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

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Nối tiếp nội dung phần 1 tài liệu Tài liệu ôn thi Đại học môn Toán - Sáng tạo và giải phương trình, bất phương trình, hệ phương trình, bất đẳng thức, phần 2 trình bày phương pháp giải hàm số trong các bài toán chứa tham số, phương pháp hàm số trong chứng minh bất đẳng thức. Mời các bạn cùng tham khảo.

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

V i dv 2: G i i i cac h$ phuong trinh sau: b) Dat ^ ^'^•^ dieu ki?n > 4P h§ phucmg trinh da cho tro thanh: P = x.y 2 2xy ^ 7 x 2 + y 2 + ^ / ^ = 8N/2 x^ + y a) c) x+y S ( s 2 - 3 P ) = 19 SP = -8S SP = -8S S =l S ^ - 3 ( 2 - 8 S ) = 1 9 ^ | S 3 + 2 4 S - 2 5 = O ' ^ 1 P = -6 7x + y = x^ - y S(8 + P) = 2 Suy ra x,y la hai nghi^m ciia phuong trinh: (x + y ) 1 + - = 5 xy x^y(l + y) + x V ( 2 + y) + x y ^ - 3 0 = 0 x 2 - X - 6 = 0c>Xj=3;X2=-2 b) d) x^y + x^l + y + y^ j + y - 1 1 = 0 V^y h? da cho c6 hai cap nghi^m (x;y) = ( - 2 ; 3 ) , ( 3 ; - 2 ) 1+ = 9 xV; 2 ( a ^ + b 3 ) = 3(a2b + b2a) Giii: c) Dat a = \/x, b = ^ h? da cho tro tharJi: a+b=6 a) Dgt Vx = a,7y = b dieu ki?n a,b > 0. fS = a + b Va^+b^+N/2ab = 8V2 D^t dieu kif n > 4P thi h^ da cho tro thanh. phuong trinh tro thanh: . Ta viet Igi h^ phuong P = ab a +b= 4 2 f s ^ - 3 S P l = 3SP [ 2 ( 3 6 - 3 P ) = 3P [S = 6 ^|ia + b)" - 4ab(a + b)^ + 2a^b^ + yflab = 8V2 trinh thanh: S=6 S=6 P =8 a +b=4 Suy ra a,b la 2 nghi^m cua phuong trinh: IS = a + b 19^ > 4P D^t -I „ , dieu ki?n x = 8 fa = 4=>x = 64 P = ab S,P>0 x 2 - 6 X + 8 = 0 o X i = 2;X2=4=> b = 4=i>y = 64 [b = 2 = > y = 8 V 2 5 6 - 6 4 P - 6 P 2 +N/2P = 8N/2 Vay h f da cho c6 hai cap nghi?m (x;y) = (8;64),(64;8) o S = P = 4«.a = b = 2 o x = y = 4 S=4 xy >0 Ngoai ra ta cung c6 the giai ngan gpn hon nhu sau: d) Dieu ki^n: . Dat ^ dieu ki^n >4P h f phuong trinh da x,y>-l P = x.y ^ 2 ( x 2 + y 2 ) + 2 7 ; ^ = 16 cho tro thanh: x + y + 27xy =16 S->/P=3 S>3;P = ( S - 3 f S + 2 + 2VS + P + l = 1 6 ^2^x^ +y^^ = x + y^{x-yf =0ox =yo2N/x=4ox =4 2^S + ( S - 3 f + 1 = 1 4 - S Vay h? CO mpt cap nghi^m duy nhat (x;y) = (4;4) 3£S
Cty TNHH M f V D W H Khattg Vift Vi > 4P,S > 0 suy ra + S - 2P > 0 . D o d o S = 1 x= : ;y = V o i X + y = 1 thay vao (2) ta du
Tdi li(u on III, ,t,u ho, s 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 2(x-y) < : : > ( x - y ) x^ + x y + y^ + 4(x - y ) + = 0 X =- V2x + l+72y + l (3-N/5 3-N/5^ 0 ? 2 -—;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
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 ta CO 16i giai nhu sau: + Hoac cac phuong trinh ciia hf khi nhan hoac chia cho nhau thi tao ra Vi X = 0 khong la nghiem cua h^ nen ta dat y = tx . Khi do h? thanh: phuong trinh dang cap. Ta thuong gap dang h^ nay 6 cac hinh thuc nhu: - 8 x = t^x-' +2tx ( l - t ^ ) = 2t + 8 ^_^3 t+ - :x i -U. ^ ax^ + bxy + cy^ = d ^ ex^+gxy + hy^ = k -^ r • ^ ^, t a ^ jax^+ bxy + cy^ =dx + ey 4 .-t-^ , < » 3 ( l - t ^ ) = (t + 4 ) ( l - 3 t ^ j « 1 2 t ^ - t - l = O o 3 gx + hxy + ky = Ix + my t = - i 4 ax + bxy + cy = d ' '(i-st^j^e gx'^ + hx^y + kxy^ + ly^ = mx + ny x = ±3 t = -=> 3 x y=±r 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: 4N/78 Phuong phap chung de giai h^ dang nay la: Tu cac phuong trinh cua h? ta x= ± 13 nhan hoac chia cho nhau de tao ra phuong trinh dang cap bac n : t = --=> 4 alx"+a,x"-^y^... + a „ y " = 0 y-+- 13 Tu do ta xet hai truang hgp: Suy ra he phuong trinh c6 cac cap nghiem: + y = 0 thay vao de tim x ^4^78 VTSI/ 4^78 VTS' (x;y)=(3,l);(-3,-l); + y 5^ 0 ta dat x = ty thi thu dugc phuong trinh: ajt" + a^t""''.... + a„ = 0 13 ' 13 ' 13 ' 13 + Giai phuong trinh tim t sau do the vao h^ ban dau de tim x,y Phuong trinh (2) ciia he c6 dang: Chii y: (Ta ciing c6 the dat y = tx ) xy|x^ + y^j + 2 = x^ +y^ + 2xy o |x^ + y^ j(xy -1) - 2(xy -1) = 0 • Vi d\ 1: Giai cac h | phuong trinh sau: xy = 1 x ^ - 8 x = y^ + 2y ( x y - l ) ( x ^ +y^ - 2 ) = 0 x2+y2=2 a) x 2 - 3 = 3 I..2 y^+1 5 x ^ ' - 4 x v ' + 3v•^-2(x + y) = 0 [x = l , [ x = - l 3^^Y THI: , ^ '
Tdi li?u on thi dai hQC sang tao vd giai PT, hat PT, hf fl, mi t>J -histuySiTrw^^^_ lang vtfT Tu do ta CO loi giai nhu sau: Ta thay y = 0 khong la nghi^m ciia h?. i ,\ . Dlit Vy = ^ y = t^x^ thay vao (1) ta du(?c: + = "t^^ Xet y ^ 0 d§t X = ty thay vao h? ta c6: St^y^ - 4ty^ + 3y^ = 2 (ty + y) Riit gpn bien x ta dua ve phuong trinh an t: -r,,rs;,u t2y2+y2 =2 ( t - 2 f (t2 + t + lj = 0 o t = 2 0 . ' ' Chia hai phuoiig trinh aia h? ta Avtqc: Thay vao (2) ta du^c: 5^!zilll = l l i o t 3 - 4 t 2 + 5 t - 2 . 0 t^+l 4x2+8x = V2x + 6 «.4x2+10x + — = 2x + 6 + V2x + 6 + - 2N/2 2^/2 V2x + 6 + i 4 4 t=l x-y x = l fx = - l X =• 2j 1 » x = —y Giai ra ,ta dug-c . x = >/l7-3 => y = 13-3N/I7 y = l [y = - l y=- s/5 —• Vi dv 2: Giii cic phucmg trinh sau: V|iy nghifm ciia hf (x;y) = ^ / l 7 - 3 13-3N/I7^ 7x2+2y+3+2y-3 =0 2(2y3 + x^) + 3y (x + i f + 6x(x +1) + 2 = 0 Vi dv 3: Gidi cac hf phuong trinh sau: 3x3-y3=^ x^^y + l - 2 x y - 2 x = l 1 2 x _ x + 7y a) \ + y b) x^ - 3x - 3xy = 6 b) 3x 3y 2x2+y x2+y2=l 2(2x + 7y) = V2x + 6 - y Giii: Giii: a) Ta CO the viet lai h? thanh: 3x3-y3)(x + y) = l (1) a) Dieu ki^n: x^ + 2y + 3 ^ 0. x2+y2=l Phuong trinh (2) tuoTig duong: Ta thay ve trai ciia phuong trinh (1) la bac 4. De tao ra phucmg trinh dang 2(2y3 + x^) + 3y(x +1)^ + 6x2 + 6x + 2 = 0 o 2(x + i f + ^. ^ ^Q cap ta se thay ve phai thanh (x^ + y^ )2. Day la phuong trinh diing cap giua y va x + 1 . Nhu vay ta c6: + Xet y = 0 h^ v6 nghi^m x2+y2 0 2x''+3x3y-2x2y2-xy3-2y''=0 ' + Xet y 7t 0. D§t x +1 = ty ta thu dug-c phuong trinh: 2t^ + St^ + 4 = 0 3x3-y3 ^^^^y Suy ra t = -2 X +1 = -2y x=y o (x - y)(x + 2y)(2x2 + xy + y^) = 0 x = -2y Thay vao phuong trinh (1) ta duQc: Vx^ - x + 2 = x + 4 o x = — ^ => y = , 2x2 + xy + y2 = 0 9 * 18 V$y h^ CO mpt c^p nghi^m: (x; y) = 14._5_ Neu 2x2 + xy + y2 = 0 —x2 + / N2 X+y = O o x = y = 0 khongthoaman. I 9'is; 2 b) De thay phuong trinh (1) ciia hf la phuong trinh dang cap ciia x va yjy Neu X = y ta c6 2x2 = 1 o x = ±
V e t r a i ciia cac p h u o n g t r i n h t r o n g h? la p h u o n g t r i n h d i n g cap bac 3 doi •u vol x,yjy .De thay y > 0 . Ta dat x = t ^ thi t h u d u g c h?: + Neu x = - 2 y < » 5 y ^ = l » y = + — T o m lai p h u o n g t r i n h c6 cac cap nghi?m: ^ ( 2 t + t^) = 3 t2+2 3 ,2 . . . '2N/5 -^/5^ f-275 4£ = - o 2 t ^ - 3 t + l = 0 t= l (x;y) = 2 2 ' 2 5 ' 5 5 ' 5 + Ne'u t = 1 t h i X = ^y o X = 1 => y = 1 x2 7 y 7 T - 2 x ( y + l ) = l 5 b) Dieu ki?n y > - 1 . Ta viet lai h ^ thanh: x'^''-3x(y + l ) = 6 + l t hthi Ne'u t = — i xx = —l ^ y < : > y = 4xx"'= — o x = 2 2 ^ ^ ^ ^ ^ Ta thay cac p h u o n g t r i n h ciia h$ 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 1 4 T o m lai he c6 cac nghiem: ( x ; y ) = ( l ; l ) . 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 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 t= 0 hai ta t h u dugc: » t ^ - 3 t - 6 ( t 2 - 2 t ) = 0 0 . 3y 4 T r u o n g hop 2xy + x^ = 3 ta c6 h?: 2xy + x"' = 3 P h u o n g t r i n h (2) t u o n g d u o n g : ,t. ,if .. xy + X'
4x + 3y n , ; — ft ^4 t^4 Sy"" 6 V 12y 8y l 6 6 J V8t^-3t2+4 =4-to t ^ l phuong trinh tro x^ x-* Vx Vxj X X = 6y X 4x + 3y 2 • By 6 thanh: t^ + 31^ - 1 = 3(Vt - y f t ^ f o ( t ^ + 3t^ - l ) ( V t + yft^f =3 T H l : X = 6y thay vac (1) ta c6: Xet ham so' f(t) - (t^ + 3t^ - lj(>/t + r / T ^ ta thay t = 1 th6a man phuong trinh. 28 168,-, y= =>x = (L) | y 2 + y 2 - 1 6 y = 16 ^ 3 7 37 ^ ' Xet t > l . T a c 6 f ( t ) = f3t^ + 6t]fN/F + V T ^ f + - ^ ^ . ^ ^ ^ — j + 3t^ - 1 ] > 0 4 24 \2 Vt.Vt-1 ^ ' y = —=>x = — 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 T H 2 : X = - - y thay vao (1) ta c6: ra phuong trinh c6 nghi?m khi va chi khi t = 1 o x = 1 4 , , Tom lai h^ phuong trinh c6 nghi^m ( x ; y ) - ( l ; l ) y= -^(L) - y + y - 1 6 y = 16 C h u y: Ta cung c6 the tim quan h? x,y d y a vao phuong trinh thu hai ciia y = 12=>x = - 8 ( T M ) h^ theo each: r24 4^ Vay h? C O nghi?m (x; y ) = ,(-8;12). Phuong trinh c6 d^ing: . 7 7 V8x^-3xy^4y^-3y^7;:^-y = 0 o , (^"yK^^^^y) ^(pOy^p xy >0 [x,y>0 78x2-3xy + 4y2+3y V'^y + y b) Dieu ki^n: x 0 nen ta suy ra x = y . De y rang phuong trinh thu hai cua h? la phuong trinh dang cap doi voi ^ / 8 x 2 - 3 x y + 4y2 + 3 y V^y+y x , y . T a thay neu y = 0 thi tit phuong trinh thu hai cua h^ ta suy ra x = 0, cap nghi^m nay khong thoa man h?. Xet y>0. T a chia phuong trinh thu hai cua h? cho y ta thu dug-c: x ''•.••.in -3-+4+ -=4. D|t thu duQfC phuong trinh y y
TTFTr PHLTONG P H A P BIEN O O l TL/ONG DLTONG Dat t = Vx + 1 + 7 4 - x > 0 => \lx + l.y/4-x = - y ^ • Thay vao p h u o n g t r i n h t2-5 t = -5 Bie'n dot titvng ditong la phteang phdp gidi he dua tren nhimg ky thudt ca ban ta c6: t + = 5ci> t'^ + 2 t - 1 5 = 0 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 * Ta xet cac v i d u sau: x =3 V i d u 1: G i a i cac 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 ^ + V 4 - 2 y + x/5 + 2 y - ( x - l ) 2 ^5 a) 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 3 x ^ + ( x - y ) 2 = 6 x 3 y + y2 (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 M a t khac theo bat d^ng thuc Co si ta c6 b) + 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) x-''-12x = ( y - 2 ) 3 - 1 2 ( y - 2 ) \^ +4y^ = 3 x + 6y^ - 4 b) He viet lai d u o i dang X + x ( y - 4 ) + (y-3r =0 y^-7x-6-^y(x-6)=l Dat t = y - 2 . Ta c(S he : d) sJ2(x-yf + 6x - 2y + 4 - 7y = Vx+T x^^-12x = t - ' ' - ] 2 t (x-tKx^+t^+xt-i2)=o n . Giai: x^ + x ( t - 2 ) + ( t - l ) ^ = 0 x ^ + t ^ + x t - 2 ( x + t) + l = 0 (2*) x>-l x^+t^+ xt-12 = 0 (3*) T u (*) suy ra a). D i e u k i ^ n s y < 2 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: 3x''-6x-V + { x - y ) 2 - y 2 = 0 T u day suy ra 2 n g h i e m cua h ^ la ( x ; y ) = ( l ; 3 ) . '1 V 3'3 x=0 Vol (3*) ke't h o p v o l ( 2 * ) ta c6 h ^ X = 2y X+t = — 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 ( V N ) . D o (x + t f < 4 x t Theo ba't dang thuc Cauchy-Schwarz ta c6 (x + t)^ - x t - 2 ( x + t) + l = 0 = 0 121 xt = 7 4 - 2 y + 74 + 2 y ) ^ < 2(4 - 2y + 4 + 2y) = 16 7 4 - 2 y + 74 + 2y < 4 'I 7^ 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 [3'3) H ? CO n g h i e m : ( 0 ; 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 ) 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)-^ NATTT + V 4 - X + ^5 + \-{x-lf =5 + V 4 - X + 7(x + l ) ( 4 - x ) = 5 (*)
D$t: a = x + l; b = 2y-l. 2(x + l - y ) 2 + ( V ^ - V y ) ^ = 0 « | ' ' / ^ ^ ^ < » x + l = y Khi do ta thu dug-c h§ phuong trinh: [ Vx +1 = ^ y ab = 2 ab = 2 ^y,, Thay vao phuong trinh (2) ta c6: ,3+ib3=3a2+|b-5' 2a^+b^=6a^+3b-10 ^ 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 h# nay c6 nghi?m khi: a = 2; b = 1 a>-l [(a-2)b = 2 ( l - b ) a^-1 a =0 Dod6tasephantichheved,ng:|^^_^^,^^^^^^^^_^^,^^^^^^ Va^ +1 = a +1 < o a^-a2-2a = 0 a = -l ^ 2(l-b) a=2 y = 0=>x = - l Vi ta luon c6: b ^ 0 nen tu phuong trinh tren ta rut ra a - 2 = — - — Voi a = 0: y = 7=>x = 6 The xuong phuong tririh duoi ta dugc: 7-3S 5-3S y= =>x = — i ^ ^ ^ ( a +1) = (b - l)Hh + 2) o (b - if [4(a +1) - b2(b + 2)] = 0 Vol a = - l = > y 2 - 7 y + l = 0 ^ 2 2 7 + 375 5 + 3>/5 b=l y = —-—=>x = ^ 4(a + l) = b2(b + 2) y=-l (L) Voi a = 2 r : > y 2 - 7 y - 8 = 0 0 Voi: b = 1 => a = 2 , suy ra: x = y = 1; . b+2 y = 8=>x = 7 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 = - ^ 5-3N/5 7 - 3 V 5 ' 5 + 375 7 + 375 (x;y) = (-l;0),(6;7). ,(7; 8) • The len phuong trinh tren taco: b = -2 => a = -1 o X = -2; y = - - i ( ^ = b^(b.2). b Vi 2: Giai cac phuong trinh sau b^ = 4 (Khong TM) ,2 x^-(2y + 2 ) x - 3 y 2 = 0 x^ - 2 x y + 2y^ +2y = 0 a) V^y h? da cho c6 2 nghi^m la: (x;y) = (1;1), x^ + 2xy2 - (y + 3)x - 2y^ - 6y2 +1 = 0 x^-x2y + 2y2+2y-2x =0 x>-l x^ + x y + 9y = y^x + y^x^ + 9x xy^ - 3x^y - 4yx^ - y + 3x^ = 0 d) Dieu ki^n: . Ta Viet lai h? phuong trinh thanh: c) d) y >0 x(y3-x^) = 7 3x^y-y2+3xy + l = 0 ^ 2 ( x - y ) 2 + 6 x - 2 y + 4 - ^ = Vx+T Giii: a) Cach 1: Lay phuong trinh thu hai tru phuong trinh thu nhat theo ve ta ^ 2 ( x - y ) ^ + 6 x - 2 y + 4 = ^y + Vx + 1 . Binh phuong 2 ve ta thu dugc: du(?c: 2xy^ - (y + 3)x - 2y^ - 6y^ +1 + (2y + 2)x + 3y2 = 0 2x^ - 4 x y + 2y^ + 6x - 2y + 4 = x + y +1 + 2^y(x +1) O 2xy2 + xy - 2y3 - 3y^ +1 - X = 0
1 2 3±2-\/3 + Neu y = - thay vao phuong trinh (1) ta c6: 4x - 12x - 3 = 0 x = ^3 - X - X"^ = 7. (5). D|itt = V ^ ( t > 0 ) . (5) CO d^ng nhu sau + Neu y = X - 1 thay vao phuong trinh (1) ta c6: - 2x^ - 3(x -1)2 = 0 o -4x2 + 6x - 3 = 0 . V6 nghi^m. l-t3 -t^ = 7 « t ^ - ( 3 - t ^ ) 3 + 7t = 0. • 3 - 2 V 2 1^ 3 + 2N/2 t Ketluan: ( x ; y ) = ( ^ ^ ; l ) , ( - ^ / 3 ; l ) , 2 '2 2 '2 Xet ham so £(t) = t^ - (3 - 1 ^ )3 + 7 t (t > 0). Ta c6 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
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) 36x2 ^ _ ;j5jj^2 ^ _ j5 a) b) x^ 2x - y^ + 9y = x(9 + y - y ^ +— = Xet phuang trinh (*) 36x2 ^ _j g j p ^ _ j 8y 3 4 2 3x-6V2x-4=473y-9-2y x ^ y - 8 y ^ + 3x^y = - 4 Vi x = 0 khong phai la nghi^m. Ta chia hai ve phuang trinh cho x^ ta c6: c) 6 x ^ - 3 x ^ y + 2xy + 4 = y^+4x + 6x^ [2xy + y - y ^ = 2 15 t=2 36 = X x + 16- Dat X - — = t = > t 2 + 1 6 t - 3 6 = 0 x t = -18 Giki: x=5 a) Tir phuong trinh (2) ciia h§ ta c6: + Neu t = 2x- — = 2 < : > x 2 - 2 x - 1 5 = 0 ( x - y ) ^ x + y^ - 9 x + y3-9 = 0 + N e u t = -18 Vi y < 1 va ^ 1 + x + ^ 1 - y = 2 nen ^ 1 + x < 2 o x < 7 . X = -9-4N/6 O X - — = - 1 8 < » x 2 + 1 8 x - 1 5 = 0 Do do x + y'^ - 9 < - 1 < 0 nen x + y'' - 9 = 0 v6 nghi^m. X X = - 9 + 476 Ta chi can giai truong hgp x = y . The vao phuong trinh ban dau ta _9-4V^;^^±1276] N g h i f m ciia h f da cho la: (x; y) = dugc. ^ 1 + x + y J T ^ = 2. D l t a = ^ l + x;b = V T o c (b > O) thi TH 2: x = 2y Thay vao phuang trinh thii hai ciia h$ ta c6: .^"^'^"^ =^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 / x^ 2x 2x^ x^ X 7 llx^ -+ o—x =. X = 0 (loai) (do dieu ki§n y^O) Tir do suy ra nghi^m cua phvrong trinh ban dau ; 4x 3 V 3x 4 4 6 V 12 x = 0;x = - 1 1 + 6N/3;X = - 1 1 - e V s / KL: Nghi^m ciia h$ da cho la: (x;y) = f Vay h f da cho c6 3 nghi^m la x = y = 0;x = y = - l l + 6V3; x = y = -11 - 6\/3 V 2y = x x>2 b) Phuang trinh t h u nhat ciia h? (2y - x)[\^ - 12y - is) = 0 o ) Dieu ki§n x^-15 y>3 y=- 12 Phuang trinh (2) ciia h^ tuang duong vai: x^ -15 THI: y = thay vao phuang trinh t h u hai cua h | ta duQc: y = 2x-2 12 ( 2 x - 2 - y ) ( 3 x 2 + y - 2 ) = 0 y = 2-3x2 3x^ 2x 4x^ x2 x2-15 • + — = , Voi y = 2x - 2 the vao phuang trinh (1) ta dugc: 2(x2-15) 3 Vx2-15 24 (1)«7X-6N/2X-4-4V6X-15-4 =0 (3) 36x^ Den day su dung bat dSng thuc Co si ta c6: •-12, x2+16x-15) + (x2+16x-15) = 0 x^-lS ^'"Vx^-lS f6N/2x-4=3.2V2(x-2)0 x2+16x-15^0 , 1— =i>6V2x-4+4V6x-15^7x-4 4V6x - 1 5 = 2.273(2x - 5) < 2(2x - 2) 6 36 = x^ +16X-15 Dau " = " xay ra khi chi khi x = 4 , x2-15 Tir (3) suy ra x = 4 la nghi^m duy rthat. V^iy h? c6 nghi?m (x;y) = (4; 6)
j:tyTNHHMlV DWHKhangVift - V o i y = 2 - 3 x ^ 3 Ta c6: x^+4+x vx^+4-x =4; y y ^ + 4 + y y y ^ + 4 - y = 4 nen ta 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 duQC phuong trinh : •y/x^+4+x = - ^ y ^ + 4 + y suy ra: • X = y . - x^y - 8 y * + Sx^y = - 2 ( 2 x y + y - y^) (x^ - 8 y ^ + 3x^ ) y = (-4x - 2 + 2 y ) 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^ + Sx^ + 4x = 8y^ + 2 y - 2 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 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 : x2 + 4x + 4 - 6(2x - 1 ) - ^(x^ + 4 x + 4 ) ( 2 x - l ) = 0 . Chia phuong trinh cho + 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 , x^ + 4 x + 4 Ix^ + 4 x + 4 x^ + 4x + 4 > 0 . Ta CO - 6 = 0. 2x-l 2x-l 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 =>x = l t= 3 DMt t = J — > 0 t h u d u g c p h u o n g t r i n h : t^ - 1 - 6 = 0 =>t = 3 3 y ^ - y - 2 = 0 b) 2y-yx^+2y + l = ( x - y ) y(y-x) = 3-3y +2y + l = x + y 2x + ( 3 - 2 x y ) y 2 = 3 THl: ^ + 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: c) 2x2-x3y = 2x2y2-7xy + 6 3y>x f3y.: x = l ; y = l(TM) 6xy = 9y^ - 2 y - l o 415 1 7 , ^ „• x^ + 2xy + 6 y - (7 + 2 y ) x ^ = - 9 x'' + 2y + 1 = 9y^ - 6xy + x^ x = — ; y = -(TM) d) xy = y^ + 3 y - 3 Giai: TH2: ^ + 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 : 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: x + y>0 x + y >0 x = l;y = l logj x + Vx +4 + l o g 2 yjy +i-y =log2 4 2xy = - y ^ + 2y + 1 o x = —;y = —(L) V / V J xy = y^ + 3y - 3 21^ 3 x^ + 2 y + 1 = x^ + 2 x y + y^ o x +Vx^ +4 .^yjy^ + 4 - y = 4 415,17 V | y h ? c 6 n g h i # m (x;y) = ( l ; l ) . 51 ' 3
CtyTNHH. angViet c) Tu phuong trinh (1) ta thay: 2x(l - y^) = sjl - y^). + xy + -3 = y^ 3 x2 + y2 + X= 3 THI: y = l thay vao (2) ta c6: -7x+ 6 = 0 x = l;x = 3;x =-2. c) d) - 4 y 2 . ^ =_l f2x + 2xy + 2xy2 =3 + 3y (•) (xy + 2 f + - l - = 2y + i x + y-1 TH2: Ket h
Cty TNHH MTV D W H miang Vift Tai li$u on thi dai hoc sdng Uio va giai PT, bat PI, he PI. bai Pi \^^„i,c,t intn^]^'l 4(x + y + 2)r(x + 2)^-(x + 2)y + y2 =3y(y2+2y + x y - 4 + 4) = 3y2(x + y + 2) TH2: Ket h(?p vol (1) ta c6 h# moi: • ''^ xy + y +1 [x +y^+x = 3 / (x + y + 2)(^4(x + 2^ -4(x + 2)y + y2 = 0. Giai bang each: P T ( l ) - P T ( 2 ) « 3 y 2 + x y + x - y - 4 = 0
16-x x=l / , 7= + 1 = > 0 . Do do X = y thay vao phuone trinh (1) 473x(x + 2) = 3(x + 3) 27 thu dup-c: 2x = 3(Vx + 3 + V x - S J '' ' ' Tom lai c6 nghi?m duy nhat: ( x ; y ) = ( l ; l ) b) Dieu kifn: x , y > 0. Ta viet lai phuong trinh (1) cua h ? thanh: ( x - 6 f (x2+3x + 9) = 0 o x = 6 V|y h0 CO nghi?m x = y = 6. ^ x y - ( x - y ) ( ^ - 2 ) - y + >^-7y =0 (*). De thay x = y = 0 khong thoa x^y man h^. T a xet x^ + y^ 0 d) Dieu ki$n: 1 7 x>-;x^ - x - y > 0 NhMenh^pntaco: - J - V ) ^ - ! ^ - ^ ) ' . ^ ^ =0 ^xy + ( x - y ) ( ^ - 2 ) + y Vx+^y Phuong trinh dau ciia h? duq>c viet l^ii n h u sau: .(x-y) =0 x-y-1 x^ - y^ - x - y + , ^ =0 '•J I ' , ' Tir phuong trinh thu hai ciia h^ ta c6: i('^-y)'+^+i 2 yjx^ - x-y +y 4 2 /— 4 -5 ( x - l ) (x + 2) x + 1- + X - X . y^ + J^^^- 2 = — x+1 + x2-x-2--^^ x+1 ^>0 «(x-y-l) =0 suy ra X = y thay vao phuong trinh thu hai cua h^ ta c6: Si^-yf +^x-y +l V x 2 x- x+ -yy + 'x = l (x + l ) ( 3 x - x ^ ) = 4 o l±Vi7 Mat khac tu phuong trinh (1) ciia h f ta c6: ^ > 0. X =• ^N-y ' l + ^/l7 1 + N/I7 Neu y3/;r7
-x^ - 3 x y - 8 x + 4y^ +13y + 9 = 0 x^ + (3y + 8)x - (4y2 + 13y + 9J = 0 V i dv 7) Giai phuong trinh v6i nghi^m la so thyc: +2y^ +2x + 8y + 6 = 0 2x^ + 2xy + y - 5 = 0 Ta C O A = (3y + 8)^ + 4(4y^ + 13y + 9) = 25y^ + lOOy +100 = (5y + lO)^ b) a) y2 + xy + 5x - 7 = 0 + xy + y + 4x + l = 0 3y + 8-(5y + 10) x= ^ = -V y. -1 Giai: 2 Tu do tinh du^c: 3y + 8 + (5y + 10) x = u + a thay vao phuong trinh (1) cua h? ta c6: x= = 4y + 9 * Cachl:Dat y=v+b Phan vi?c con lai la kha don gian. n,u (u + a)2+2(v + b)2+2(u + a) + 8(v + b) + 6 = 0 v * >; b) Lay phuang trinh (1) tru phuang trinh (2) ta thu dugc: .s ^ o +2v^+2(a + l)u + 4v(b + 2) + a^+2b2+2a + 8b + 6 = 0. 2x*^ + 2xy + y - 5 - ^y^ + xy + 5x - 7j = 0 o 2x^ + (y - 5)x - y^ + y +12 = 0 Ta mong muon khong c6 so h^ng b^c nhat trong phuang trinh nen dieu a+l= 0 a=-l ki?n la: b + 2=:0 b = -2 x=-y+2 X = u-1 Tu do ta CO cac h dat an phu nhu sau: Dgt thay vao h$ ta c6: Nhan xet: Khi gap cac he phuong trinh dang: y = v-2 a j X ^ + ajxy + a3y^ + a4X + agy + =0 u2+2v2=3 b j X ^ + b2xy + b3y^ + b 4 X + b j y + b^ = 0 day la h$ dSng cap. u^ +uv = 2 + Ta dat x = u + a,y = v + b sau do tim dieu ki?n de phuang trinh khong c6 so' u = V hang bac 1 hoac khong c6 so' hang tu do. T u h? ta suy ra 2(u^ + 2v^ j = S^u^ + uvj u^ + 3uv - 4v^ = 0 u = -4v + 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. vao tinh chat. Phuang trinh ax^ + bx + c bieu dien du^c thanh dang: Cach 2:Ta cong phuang trinh (1) vai k Ian phuang trinh (2). (Ax + B)^ci>A = 0 +2y^+2x + 8y + 6 + k x^+xy + y + 4x + l =0 Doi voi cac d^i so bac 3: (l + k)x^ +(2 + 4k + ky)x + 2y^ +8y + ky + k + 6 = 0 Ta C O the van dung cac huang giai Ta CO + Bie'n doi h^ de tao thanh cac hSng dang thiic A = (2 + 4k + ky)2-4(k + l)(2y2+8y + ky + k + 6) + Nhan cac phuang trinh voi mpt bieu thiic d^i so' sau do cpng cac phuang = (k^ - 8k - 8)y2 + (4k2 - 32k - 32)y + Uk^ - 12k - 20 . Vi dy 8) Giai h^ phuang trinh vai nghi^m la so' thyc: Ta mong muon A c6 d^ng (Ay -hB)^ o A = 0 c6 nghi^m kep: x^ + 3xy^ = -49 x^ + 3x^y = 6xy - 3x -49 a) i c) x^ -8xy + y^ =8y-17x x^ - 6 x y + y^ =10y- 25x-9 o (4k2 - 32k - 32)^ - 4(k2 - 8k - 8)(l2k2 - 12k - 2o) = 0 o k = - | x3-y3=35 x^ + y^ = (x - y)(xy - 1) Tu do ta C O each giai nhu sau: b) . d) 2x^ +3y^ = 4 x - 9 y x^ - x^ + y +1 = xy(x - y - 1 ) Lay 2 Ian phuang trinh (1) tru 3 Ian phuang trinh (2) cua h? ta c6:
Giii: Trir hai phuong trinh cho nhau ta c6: y = - 1 thay vao thi h? v6 nghif m +3xy^ +49 = 0 . 1 3 + 3N/5^ 1 3-32/5' a) Phan tich: Ta viet lai nhu sau: KL: Nghi^m cua h# la: (x;y) = y2+8(x + l ) y + x2+17x = 0 2' 4 2' 4 -3y2+48 = 0 Nh?n thay x = - 1 thi tro thanh: y = ±4 iity y2-16 = 0 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$t thanh cac an phy moi lam don g i ^ cau true cua phuong trinh, h^ x^ + 3xy^ + 49 + 3(x2 - 8xy + y^ - 8y + 17x) = 0 phuong trinh. Qua do tao thanh cac h? phuong trinh m o i don gidn hon, hay quy ve cac d^ng h ^ quen thupc nhu doi xung, dla\ cap... o ( x + l)r(x + l ) 2 + 3 ( y - 4 ) 2 ] = 0 De t^o ra an phy ngudi giai can xu ly linh ho^it cac phuong trinh trong h? T u do ta de dang tim dugc cac nghi^m cua h?: {x;y) = ( - l ; 4 ) , ( - l ; - 4 ) thong qua cac ky thuat: Nhom nhan t u chung, chia cac phuong trinh theo nhung so'hang c6 sin, nhom dya vao cac hang dSng thuc, doi bien theo dac b) Lam tuong ty nhu cau a thii phuong trinh... Lay phuong trinh (1) cpng voi 3 Ian phuong trinh (2) thi thu du(?c: Ta quan sat cac VI dy sau: (x +1) (x +1)^ + 3(y - 5)^ = 0 . T u do de dang tim dugc cac nghi^m cua h^. V i dy 1: GiAi cac h^ phuong trinh sau c) Lay phuong trinh (1) t r u 3 Ian phuong trinh (2) ta thu dugc: 2 x 2 - 2 x y - y 2 =2 x*-4x^ + y ^ - 6 y + 9 = 0 ( x - 2 ) 3 = ( y + 3)^x = y + 5 a) < b) < 2x^-3x2-3xy2-y3+l = 0 x^y + x^ + 2 y - 2 2 = 0 Thay vao phuong trinh (2) ta c6: y = -3 Giii: 2(y + 5)2+3y2 ^4(y + 5 ) - 9 y o 5 y 2 + 2 5 y + 30 = 0 o a) Ta viet lai h? phuong trinh thanh: y = -2 3 x 2 - ( x + y r =2 3x2-(x + y)2=2 Vay h? phuong trinh c6 cac nghi^m la: (x;y) = (2;-3),(3;-2) [3x3+3x2y-(x + y ) 3 - 3 x 2 = - l [3x^{x + y)-(\ y f = - 1 \" d) Lay 2 Ian phuong trinh (2) tru d i phuong trinh (1) ta thu du(?c: ( x - l ) r y 2 - ( x + 3)y + x 2 - x - 2 =0 .2 a-b^ =2 D§t a = 3x ,b = x + y ta thu du^c h$ phuong trinh: o Truong hg^p 1: x 1 h# v6 nghi?m [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 Truong hgp 2: x^ +y^ = ( x - y ) ( x y - l ) thuduQc: ( b 2 + 2 ) b - b ^ - ( b 2 + 2) = - l o b 2 - 2 b + l = 0b = 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 a= 3 x2=l y=0 , , Khi b=l ' x+y=1 x=- l ^ NT- 1 3±3>/5 + Neu X = — => V = Tom l ^ i h? phuong trinh c6 2 c$p nghi?m: (x; y ) = ( l ; O),(-1; 2) - y^ - ( x - l ) y + x^ - x + 2 = 0 + Neu y ^ - ( 2x - l ) ^y + x ^ -4x + 2 = 0 tacoh^: y 2 - ( x + 3)y + x 2 - x - 2 = 0
-cry iNHHmiVDWHKftangVifr b) Ta viet l?i phuong trinh thanh: (x^-2f.(y-3f=4 Dat x^+v^ y + ~1 = a; X + y +1 = b . Ta c6: x^y + x^ + 2 y - 2 2 = 0 D | t a = x^ - 2; b = y - 3 . Ta CO h? phuong trinh sau: ab = 25 x2+y2=5(y + l) x = 3;y = l • a = b = 5 a2 + b 2 = 4 a2+b2=4 j(a + b ) 2 - 2 a b = 4 a + b = 10 x+y=4 x = - - - =11 < (a + 2)(b + 3) + a + 2 + 2(b + 3) = 22 [ab + 4(a + b) = 8 [ab + 4(a + b) = 8 V|iy CO nghiem (x; y) = (3; l). ' 23 ' 11^ a + bi=2 2 (a + b r + 8 ( a + b)-20 = 0 ab = 0 b) Dieu ki^n: x ?t y. ab + 4(a + b) = 8 a + b = -10.(L) da cho tuong duong: ab = 48 (y-x) 2(x + y f - y - x + - 25 „ a = 2,b = 0 Xet a + b = 2 a = 0,b = 2 . y-x ab = 0 x = ±^/2 ^-x + -y - x j y-x +- y-x + Neu: a = 0 , b - 2 y = 5 Dat x + y = a; y - X + = b; b > 2 h | thanh: x = ±2 + Neu a = 2,b = 0=>. y-x y =3 5 Tom lai hf c6 cac cap nghiem: (x;y) = (%/2;5J,^-\/2;5J,(2;3),(-2;3) y +x= — a + b=: — 5 ^ 4 13 3 4 a =— y - X = -2 Vi d\ 2: Giai cac phuang trinh sau 4 y +x= - X = —;y = — 2a2-b2=-^ 4 x^ +y^ +6xy L _ + ^=:0 b= -5 8^ 8 1 ^7 ^3 a) x2 + y 2 j ( x + y + l) = 25(y + l) b) (x-y)2 8 + xy + 2y^ + x - 8 y = 9 y-x=-- 2 y - i - . ^ =0 ^ x-y 4 V^y hf CO nghifm (x;y) = ^7 3Vl3 3) 8'8 8 ' 8 Giai: Vi dvi 3: Giai cac h^ phuang trinh sau a) De y rang khi y = -1 thi hf v6 nghiem xVl7-4x2 +y7l9-9y2 =3 x^ + X y^-4y2+y + l = 0 a) ( x 2 + y 2 ] ( x + y + l ) = 25(y + l) V17-4x2 + ^ 1 9 - 9 y 2 = 1 0 - 2 x - 3 y xy+ x 2 y 2 + l - ( 4 - x ^ ) y 3 =0 Xet y 7i - 1 . Ta vie't lai h? thanh: i Giai: [ x 2 + y 2 + x ( y + l) + (y + l) =10(y + l) 1 Dieuki^n: - : ^ < x < : ^ ; - ^ < y 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 ^ ^ ' 2 2 3 3 y+1 • - o x2+y2 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 x 2 + y 2 + x ( y + l) + (y + l)^ =10(y + l) + (x + y + l ) = 10 3y va - 9y2 . Va tong binh phuang cua chiing la nh&ng hang so. y +1
V i dv 4: G i i i cdc h# p h u o n g trinh s a u D l t 2x + \ / l 7 - 4 x ^ = a;3x + y-y/l^ - 9y^ = b . H | da cho tuong duong: 5y a + b = 10 6x'* - ( x ^ - x)y2 - ( y +12)x^ = - 6 =4 a = 5;b = 5 x -y x+y a ^-17 b2 - 1 9 b) = 3 a = 3;b = 7' 5x''-(x2-l)^y2-llx2 =-5 4 6 5x + y- xy 1 Giii 2x + 7l7- -4x2 "" = 2 x=2 a) Nhan thay x = 0 khong la nghi^m ciia h ^ . THl: 3y + -9y2 5±Vl3 Chia hai ve phuong trinh cho x^ ta c6: y=—7— 6 (X 1' f 2 / 6x2 y2-y-12 = 0 1^ 6 X - X - y -y =o -4x2 X 2x + \ \ TH2: (lo9i). 5 -> • /• 2 / -9y2 5x2 ( 1^ y 2 - l l = 0 3y + + 2 X 5 X - X - y2-l =0 X ' l 5 + N/I31 f 1 5->/l3' V9yh?c6nghi?m (x;y) = 6a2 - a y 2 - y=0 2' 6 2' 6 Dat X — = a . H? thanh: 5 a 2 - a 2 y 2 - i = o ^x^ + x y2 + y + l = 4y 2 X b) Ta viet l?i h# nhu sau: Chia hai vecho a2 va d^t y + — = X, — = Y g i a i ra ta du(?c xy + x2y2 + i + x^y^ =4y^ Ta thay y = 0 khong thoa man h^.Chia phuong trinh dau cho y^, phuong - i = i l±Vl7 X = x ~2 x^ +x y+1 y =l y = l y trinh thii 2 cho y^ ta dug-c: ^ a = l,y„ _=o2 1 1± Vs X x , 3 , X — = 1 X = + — + l + x-^ =4 X 1>C [y = 2 y = 2 xy + 1 x2+4 = 2 V^y h? CO nghi^m (x;y) = y Viet l ^ i duoi d^ng: y xy+ 1 =4 b). Dieu ki^n: \,y ^
3 o b) Phuang trinh (2) tuong duong: x =__,y =3 1 5 (x^-y) = x (2x - y2 ) ( y - 9x2 ) = i8x2y2 ^ g^l^l ^ jg^3 ^ y 3 ^ 2xy —+—=4 1 . 5R 1 1 thanh: a b c^a=-,b = - « x=l,y=- . (x + y 2 ) = 5y 9x2y2+18x^+y3 „ ^ ISx^ y^ >'= 5 . b + 5a = 5 ^ ^ = 29xy + +— + 2 = 4 c, xy X r 2x^ 2x 2x Vay CO nghi?m (x;y) = -1 ] 1^ fa 3^ o9x H + = 4 < » 9x + ^ y +- = 4.--- I 2' . U'2j y J yJ V i d\ 5: G i a i cac h$ phuong trinh sau 2x Dat a = 9x + ^ ;b = y + - . H ? thanh: x> 11 {xy + 3 ) % ( x + y ) = 8 9x + ^ = 4 a) X y _ 1 b) a+2b=4 9x + y = 4x 2x a = 2;b = l o x^+1 y^+1 -1 = 18 ab = 2 2x , y^ +2x = y GiAi x = 0(L) y = 4x-9x^ a) Trien khai phuang trinh (1) 1 1 • (4x-9x2j^ +2x = 4x-9x^ X = —=> y=— (1) o x^y^ + 6xy + 9 + x^ + 2xy + y^ = 8 o x^y^ + x^ + y^ +1 = - S x y 9 ^ 3 o ( x 2 + l ) ( y ^ + l ) = -8xy. Vay h§ CO nghi^m (x; y ) = 9'3 Nhan thay x = 0, y = 0 khong la nghi^m cua h?. V i d y 6: G i a i cac h ? phuong trinh sau Phuang trinh (1) khi do la: ^ ^ ^ - ^ " ^ = • XN/X^ + 6 + yVx^ + 3 = 7 x y 2x2y + y ^ = 2 x * + x^ a) . b) • xVx^ + 3 + y-y/y2 + 6 =x^ +y^ + 2 (x + 2 ) 7 y + l = ( x + l ) ' D a t - ^ = a ; ^ =- b . H ? da cho tuang duang vai: x^+l y2+l Gi4i Giai he:. r 1 H f phuang trinh tuang duang voi : a = — x^+1 : 2 x =- l y _ i / y N x'^ + 3 + X a +b = — b=i 2,1 4 y = 2±V3 x J y ^ + 6 + y + y Vx^ + 3 + X = 9xy y +6 + y 4 4 y + J V / =9 ; 1 x=2+±V3 Vx^+3-x a =— X +y =2 4 4 < ) Vx^+3-x +yf^y2+6- =2 y=-i . b=-i y _ 1 - 2 y^+1 2 T =9 \/y^+6-y x^+3-x V^y h ? CO nghifm (x;y) = (-1;2 - ),(-1;2 + >/3),(2 - VS; -l),(2 + V S ; - l ) , |x V x 2 + 3 - x j + y[^7y2+6-y = 2 " O K I (1), cf-fln? ;