. x^, 4x +3 y, Ta thay phuong trinh c6 nghỉm khi va chi khi — va
8) Bieh doi phuong trinh (1) d! (2) liiuri grroridi)
(x + 3)7(y + 4)2 + 4 = - ( y + 4)7(x + 3)^+1 (*) + x = -3 y = -4 ta thay khong thoa man.
+ x 9 i - 3 => y 7 t - 4 thi binh phuong hai ve phuong trinh (*) j
(x + 3)(y + 4 ) < 0
- < » y + 4 = -2(x + 3 ) o y = - 2 x - 1 0
( y ^ 4 ) ^ = 4 ( x + 3)2 ,.„.x,*
Thay vao phuong trinh (2) va riit gpn ta dugc:
4x2 + 28x + 51 + 3^4x +15 = 0
1 tij neu On rnt a«i npc san^ luv r*/ r i , TT^TT imt T J I ~ rs^yeti i T T W ^ W * ^ .4(x + 4 f 27(4x + 15)-(4x + 13) .4(x + 4 f 27(4x + 15)-(4x + 13) •4(x + 4 f - 9^(4x + 15)^ + 3(4x + 13)^4x + 15 + (4x +13^ 16(4x + 7)(x + 4 f = 0 1 - 9^(4x + 1 5 f + 3(4x + 13)^4x + 15, + (4x +13)^ 4(4x + 7) 9^(4x +15)^ + 3(4x + 13)^4x + 15 + (4x +13)^ = 0 = 0 1 - - 4(4x + 7) 9^(4x + 15)^ + 3(4x + 13)^4x + 15+ (4x +13)' x = -4 Voi x = - 4 = > y = - 2 4(4x + 7) = 0 - V o i l - - 9^(4x + 1 5 f + 3(4x + 13)^4x + 15+ (4x +13)
Ta se chung minh phuong trinh nay v6 nghiem nhu sau: De thay voi mpi x thi 4x'^ + 28x + 51 > 0
= 0 (3)
Do do phuong trinh(**)c6 nghiem khi 3^4x + 15 < 0 = ^ x < - — . T u d o suy 4
ra ve trai cua (3) luon duong, dan den phuong trinh nay v6 nghiem. K L : ( x ; y ) = ( - 4 ; - 2 )
9) Tu phuong trinh (2) ta thu duqc: y ^ = 2 - x ^ - y - ^
Thay vao phuong trinh (1) ta c6:
2x^ + x + x^ - 2y = 4 => x^ + 2x - xy - ^ + x^ - 2y = 4 < » ( x - 2 ) ( x ^ +2x + 4) + x(x^ +2x + 4 ) - y ( x ^ +2x + 4) = 0
<=> 2x^ + 2x^ + 4x - x^y - 2xy - 4y = 8
« ( x 3 - 8 ) + (x3+2x2+4x)-(x2y + 2xy + 4y) = 0=i>(2x-2-y)(x2+2x + 4) = 0 <=>y = 2 x - 2
Thay y = 2x - 2 vao phuong trinh (2)va rut n ta dugc
x{6x-7) = 0<^ X = 0 => y = - 2 7 1
6 ^ 3
Vay h | phuong trinh c6 2 nghiem (x;y) = (0;-2), (7_ l\
6'3
10) Voi dieu kien x > 0 h^ phuong trinh da cho tuong duong voi he: x^y^ + 8xy^ - 6xy - 12y^ - 7y + 8 = 0
13y^ + y + l - 6 x y ^ = 0 • ; 0-
Lay (1) + (2) ta c6 dugc phan tich sau: ,
x^y^ + 2xy2 + y^ - 6xy - 6y + 9 = 0 o [y(x + l)f - 6y(x +1) + 9 = 0 j Tadugc y(x + l ) = 3<»19y^-17y + l = 0
17 + V213 49-3sf2U
• ^ " ' ^ = - 3 ^ ^ ' ^ = 2 I
17-N/2T3 49 + 37213
- Vol y = ; X =
^ 38 2 :• Vay h^ phuong trinh da cho c6 bo nghifm la: ^
49-3N/213 17 + 7213 ) f 49 + 37213 17 - 72131 (x;y) =
38
11). Dieu kien: y ;^ 0
Voi y ^ 0 ta bien doi he phuong trinh thanh
38 4 2 xy = — y xy 2 2xy + y^ = 2 . y , Dat a =: b = xy h^ phuong trinh tren tro thanh
' 2 a - b = = l b 2 a - 2 b + — = 2 a 2 a b - b 2 = 4 (3) 2â-2ab + b^ =2a (4)
Cpng (3) va (4) theo ve va thu gpn ta dugc
a - a - 2 = 0<=> a = - l a = 2
TH2 : a = 2 => b = 2 ta CO phuang trinh — = 2 y y xy = 2