2x1
x+ y= 3
x= 1, y = 2
x=1
2, y = 1
Vy h cho c nghim (x; y) = (1; 2),
1
2; 1
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34 Tuyn tp nhng bi h c sc
Cu 53
x2 + 2y2 =xy + 2y2x3 + 3xy2 = 2y2 + 3x2y
Gii
mt cht y l h bn ng cp. Nu ta vit li nh sau x2 + 2y2 xy= 2y2x3 + 3xy2 3x2y= 2y2
T ta c
2y2(x2 + 2y2 xy) = 2y 2x3 + 3xy2 3x2y 4y (y x) x2 xy+y2 = 0TH1 : y= 0 x= 0
TH2 : x= y = 0TH3 : x= y thay vo (1) ta c
2y2 = 2y
x= y = 0x= y = 1
Vy h cho c nghim (x; y) = (0; 0), (1; 1)
Cu 54 2x2y+y3 = 2x4 + x6
(x + 2)y+ 1 = (x + 1)2
Gii
iu kin : y 1Khai thc t (1). C v nh l hm no . Chn chia cho ph hp ta s c mc ch, ys chia cho x3 v x= 0khng l nghim ca h. PT(1) khi s l
2y
x+ y
x3
= 2x + x3 yx
=x y = x2
Thay vo (2) ta s c
(x + 2)
x2 + 1 = (x + 1)2 (x + 2)2 x2 + 1 = (x + 1)4 x= 3, y= 3(T M)x= 3, y = 3(T M)
Vy h cho c nghim : (x; y) = (3;3)
Ta s n mt cu tng t n
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2.2 Cu 31 n cu 60 35
Cu 55
x5 + xy4 =y10 +y64x + 5 +
y2 + 8 = 6
Gii
iu kin : x 54
Thy y= 0khng l nghim ca h. Chia 2 v ca (1) cho y5 ta c
x
y
5+
x
y =y5 + y x
y =y x= y2
Thay vo (2) ta c
4x + 5 +
x + 8 = 6 x= 1 y = 1
Vy h cho c nghim (x; y) = (1;1)
Cu 56
xy+ x + 1 = 7yx2y2 + xy+ 1 = 13y2
Gii
y l cu Tuyn sinh khi B - 2009. Cc gii thng thng nht l chia (1) cho y , chia (2)choy2 sau khi kim tra y= 0khng phi l nghim. Ta s c
x +x
y+
1
y= 7
x2 +x
y+
1
y2 = 13
x +1
y+
x
y = 7
x +1
y
2 x
y = 13
a + b= 7a2 b= 13
a= 4, b= 3a= 5, b= 12
x +1
y = 4
x= 3y x +1y = 5x= 12y
x= 1, y=
1
3x= 3, y= 1
Vy h cho c nghim : (x; y) =
1;1
3
, (3; 1)
Tip tc ta n thm mt cu tuyn sinh na
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36 Tuyn tp nhng bi h c sc
Cu 57
x4 + 2x3y+ x2y2 = 2x + 9x2 + 2xy= 6x + 6
Gii
tht k nu ta th kho lo xyln (1) s ch cn li phng trnh n x. D s l bc 4nhng liu th n nhiu. H vit li
x4 + 2x2(xy) +x2y2 = 2x + 9
xy=6x + 6 x2
2
T (1) s tng ng
x4
+ x2
(6x + 6 x2
) +6x + 6 x
2
22
= 2x + 9 x= 4x= 0 y=17
4V L
Vy h cho c nghim (x; y) =4;17
4
Cu 58
3
1 +x +
1 y = 2x2 y4 + 9y= x(9 +y y3)
Gii
iu kin : y 1Khng lm n g c t (1). Xt (2). 1 to th (2) c th phn tch c thnh
(x y) (9 x y3) = 0
x= yx= 9 y3
Vi x= y thay vo (1) ta s c
3
1 +y+
1 y= 2 a + b= 2a3 +b2 = 2
b 0 a= 1, b= 1a= 1 3, b= 3 + 3
a=
3 1, b= 3 3 y= 0y= 63 11
y= 63 11
Vi x= 9 y3 thay vo (1) ta s c3
10 y3 +
1 y = 2
Ta c3
10 y3 +
1 y 3
9> 2
Vy h cho c nghim : (x; y) = (0; 0), (63 11;63 11), (63 11;63 11)
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2.2 Cu 31 n cu 60 37
Cu 59
xy+
1 y= y2
y
x 1y = 1
Gii
iu kin : x 1, 0 y 1Thot nhn bi ton ta thy nh lc vo m cung nhng cn thc. Tuy nhin ch vi nhngnh gi kh n gin ta c th chm p bi tonVit li phng trnh (2) nh sau
2
y
x 1 = y 1
T iu kin d thy V T 0 V PDu bng xy ra khi x= y = 1Vy h cho c nghim (x; y) = (1; 1)
Cu 60
x
17 4x2 +y
19 9y2 = 317 4x2 +
19 9y2 = 10 2x 3y
Gii
iu kin :172 x
172
,193 y
193
Bi ton ny xut hin trn thi th ln 2 page Yu Ton hc v ti l tc gi ca n. tng ca n kh n gin, ph hp vi 1 thi tuyn sinh x
17 4x2 lin quan n 2xv17 4x2, y
19 9y2 lin quan n 3yv19 9y2.
V tng bnh phng ca chng l nhng hng s. y l c s ta t nt 2x +
17 4x2 =a, 3x +
19 9y2 =b. H cho tng ng
a + b= 10a2 17
4 +
b2 196
= 3
a= 5, b= 5a= 3, b= 7
TH1 :
2x +17 4x2 = 53y+
19 9y2 = 5
x= 1
2x= 2
y=5 13
6
TH2 :
2x +
17 4x2 = 33y+
19 9y2 = 7 (Loi)
Vy h cho c nghim :(x; y) =
1
2;5 +
13
6
1
2;5 13
6
2;
5 +
13
6
2;
5 136
V y l tng gc ca n. Hnh thc n gin hn mt cht
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38 Tuyn tp nhng bi h c sc
2.3 Cu 61 n cu 90
Cu 61
x
5 x2 +y
5 4y2 = 15 x2 +5 4y
2 =x 2y
Nghim : (x; y) = (1;1),
2;12
Cu 62
x3 xy2 + y3 = 14x4 y4 = 4x y
Gii
R rng l mt h a v c dng ng cp bng cch nhn cho v vi v. Tuy nhin, biny nu s dng php th tt ta s a v mt kt qu kh p mtPhng trnh (2) tng ng
4x(x3 1) =y(y3 1)n y ta rt x3 1vy3 1t (1). C th t (1) ta c
x3 1 =y3 y2xy3 1 =xy2 x3
Thay tt c xung (2) v ta thu c
4xy2(y x) = xy(x2 y2)
x= 0y = 0x= y4y= y + x
y= 1x= 1x= y = 1
y= 13
25, x=
33
25
Vy h cho c nghim (x; y) = (0; 1), (1; 0), (1; 1),
13
25;
33
25
Cu 63
x +
x2 y2x
x2 y2 +
xx2 y2x +
x2 y2 =
17
4
x(x +y) +
x2 + xy+ 4 = 52
Gii
iu kin : x =
x2 y2, x2 y2 0, x2 +xy+ 4 0Hnh thc bi h c v kh khng b nhng nhng tng th l ht. Ta c th khai thc c2 phng trnh. Pt(1) c nhiu cch x l : ng cp, t n, lin hp. Ti s x l theo hngs 3. (1) khi s l
x +
x2 y22
x2 (x2 y2) +
x
x2 y22
x2 (x2 y2) =17
4 2 (2x
2 y2)y2
=17
4 y= 4x
5
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2.3 Cu 61 n cu 90 39
Tip tc khai thc (2). D thy t
x2 +xy+ 4 =t 0th (2) tr thnh
t2 + t= 56
t= 7t= 8(L) x
2 +xy = 45
Kt hp li ta c y= 4
5x
x2 + xy= 45
x= 5, y = 4x= 5, y= 4x= 15, y= 12x= 15, y= 12
Vy h cho c nghim : (x; y) = (5;4), (5; 4), (15; 12), (15;12)
Cu 64x +y+x
y = 2
y+ xy x= 1Gii
iu kin : x, y 0,y min{x},x min{y}Khng tm c mi lin h g t c hai phng trnh, ta tin hnh bnh phng nhiu ln ph v ton b cn thc kh chu. Phng trnh (1) tng ng
2x + 2
x2 y= 4
x2 y= 2 x x2 y= x2 4x 4 4x y= 4Lm tng t phng trnh (2) ta s c : 4x
4y =
1. Kt hp 2 kt qu li d dng tm
c x,yVy h cho c nghim : (x; y) =
17
12;5
3
Cu 65
x + 2xy
3
x2 2x + 9 =x2 + y
y+ 2xy
3y2 2y+ 9=y2 +x
Gii
Hnh thc ca bi h l i xng. Tuy nhin biu thc kh cng knh v li nhn xt thyx= y = 1l nghim ca h. C l s nh giCng 2 phng trnh li ta c
x2 + y2 = 2xy
1
3
x2 2x + 9 + 1
3
y2 2y+ 9
T ta nhn xt c nghim th xy 0v l 3
t2 2t+ 9 2nn ta nh gix2 + y2 2xy
1
2+
1
2
(x y)2 0
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40 Tuyn tp nhng bi h c sc
Du bng xy ra khi (x; y) = (1; 1)
Cu 66
6
x
y 2 = 3x y+ 3y2
3x +
3x y= 6x + 3y 4
Gii
iu kin : y= 0, 3x y, 3x +3x y 0Phng trnh (1) khi s tng ng
6x
2y=y3x y+ 3y2
2 (3x
y)
y3x y 3y
2 = 0
3x y = y
3x y =3y
2
TH1 :
3x y= y. T y suy ra y 0v3x= y2 + ythay tt c vo (2) ta c
2
y2 + y y= 2 y2 + y+ 3y 4 2y2 + 7y 4 = 0y 0 y = 4 x= 4
TH2 :
3x y=3y2
. T y suy ra y 0v3x= 9y2
4 + ythay tt c vo (2) ta cng s tm
c y =8
9x=
8
9Vy h cho c nghim (x; y) = (4;4),
8
9;8
9
Cu 67
(3 x)2 x 2y2y 1 = 03
x + 2 + 2
y+ 2 = 5
Gii
iu kin : x 2, y12
Phng trnh (1) tng ng
(2 x)2 x +2 x= (2y 1)
2y 1 +
2y 1 f(2x 1) =f(
2y 1)
Vi f(x) =x3 +xn iu tng. T suy ra
2 x= 2y 1 x= 3 2ythay vo (2)ta c
3
5 2y+ 2
y+ 2 = 5
a + 2b= 5a3 + 2b2 = 9
a= 1, b= 2
a=3 654
, b= 23 + 658
a=
65 3
4 , b=
23 658
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2.3 Cu 61 n cu 90 41
y= 2
y=233 + 23
65
32
y=233 2365
32
Vy h cho c nghim
(x; y) = (1;2),
23
65 18516
;233 2365
32
23
65 + 185
16 ;
233 + 23
65
32
S dng tnh n iu ca hm s cng l mt hng kh ph bin trong gii h phng trnh.Ch cn kho lo nhn ra dng ca hm, ta c th rt ra nhng iu k diu t nhng phngtrnh khng tm thng cht no
Cu 681 +xy+ 1 +x + y= 2
x2y2 xy= x2 + y2 + x + y
Gii
iu kin : xy 1, x + y 1Mt cht bin i phng trnh (2) ta s c
x2y2 + xy= (x +y)2 + x + y (xy x y)(xy+ x + y+ 1) = 0
x +y =xyx +y = xy 1
TH1 : xy=x + ythay vo (1) ta c2
1 +xy = 2 xy= 0 x= y = 0TH2 : x + y = xy 1thay vo (1) ta c
1 +xy+xy= 2(V L)
Vy h cho c nghim : (x; y) = (0; 0)
Cu 69
x + 3x yx2 + y2
= 3
y x + 3yx2 +y2
= 0
Gii
Ti khng nhm th bi ton ny xut hin trn THTT, tuy nhn hnh thc ca h kh pmt v gn nh nhng khng h d gii mt cht no. Hng lm ti u ca bi ny l phcha. Da vo tng h kh i xng ng thi di mu nh l bnh phng ca Moun mta s dng cch ny. Hng gii nh sau
PT(1)+i.PT(2) ta s c
x + yi +3(x yi) (xi + y)
x2 + y2 = 0
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42 Tuyn tp nhng bi h c sc
t z=x +yikhi phng trnh tr thnh
z+3z iz|z|2 = 3 z+
3z izz.z
= 3 z+3 iz
= 3
z= 2 + iz= 1 i
Vy h cho c nghim (x; y) = (2; 1), (1;1)Hnh thc ca nhng bi h ny kh d nhn thy. Th lm mt s cu tng t nh.
Cu 70
x +5x + 7
5y
x2 +y2 = 7
y+7
5x 5yx2 + y2
= 0
Cu 71
x + 5x yx2 + y2
= 3
y x + 5yx2 +y2
= 0
Cu 72
x +16x 11y
x2 +y2 = 7
y 11x + 16yx2 + y2
= 0
Cu 73
(6 x)(x2 + y2) = 6x + 8y(3 y)(x2 +y2) = 8x 6y
Gi : Chuyn h cho v dng
x +6x + 8y
x2 +y2 = 6
y+8x 6yx2 + y2
= 3
Nghim : (x; y) = (0; 0), (2;1), (4; 2)
Phc ha l mt phng php kh hay gii h phng trnh mang tnh nh cao. Khngch vi loi h ny m trong cun sch ti s cn gii thiu mt vi cu h khc cng s dngphc ha kh p mt.
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2.3 Cu 61 n cu 90 43
Cu 74
4x2y2 6xy 3y2 = 96x2y y2 9x= 0
Gii
y l mt bi ton cng kh p mt. Thy x= 1l nghim ca h . Ta suy raP T(1) +P T(2) (x 1)(4y2(x + 1) + 6xy 9) = 0
TH1 : x= 1 y= 3TH2 : 4y2(x + 1) + 6xy 9 = 0V x= 0khng l nghim. Suy ra 4y2x(x + 1) + 6x2y 9x= 0(*)V sao nhn xvo y. UCT chng ? Ti ch gii thiu cho cc bn UCT nng cao thi chti ch dng bao gi. L do ch n gin ti mun xut hin 6x2y 9x= y2 t (2) thiVy (*)
4y2x(x + 1) +y2 = 0
y2(2x + 1)2 = 0
TH1 : y= 0v nghimTH2 : x= 1
2 y= 3, y= 3
2
Vy h cho c nghim : (x; y) = (1; 3),1
2; 3
,
1
2;3
2
Cu 75
x2
(y+ 1)2+
y2
(x + 1)2 =
1
2
3xy=x + y+ 1
Gii
iu kin x, y= 1Bi ton ny c kh nhiu cch gii. Ti xin gii thiu cch p nht ca bi nyp dng Bt ng thc AMGMcho v tri ca (1) ta c
V T 2xy(x + 1)(y+ 1)
= 2xy
xy+ x + y+ 1=
2xy
xy+ 3xy =
1
2
Du bng xy ra khi (x; y) = (1; 1),13 ;13
Cu 76
3y2 + 1 + 2y(x + 1) = 4y
x2 + 2y+ 1y(y x) = 3 3y
Gii
iu kin : x2 + 2y+ 1
0
Khng lm n g c t (2). Th bin i (1) xem sao. PT(1) tng ng
4y2 4y
x2 + 2y+ 1 +x2 + 2y+ 1 =x2 2xy+ y2
2y
x2 + 2y+ 12
= (x y)2
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44 Tuyn tp nhng bi h c sc
x2 + 2y+ 1 = 3y xx2 + 2y+ 1 =x + y
C v hi o nh ? Nhng mt cht th (1) c vc dng ca cc hng ng thc nn tangh n hng ny
By gi x l hai trng hp kia th no ? Chc bnh phng thi. Tt qu ! Phng trnh sch cn li xyv ym nhng ci th (2) c cTH1 :
x2 + 2y+ 1 = 3y x
3y xx2 + 2y+ 1 = 9y2 6xy+ x2
3y x6xy= 9y2 2y 1xy= y2 + 3y 3(2)
x= 1, y= 1(T M)
x=415
51 , y=
17
3(T M)
TH2 :
x2 + 2y+ 1 =x +y
x +y 0x
2
+ 2y+ 1 =x
2
+ 2xy+ y
2
x + y 02xy=
y2 + 2y+ 1
xy= y2 + 3y 3 x= 1, y= 1
x=
41
21 , y = 7
3(L)
Vy h cho c nghim : (x; y) = (1; 1),
415
51;
17
3
Nh chng ta bit. Tam thc bc hai c kh nhiu ng dng trong gii ton v h cngkhng phi l ngoi l. Ch vi nhng nh gi kh n gin : t iu kin ca tamthc c nghim m ta c th tm ra cc tr ca cc n. T nh gi v gii quyt nhngbi ton m cc phng php thng thng cng b tay. Loi h s dng phng php ny
thng cho di hai dng chnh. Th nht : cho mt phng trnh l tam thc, mt phngtrnh l tng hoc tch ca hai hm f(x) v g(y). Th hai : cho c 2 phng trnh u lphng trnh bc hai ca 1 n no . Hy th lt qua mt chm h loi ny nh.
Cu 77
x4 + y2 =698
81x2 + y2 + xy 3x 4y+ 4 = 0
GiiHnh thc ca h : mt phng trnh l tam thc bc hai mt c dng f(x) +g(y)v mt skh khng b. Ta hy khai thc phng trnh (2) bng cch nh gi Vit li phng trnh (2) di dng sau
x2 + (y 3)x + (y 2)2 = 0()y2 + (x 4)y+ x2 3x + 4 = 0()
(*) c nghim th x 0 (y 3)2 4(y 1)2 0 1 y73
(**) c nghim th y 0 (x 4)4 4(x2 3x + 4) 0 0 x 43
T iu kin cht ca hai n gi ta xt (1) v c mt nh gi nh sau
x4 + y2
4
3
4+
7
3
2=
697
81 0 V Pnn v nghimVy h cho c nghim : (x; y) = (0; 1), (1;1)
Cu 86
x3(4y2 + 1) + 2(x2 + 1)
x= 6
x2y(2 + 2
4y2 + 1) =x +
x2 + 1
Gii
iu kin : x 0Hnh thc ca bi h r rng l kh rc ri. Tuy nhin, (2) nu ta chia c 2 v cho x2th s c lp c xvyv hi vng s ra c iu g.
Nhn thy x= 0khng l nghim. Chia 2 v ca (2) cho x2
ta c
2y+ 2y
4y2 + 1 = 1
x+
1
x
1
x2+ 1
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48 Tuyn tp nhng bi h c sc
R rng 2 v u c dng f(t) =t +t
t2 + 1v hm ny n iu tng. Vy t ta suy ra
c 2y= 1
xthay vo (1) ta c
x31x2 + 1
+ 2(
x2
+ 1)
x
= 6
x3 + x + 2(x2 + 1)x= 6R rng v tri n iu tng vi iu kin ca x. Vy x= 1l nghim duy nht
Vy h cho c nghim : (x; y) =
1;1
2
Cu 877x + y+ 2x + y = 52x + y+ x y= 2
Gii
y l cu trong VMO 2000-2001. Khng hn l mt cu qu khiu kin : y min{2x;7x}Xut hin hai cn thc vy th t
7x + y = a ,
2x +y = bxem
Nhng cn x yth th no ? Chc s lin quan n a2, b2. Vy ta s dng ng nht thc
x y = k(7x + y) +l(2x + y) k= 35 , l= 85Vy h cho tng ng
a + b= 5
b +3a2
5 8b
2
5 = 2
a, b 0
a=15 77
2
b=
77 5
2
7x + y=151 1577
2
2x + y=51 577
2
x= 10 77y =
11 772
Vy h cho c nghim : (x; y) =
10
77;
11
77
2
Mt cch khc cng kh tt. t
7x + y = a,
2x + y= bv ta xy dng mt h tm sau a + b= 5a2 b2 = 5x
a +b= 5a b= x b=
5 x2
Thay vo (2) v ta c5 x
2 + x y= 2 x= 2y 1
n y thay li vo (2) v ta cng ra kt qu
Mt v d tng t ca bi ny
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2.3 Cu 61 n cu 90 49
Cu 88
11x y y x= 17
y x + 6y 26x= 3
Nghim : (x; y) = 37
20;81
10
Cu 89
3x
1 +
1
x + y
= 2
7y
1 1
x +y
= 4
2
Gii
y l cu trong VMO 1995-1996. Mt tng kh p mt m sng toiu kin : x, y 0, x + y >0H cho tng ng
1 + 1
x +y =
23x
1 1x + y
=4
27y
1
x + y =
13x2
2
7y
1 = 1
3x+
2
27y
1x + y
=
1
3x 2
2
7y
1
3x+
2
27y
1x + y
= 1
3x 8
7y 21xy= (x + y)(7y 3x)
(y 6x)(7y+ 4x) = 0 y= 6xThay vo phng trnh u ta c
1 + 1
7x=
23x
x= 11 + 4
7
21 y = 22
7 +
87
Mt cch khc c th s dng trong bi ny l phc ha. N mi xut hin gn ytx= a >0,y= b >0. Ta c h mi nh sau
a +
a
a2 + b2 =
23
b ba2 + b2
=4
27
P T(1) +i.P T(2) (a +bi) + a bia2 + b2
= 2
3+
4
27
i
t z=a +biphng trnh cho tr thnh
z+1
z =
23
+4
27
i z a, b x, y
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50 Tuyn tp nhng bi h c sc
Vy h cho c nghim : (x; y) =
11 + 4
7
21 ;
22
7 +
87
Bi h ny c kh nhiu d bn phong ph. Ti xin gii thiu cho cc bn
Cu 90
x
3
1 +
6
x + y
=
2
y
1 6
x +y
= 1
Nghim : (x; y) = (8; 4)
2.4 Cu 91 n cu 120
Cu 91
x
1 12
y+ 3x
= 2
y
1 +
12
y+ 3x
= 6
Nghim : (x; y) = (4 + 2
3; 12 + 6
3)
Cu 92
10x
1 +
3
5x + y
= 3
y
1 3
5x +y
= 1
Nghim : (x; y) =
2
5; 4
Cu 93
4
x
1
4+
2
x +
y
x +y
= 2
4
y
1
42
x +
y
x + y
= 1
Tip theo ta n mt vi v d v s dng phng php lng gic ha trong gii h phng trnh
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2.4 Cu 91 n cu 120 51
Cu 94
x
1 y2 +y1 x2 = 1(1 x)(1 +y) = 2
Gii
iu kin :|x| 1,|y| 1iu kin ny cho ta tng lng gic ha. t x= sina, y= sinbvi a, b
2;
2
Phng trnh u tng ng
sinacosb + sinbcosa= 1 sin(a + b) = 1 a + b= 2
Phng trnh (2) tng ng
(1
sina)(1 +sinb) = 2
(1
sina)(1 +cosa) = 2
a=
2
a= 0 b=
b=
2
x= 1, y= 0(L)x= 0, y= 1
Vy h cho c nghim : (x; y) = (0; 1)
Cu 95 2y=x(1 y2)3x
x3 =y(1
3x2)
Gii
Thot nhn ta thy c v h ny cng xong, ch c g khi vit n di dng xy2 =x 2yx3 3x2y= 3x y
a n v dng ng cp, nhng ci chnh y l nghim n qu l. Vy th hng khcxem. Vit li h cho sau khi xt
x= 2y1 y2y =
3x x31 3x2
Nhn biu thc v phi c quen thuc khng ? Rt ging cng thc lng gic nhn i vnhn ba ca tan. Vy tng ny rat x= tanvi
2;
2
. T PT(2) ta s c
y=3tan tan3
1
3tan2
= tan 3
M nh th theo (1) ta s c
x= 2tan3
1 tan23 = tan 6
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52 Tuyn tp nhng bi h c sc
T suy ra
tan = tan 6 = k5 =
2
5 ;
5; 0;
5;2
5
Vy h cho c nghim : (x; y) = tan 25
;tan6
5 ,tan
5
;tan3
5 , (0; 0)
Lm mt bi tng t nh.
Cu 96
y =3x x31 3x2
x=3y y31 3y2
S dng phng php lng gic ha trong gii h phng trnh cn phi nm r cc hngng thc, ng thc, cng thc lng gic, v cn mt nhn quan tt pht hin mt biuthc no ging vi mt cng thc lng gic.
Cu 97 x3y(1 +y) +x2y2(2 +y) +xy3 30 = 0x2y+x(1 +y+y2) +y
11 = 0
Gii
y l mt h kh mnh nhng hay. Nhn vo 2 phng trnh ta thy cc bin "kt dnh" vinhau kh tt v hng s c v nh ch l k ng ngoi. Vy hy vt hng s sang mt bn vthc hin bin i v tri. H phng trnh cho tng ng
xy(x + y)(x +y+ xy) = 30xy(x + y) +x + y+ xy= 11
n y tng r rng. t a= xy(x + y), b= xy + x + yv h cho tng ng
ab= 30a + b= 11
a= 5, b= 6a= 6, b= 5
xy(x +y) = 5xy+ x + y= 6
xy(x +y) = 6xy+ x + y= 5
TH1 :
xy(x + y) = 6xy+ x + y= 5
xy= 2x + y = 3
xy= 3x + y = 2
(L)
x= 2, y= 1x= 1, y= 2
TH2 :
xy(x + y) = 5xy+ x + y= 6
xy= 5
x + y = 1 (L) xy= 1x + y = 5
x= 5 212 , y = 5 + 212
x=5 +
21
2 , y=
5212
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2.4 Cu 91 n cu 120 53
Vy h cho c nghim : (x; y) = (1; 2), (2; 1),
521
2 ;
5 212
Tc gi ca n rt kho lo trn nhiu ln cch t n tng tch vo mt h, gy nhiu kh
khn cho ngi lm
Cu 98
sin2x +
1
sin2x+
cos2y+
1
cos2y =
20y
x + ysin2y+
1
sin2y+
cos2x +
1
cos2x=
20x
x + y
Gii
Bi ton xut hin trong VMO 2012-2013. Hnh thc bi h c s khc l khi c c hmlng gic chen chn vo. Vi kiu h ny nh gi l cch tt nhtTa s cng hai phng trnh vi nhau v s chng minh V T 210 V Pp dng Bt ng thc Cauchy Schwarzcho v phi ta c
20y
x + y+
20x
x + y
2
20y
x + y+
20x
x +y
= 2
10
Gi ta s chng minh : V T
2
10tc l phi chng minhsin2x +
1
sin2x+
cos2x +
1
cos2x
10
V T =
sin x 1
sin x
2+
22
+
cos x 1
cos x
2+
22
1
sin x+
1
cos x (sin x + cos x)
2+
2
22
Hin nhin ta c sinx + cosx
2nn1
sin x+
1
cos x (sin x + cos x) 4
sin x + cos x
2 42
2 =
2
VyV T 2 + 8 = 10. Tng t vi bin yv ta c iu phi chng minhng thc xy ra khi x= y =
4+ k2
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54 Tuyn tp nhng bi h c sc
Cu 99
x
xx= yy+ 8yx y= 5
Gii
iu kin : x, y 0 h ny cho mt phng trnh n gin qu. Th thng ln (1) chng ? Khng nn ! Bin i1 to ri hy th. Hng bin i kh n gin l lm ph v cn thcPhng trnh (1) tng ng
x(x 1) = y(y+ 8) x(x 1)2 =y(y+ 8)2
n y thc hin th x= y + 5 ln (1) v ta c
(y+ 5)(y+ 4)2 =y(y+ 8)2
y= 4
x= 9
Vy h cho c nghim (x; y) = (9; 4)
Cu 100
1x
+y
x=
2
x
y + 2
y
x2 + 1 1 = 3x2 + 3Gii
iu kin : x >0, y= 0R rng vi iu kin ny th t (2) ta thy ngay c nghim th y >0Phng trnh (1) tng ng
x + y
x =
2 (
x +y)
y
x +y = 0(L)
y= 2x
Vi y= 2xthay vo (2) ta c
2xx2 + 1 1 = 3x2 + 3 2x3x2 + 1 = 2x x2 + 1 = 2x2x 3
R rng v tri n iu tng v v phi n iu gim nn phng trnh ny c nghim duynht x=
3 y = 23
Vy h cho c nghim (x; y) = (
3; 2
3)
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2.4 Cu 91 n cu 120 55
Cu 101
y= x3 + 3x + 4x= 2y3 6y 2
Gii
Hnh thc bi h kh gn nh nhng cng khin nhiu ngi phi lng tng. Nhn xtx= y = 2l nghim. Ta tin hnh tch nh sau
y 2 = (x + 1)2(x 2)x 2 = (y+ 1)2(y 2)
n y nhn cho v vi v ta c
2(y 2)2(y+ 1)2 = (x + 1)2(x 2)2
D thy V T 0 V P. y ng thc xy ra khi x= y = 2
Cu 102
x3 xy2 + 2000y= 0y3 yx2 500x= 0
Gii
D dng a c v h ng cp. Nhng ta bin i mt to n ti u.
H cho tng ng
x (x2 y2) = 2000yy(x2 y2) = 500x 500x
2(x2 y2) = 2000y2(x2 y2)
x= yx= yx= 2yx= 2y
Thay li vi mi trng hp vo (1) v ta c
y = 0, x= 0
y = 1010
3 , x= 20103y = 10
10
3 , x= 20
10
3
Vy h cho c nghim : (x; y) = (0; 0),
20
10
3;10
10
3
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56 Tuyn tp nhng bi h c sc
Cu 103
3
x2 + y2 1+ 2y
x= 1
x2 +y2 + 4x
y = 22
Gii
tng t n ph r rng. t x2 + y2 1 =a , yx
=b . H cho tng ng
3
a+ 2b= 1
a +4
b = 21
a= 7, b=
2
7
a= 9, b=1
3
x2 +y2 = 82x= 7y
x2 +y2 = 10x= 3y
y= 4
2
53, x= 14
2
53x= 3, y = 1
Vy h cho c nghim : (x; y) = (3;1)14253 ;4253
Cu 104
x +1
y+
x + y 3 = 3
2x + y+1
y = 8
Giiiu kin : y= 0, x +1
y 0, x + y 3
tng t n ph cng kh r rng.
t
x +1
y =a 0,x + y 3 =b 0. H cho tng ng
a + b= 3a2 + b2 = 5
a= 1, b= 2
a= 2, b= 1
x +1
y = 1
x + y
3 = 4
x +1y
= 4
x + y 3 = 1
x= 410, y= 3 + 10x= 4 +
10, y = 3
10
x= 3, y= 1x= 5, y= 1
Vy h cho c nghim : (x; y) = (3; 1), (5;1)(4 10;310)
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2.4 Cu 91 n cu 120 57
Cu 105
x3(2 + 3y) = 8x(y3 2) = 6
Gii
y l mt cu kh ging cu s 37Nghim : (x; y) = (2;1), (1;2)
Cu 106
2x2y+ 3xy= 4x2 + 9y7y+ 6 = 2x2 + 9x
Gii
Bi ny nu li ngh c th dng mn v th thn chng y vo PT(1). Nhng hy dng UCT y s tt hn.Nhn thy y = 3l nghim (ci ny gi li nh, ti khng gii thch na), thay y = 3vo hta c
2x2 + 9x 27 = 027 2x2 + 9x= 0
Nh vy hng ca ta s cng hai phng trnh ban u li v nhn ty 3s xut hin. Vy
P T(1) +P T(2) (3 y) 2x2 + 3x 2 = 0
n y d dng gii ra (x; y) =2;16
7
,
1
2;1
7
,
3(3
33)4
; 3
Cu 107
x2 + 3y= 9y4 + 4(2x 3)y2 48y 48x + 155 = 0
Giiy l mt cu kh hc, khng phi ai cng c th d dng gii n c.Th 3y= 9 x2 t (1) xung (2) ta c
y4 + 8xy2 12y2 16(9 x2) 48x + 155 = 0
y4 + 8xy2 + 16y2 12(y2 + 4x) + 11 = 0
y2 + 4x= 1y2 + 4x= 11
TH1 :
y2 + 4x= 11
9 x23
2
+ 4x= 11
x4
18x2 + 36x
18 = 0
x4 = 18(x 1)2
x2 32x + 32 = 0x2 + 3
2x 32 = 0
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58 Tuyn tp nhng bi h c sc
x=
3
2
18 1222
y= 12
2 6
36 24212
x=32
18 122
2 y=12
2 6
36 242
12
TH2 :y2 + 4x= 1
9 x2
3
2+ 4x= 1 x4 18x2 + 36x + 72 = 0
x2 6x + 12 x2 + 6x + 6 = 0 x= 3 3 y = 1 23Vy h c c thy 6 nghim nh trn
Mt thc mc nh l TH2 v sao x4 18x2 + 36x+ 72 = (x2 6x+ 12)(x2 + 6x+ 6). Tchnhn t kiu g hay vy ? Casio truy nhn t chng ? C th lm. Nhng thc ra phng trnhbc 4 c cch gii tng qut bng cng thc Ferrari. i vi v d trn ta lm nh sau
x4 18x2 + 36x + 72 = 0 x4 2ax2 + a2 = (18 2a) x2 36x + a2 72Ta phi tm asao cho v phi phn tch c thnh bnh phng. Nh th ngha l
182 = (18 2a) a2 72 a= 9Nh vy
x4 18x2 + 36x + 72 = 0 (x2 + 9)2 = 9(2x 1)2 (x2 6x + 12)(x2 + 6x + 6) = 0
Chi tit v gii phng trnh bc 4 cc bn c th tm d dng trn google. Gi ta tip tc cc
bi h. Tip theo l mt chm h s dng tnh n iu ca hm s kh d nhn.
Cu 108
x +
x2 + 1
y+
y2 + 1
= 1
y+ yx2 1 =
35
12
Gii
iu kin : x2 >1Khng th lm n c g t (2). T (1) ta nhn xt thy hai hm ging nhau nhng chngli dnh cht vi nhau, khng chu tch ri. Vy ta dt chng ra. Php lin hp s gip taPhng trnh (1) tng ng
x +
x2 + 1
y+
y2 + 1
y2 + 1 y
=
y2 + 1 y x +
x2 + 1 = y +
y2 + 1
Tch c ri nhng c v hai bn khng cn ging nhau na. Khoan !! Nu thay y2 = (y)2th sao nh. Qu tt. Nh vy c hai v u c dng f(t) =t+
t2 + 1v hm ny n iu
tng. T ta rt ra x=
y
Thay li vo (2) ta cy+
yy2 1 =
35
12
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2.4 Cu 91 n cu 120 59
y thc ra l mt phng trnh kh kh chu. Thot tin khi thy loi ny ta s bnh phng2 v ln. iu kin bnh phng l y >0khi ta c
y2 + 2y2
y2 1+
y2
y2
1
= 35
122
y4 y2 + y2
y2
1
+ 2y2
y2 1=
35
122
n y kh r rng . t y2
y2 1 =t >0v phng trnh tng ng
t2 + 2t
35
12
2= 0
t=
49
12(L)
t=25
12
y2
y2 1 =25
12
y=
5
4
y= 53
i chiu iu kin bnh phng ch ly 2 gi tr dng.
Vy h cho c nghim : (x; y) =54;54 ,53;53
Cu 109
(4x2 + 1)x + (y 3)5 2y= 04x2 + y2 + 2
3 4x= 7
Gii
iu kin : y5
2 , x 3
4 Vit li phng trnh (1) nh sau
(4x2 + 1)x= (3 y)
5 2y (4x2 + 1)2x= (6 2y)
5 2y f(2x) =f
5 2y
Vi f(t) =t3 + tl hm n iu tng. T ta c 2x=
5 2y x 0thay vo (2) ta c
4x2 +
5
2 2x2
2+ 2
3 4x= 7
Gi cng vic ca ta l kho st hm s v tri trn 0;34v chng minh n n iu gim.
Xin nhng li bn cVi hm s v tri n iu gim ta c x=
1
2l nghim duy nht y = 2
Vy h cho c nghim : (x; y) =
1
2; 2
Hy k mi tng quan gia cc biu thc trong mt phng trnh va ta s t mc ch
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60 Tuyn tp nhng bi h c sc
Cu 110
y3 + y = x3 + 3x2 + 4x + 21 x2 y= 2 y 1
Gii
iu kin : 0 y 2,1 x 1Phng trnh (1) tng ng
y3 +y = (x + 1)3 + (x + 1) y = x + 1
Thay vo (2) ta c 1 x2 1 +x= 1 x 1
Phng trnh ny khng qu kh. t t =
1 +x+
1 x 1 x2 = t2 2
2 . Thay vo
phng trnh ta c
t2 22
=t 1
t= 0t= 2
1 x + 1 +x= 01 x + 1 +x= 2 x= 0, y = 1
Vy h cho c nghim :(x; y) = (0; 1)
Nhng bi ny thng s nng v gii phng trnh v t hn.
Cu 111x + 1 + x + 3 + x + 5 = y 1 + y 3 + y 5
x + y+x2 + y2 = 80
Gii
iu kin : x 1, y 5Phng trnh u c dng
f(x + 1) =f(y 5)Vi f(t) =
t +
t+ 2 +
t+ 4l hm n iu tng. T ta c y= x + 6thay vo (2) ta
c
x + x + 6 + x2 + (x + 6)2 = 80 x= 55 72 y= 55 + 5
2
Vy h cho c nghim : (x; y) =
5
5 72
;5
5 + 5
2
y ti a ra mt s cu h s dng tnh n iu ca hm s kh n gin. Ni l ngin v t mt phng trnh ta nhn thy ngay hoc mt cht bin i nhn ra dng cahm cn xt. Ti s cn gii thiu kh nhiu nhng bi cn bin i tinh t nhn ra dnghm, nhng cu sau ca cun sch.
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2.4 Cu 91 n cu 120 61
Cu 112
x + 4
32 x y2 = 34
x +
32 x + 6y = 24
Gii
iu kin : 0 x 32C v y l mt h khc rc ri khi xut hin cn bc 4. Ta s dng cc nh gi giiquyt ci h nyCng 2 phng trnh cho nhau ta c
x +
32 x + 4x + 432 x= y2 6y+ 21
Hin nhin ta c : V P 12Gi ta tin hnh nh gi v tri. p dng bt ng thcCauchySchwarzcho v tri ta c
x +
32 x
(1 + 1)(x + 32 x) = 8
4x + 432 x (1 + 1)(x +32 x) 4VyV T V PDu bng xy ra khi (x; y) = (16; 3)
Ti cn mt cu tng ging bi ny nhng hi kh hn mt cht. Bn c c th gii n
Cu 113
2x + 2 4
6
x
y2 = 2
2
42x + 26 x + 22y = 8 + 2
Nghim : (x; y) = (2;
2)
Cu 114
x2(y+ 1)(x + y+ 1) = 3x2 4x + 1xy+x + 1 =x2
GiiBi ny c l khng cn suy ngh nhiu. C th y+ 1 ln (1) coi saoNhn thy x= 0khng l nghim. Phng trnh (2) tng ng
x(y+ 1) =x2 1 y+ 1 = x2 1
x
Thay ln (2) ta s c
x(x2 1)x +x2 1
x = 3x2 4x + 1
x= 2 y= 52
x= 1 y = 1Vy h cho c nghim : (x; y) = (1;1),
2;5
2
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62 Tuyn tp nhng bi h c sc
Cu 115
4xy+ 4(x2 +y2) + 3
(x + y)2 = 7
2x + 1
x + y = 3
Giiiu kin : x + y= 0y l mt bi h khng n gin cht no. Tuy nhin ta c mt nhn xt kh tt sau y :
a(x2 +y2) +bxy =k(x + y)2 + l(x y)2
Gi hy phn tch4x2 + 4y2 + 4xy= k(x + y)2 + l(x y)2Cn bng h s ta thu c : 4x2 + 4y2 + 4xy= 3(x + y)2 + (x y)2Nh vy tng s l t n ph tng-hiu chng ? Cng c c s khi2x= x + y + x y. Nhvy tng s b l th. Bin i h thnh
3(x + y)2 + (x y)2 + 3
(x + y)2 = 7
x + y+ 1
x +y+x y= 3
ng vi t ngay. mt cht 3(x+ y)2 + 3
(x + y)2 = 3
x + y+
1
x + y
2 6. Nh vy
cch t n ca ta s trit hn.t x + y+
1
x + y =a, x y = b ta thu c h mi
b2 + 3a2 = 13a + b= 3|a| 2
a= 2, b= 1
a= 12
, b=7
2(L)
x + y+
1
x + y = 2
x y = 1
x + y= 1x y= 1
x= 1y= 0
Vy h cho c nghim (x; y) = (1; 0)
OK cha ? Tip tc thm mt cu tng t nh
Cu 116
x2 +y2 + 6xy 1(x y)2 +
9
8= 0
2y 1x y +
5
4= 0
Gii
iu kin : x =yH cho tng ng
2(x + y)2
(y x)2
1
(y x)2 +9
8= 0y x + 1
y x
+ (x + y) +5
4= 0
2(x +y)2 y x + 1y x2
+25
8 = 0y x + 1
y x
+ (x + y) +5
4= 0
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2.4 Cu 91 n cu 120 63
t x + y= a, y x + 1y x =b, |b| 2ta c h mi
a + b=
5
42a2 b2 = 25
8
a=
5
4b= 5
2
y+ x=
5
4y
x=
2
y+ x= 5
4
y x= 12
x=13
8 , y=
3
8x=
7
8, y=
3
8
Vy h cho c nghim : (x; y) =
7
8;3
8
,
13
8;3
8
Ti s a thm 2 cu na cho bn c luyn tp
Cu 117
3(x2 + y2) + 2xy+ 1
(x y)2 = 20
2x + 1
x y = 5
Nghim : (x; y) = (2; 1),
410
3 ;
10 3
3
,
4 +
10
3 ;
3 103
Cu 118
(4x2 4xy+ 4y2 51)(x y)2 + 3 = 0(2x 7)(x y) + 1 = 0
Th ng no mt cht xem v sao li a c v ging 3 cu trn ?
Nghim :(x; y) = 5
3
2
;1 +
3
2 ,5 +
3
2
;1 3
2
Cu 119
2x
2 +x 1y
= 2
y y2x 2y2 = 2
Giiiu kin : y
= 0
Phng trnh (2) tng ng vi1
y x 2 = 2
y2
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64 Tuyn tp nhng bi h c sc
t a=1
yta chuyn h v
2x
2
+x a= 22a2 + a x= 2
x= 1, a= 1x= 1, a= 1
x=132
, a= 3 12
x=
3 1
2 , a=
1 32
Vy h cho c nghim : (x; y) = (1;1),13
2 ; 1 3
Cu 120
4x2 + y4 4xy3 = 14x2 + 2y2 4xy= 2
Gii
Hnh thc kh gn nh nhng cng rt kh chi. Mt cht tinh ta nhn thy y2 = 1 lnghim ca h. Thay vo v ta rt ra
P T(1) P T(2) y4 4xy3 2y2 + 4xy+ 1 = 0 (y2 1)(y2 4xy 1) = 0
Vi y= 1thay vo (2) ta tm c x= 0hoc x= 1Vi y= 1thay vo (2) ta tm c x= 0hoc x= 1Vi y2 = 4xy+ 1. Khng cn ngh nhiu, th tru b vo cho nhanh !!!
Ta rt ra x=y2 1
4y thay vo (2) ta c
4
y2 1
4y
2+ 2y2 + 1 y2 = 2 5y4 6y2 + 1 = 0
y= 1 x= 0y= 1 x= 0y= 1
5 x= 1
5
y= 1
5 x=
1
5
Vy h cho c nghim :(x; y) = (1; 1), (1;1), (0;1), (0;1),
15
; 15
,
1
5;
15
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2.5 Cu 121 n cu 150 65
2.5 Cu 121 n cu 150
Cu 121
x4 + x3y+ 9y = y3x + x2y2 + 9xx(y3 x3) = 7
Gii
Khng cn bit T quc ni u, chin phng trnh u
P T(1) (x y)(x(x + y)2 9) = 0Vi x= y kt hp vi (2) r rng khng thaCn li ta kt hp thnh mt h mi
x (y3 x3) = 7x(x + y)2 = 9
y l mt bi ton kh quen thuc v hp dn tng xut hin trn bo THTT, cch lmph bin nht vn l "tru b"
Trc ht c nh gi x >0v rt ra y = 3
x3 +7
x. Thay xung ta c
x
x +
3
x3 +
7
x
2= 9 x3 + 2x 3
x6 + 7x2 + 3
x(x4 + 7)2 = 9
t v tri l f(x). Ta c
f(x) = 3x2 + 2
3
x6 + 7x2 + 6x6 + 14x2
3 3
(x6 + 7x2)2
+
1
3.9x8 + 70x4 + 49
3
x2(x4 + 7)4>0
Vyf(x) = 9c nghim duy nht x= 1 y= 2Vy h cho c nghim : (x; y) = (1; 2)
Tip theo ti xin gii thiu cho cc bn mt s cu h s dng Bt ng thc Minkowskigii. Bt ng thc Minkowski l mt bt ng thc khng kh v cng thng c dng,bt ng thc cp n vn di ca vect trong khng gian m sau ny hc sinh quen
gi n l bt ng thc V ectorVi hai vectu ,v bt k ta lun c
|u |+ |v | |u +v |Nu ta ha 2 vecto ny ta s thu c
a12 +b12 +
a22 + b22
(a1+ a2)2 + (b1+ b2)
2
ng thc xy ra khi (a1, a2)v(b1, b2)l 2 b t ly l mt h qu hay dng trong gii h
Th khi no nhn vo mt bi h ta c th ngh n s dng Bt ng thc Minkowski. Thngkhi nhn thy tng hai cn thc m bc ca biu thc trong cn khng vt qu 2 th ta cth chn hng ny. Ti s nu 3 v d bn c hiu r hn
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66 Tuyn tp nhng bi h c sc
Cu 122
3x + 4y= 26x2 + y2 4x + 2y+ 5 +
x2 + y2 20x 10y+ 125 = 10
Gii
tng s dng hin r ri. Bc u tin ta lm l phn tch biu thc trong cnthnh tng cc bnh phng . V tri ca (2) khi s l
(x 2)2 + (y+ 1)2 +
(x 10)2 + (y 5)2
Tuy nhin nu ta s dng Bt ng thc Minkowskingay by gi th n s l
V T
(x 2 +x 10)2 + (y+ 1 +y 5)2
Khng phi 10 na m l mt biu thc kh phc tp. Khi ta phi xem li cch vit cc
bnh phng ca mnh nu l hng s v phi th khi cng vo ta phi lm trit tiu n i. Vy cn phi vitnh sau
V T =
(x 2)2 + (y+ 1)2+
(10 x)2 + (5 y)2
(x + 2 + 10 x)2 + (y+ 1 + 5 y)2 = 10
Ok ri. ng thc xy ra khi 10 x
x 2 =5 yy+ 1
3x 4y = 10Kt hp (1) d dng gii ra (x; y) = (6; 2)
Nh ta thy, s dng khng kh. Tuy nhin ci kh y chnh l ngh thut i du vsp xp cc hng t ca bnh phng ta t c mc ich
Cu 123
x2 2y2 7xy= 6x2 + 2x + 5 +
y2 2y+ 2 =
x2 + y2 + 2xy+ 9
Gii
Xt phng trnh (2) ta c
V T =
(x + 1)2 + 22 +
(y 1)2 + 12
(x + y)2 + 32 =V P
ng thc xy ra khi x + 1 = 2(y 1) x= 2y 3Thay vo (1) v ta d dng gii ra (x; y) =
5
2;1
4
, (1;1)
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2.5 Cu 121 n cu 150 67
Cu 124
2x2 + 6xy+ 5y2 + 5 =
2x2 + 6xy+ 5y2 + 14x + 20y+ 25x4 + 25y2 2 = 0
Gii
By gi nu chuyn cn sang v tri, hng s sang v phi l cht d. Mu cht y l g ?S 5 chng ? ng vy, ta phn tch 5 =
32 + 42 s dng bt ng thc Minkowski. Tuy
nhin cc i du v sp xp s hng nh th no. Ci ta phi quan tm n v phi chn la cho ph hp. y s l
V T =
(x + y)2 + (x + 2y)2 +
42 + 32
(x + y+ 4)2 + (x + 2y+ 3)2 =V P
ng thc xy ra khi x + y
4 =
x + 2y
3 x= 5y
Thay vo (2) v ta d dng gii ra (x; y) =
1;15
,1;15
Cu 125
2y(x2 y2) = 3xx(x2 +y2) = 10y
Gii
Mt h a v dng ng cp r rng. Tuy nhin, ta hy x l s b h ny loi mt strng hpT (2) d thy x.yphi cng du, m nu th (1) x2 y2Trc ht x= y = 0l mt nghim ca hNhn cho 2 phng trnh cho nhau ta c
20y2(x2 y2) = 3x2(x2 +y2) (x 2y)(2y+ x)(5y2 3x2) = 0
V x v y cng du nn nn t y ta suy ra x= 2yhoc x=
5
3y
n y ch vic thay vo (1). Xin nhng li cho bn cVy h cho c nghim :(x; y) = (0; 0), (2; 1), (2,1),
4
30375
6 ;
4
135
2
,
4
30375
6 ;
4
135
2
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68 Tuyn tp nhng bi h c sc
Cu 126
7 +x +
11 y = 67 +y+
11 x= 6
Gii
Cng 2 phng trnh cho nhau ta c
7 +x +
11 x +
7 +y+
11 y= 12
p dng bt ng thc Cauchy Schwarzcho v tri ta c
V T
(1 + 1)(7 + x + 11 x) +
(1 + 1)(7 +y+ 11 y) = 12Du bng xy ra khi (x; y) = (2; 2)
Cu 127
2x2y2 + x2 + 2x= 22x2y x2y2 + 2xy= 1
Gii
Bin i 1 t, h cho tng ng
2x2y2 + (x + 1)2 = 3
2xy(x + 1) x2
y2
= 1 (xy+ x + 1)2 = 4
xy= 1 xxy= 3 x
Vi xy= 1 xthay vo (1) ta c
2(1 x)2 + x2 + 2x= 2
x= 0(L)
x=2
3 y=1
2
Vi xy= 3 xthay vo (2) ta c
2(x + 3)2
+ x2
+ 2x= 3 x=
8
3y =
1
8x= 2 y = 12
Vy h cho c nghim : (x; y) =
2
3;1
2
,
8
3;1
8
,
2;1
2
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2.5 Cu 121 n cu 150 69
Cu 128
(x 1)(y 1)(x + y 2) = 6x2 + y2 2x 2y 3 = 0
Gii
Bi ny tng t n ph r rngt x 1 =a, y 1 =b ta a v h sau
ab(a +b) = 6a2 +b2