. x^, 4x +3 y, Ta thay phuong trinh c6 nghỉm khi va chi khi — va
b) Dieu kifn: x,y > 0 Ta viet lai phuong trinh (1) cua h? thanh:
^ x y - ( x - y ) ( ^ - 2 ) - y + >^-7y =0 (*). De thay x = y = 0 khong thoa
man h^. T a xet x^ + y^ 0
N h M e n h ^ p n t a c o : - J - V ) ^ - ! ^ - ^ ) ' . ^ ^ = 0
^xy + ( x - y ) ( ^ - 2 ) + y V x + ^ y . ( x - y )
Tir phuong trinh thu hai ciia h^ ta c6:
= 0 4 2 4 2 - + X - X x + 1 2 /— 4 -5 ( x - l ) (x + 2) . y + J ^ - 2 = — + x2-x-2--^^ ^>0 ^ ^ ^ x+1 x + 1 suy ra X = y thay vao phuong trinh thu hai cua h^ ta c6:
'x = l
l±Vi7 (x + l ) ( 3 x - x ^ ) = 4 o
X =•
Ket h(?p dieu ki$n ta c6: (x;y) = ( l ; l ) / ' l + ^/l7 1 + N/I7
4 ' 4
c) Dieu ki^n: x > 0,y > 5 . Ta viet Igii phuong trinh (1) cua h^ thanh:
^ x y - ( x - y ) ( 7 x y - 2 ) - y + > / x - ^ = 0 (*). De thay x = y = 0 khong thoa man h^. Ta xet x^ + y^ ;t 0 . Nhan lien h^p (*) ta c6:
/ X 1 6 - x y
<=>(x-yj , = + - T = ^ [Vx^ + i 6 ( y - x ) + 7 ^ Vxy + y
T u phuong trinh (1) ta c6: y - 5 - ^ y - 5 + x + 3 - 3Vx + 3 + 2 = 0 . T a coi day
la phuong trinh b$c 2 an yJy-5 . Dieu kỉn de phuong trinh c6 nghi^m la:
= 0
A = 9 - 4 (x + 3)-3Vx + 3 + 2 l ^ 0 o V x + 3 ^ ^ " ^ ^ ^ < 1 6 . Tir do suy ra
1 6- x
/ , 7= + 1 = > 0. Do do X = y thay vao phuone trinh (1)
thu dup-c: 2x = 3(Vx + 3 + V x - S J '' ' ' <^x''-9x3+9x2+324 = 0 c : > ( x - 6 f (x2+3x + 9) = 0 o x = 6 V|y h0 CO nghỉm x = y = 6. d) Dieu ki$n: x^y 1 7 x > - ; x ^ - x - y > 0
Phuong trinh dau ciia h? duq>c viet l^ii n h u sau:
« ( x - y - l ) x - y - 1 i( ' ^ - y ) ' + ^ + i Si^-yf + ^ x - y + l V x 2 - x - y + x^ - y^ - x - y + , ^ =0 yjx^ - x-y + y x + y '•J I ' , ' = 0 Mat khac tu phuong trinh (1) ciia hf ta c6: ^ > 0.
^ N - y
Neu y<0=>3/;r7<oox<y<0 v6 ly do x ^ i . N h u v|iy h? c6 nghỉm
khi y > 0 . Do do X + • ->0
^ ( x - y f + ^ x - y +1 ^]x^-x-y+y
Vay X - y -1 = 0 thay vao phuong trinh (2) ta c6: 4x2 - 4x - 9 - 3 7 2 x ^ = 0
o ( 2 x - l ) 2 - 3 V 2 x - l -10 = 0 Dat V2x-1 = t > 0 ta CO t ^ - 3 t - 1 0 = 0 o ( t - 2) ( t 3+ 2 t 2 + 4 t + 5) = 0<:>t = 2 o x = | Vay h? CO mpt nghỉm la (x;y) = '5 3^ 2 ' 2 i Íll'
Vi dv 7) Giai phuong trinh v6i nghi^m la so thyc: a) +2y^ +2x + 8y + 6 = 0 + xy + y + 4x + l = 0 b)
2x^ + 2xy + y - 5 = 0
y2 + xy + 5x - 7 = 0 Giai:
* Cachl:Dat x = u + a thay vao phuong trinh (1) cua h? ta c6: y = v + b
(u + a)2+2(v + b)2+2(u + a) + 8(v + b) + 6 = 0 v * >; o +2v^+2(a + l)u + 4v(b + 2) + â+2b2+2a + 8b + 6 = 0.
Ta mong muon khong c6 so h^ng b^c nhat trong phuang trinh nen dieu kỉn la: a + l = 0 a = - l b = -2
b + 2=:0
Tu do ta C O cac h dat an phu nhu sau: Dgt Xy = = u - 1 v- 2 thay vao h$ ta c6:
u2+2v2=3 day la h$ dSng cap. u^ +uv = 2
Tu h? ta suy ra 2(u^ + 2v^ j = S^u^ + uvj <=> u^ + 3uv - 4v^ = 0 <=> Cong vif c con lai la kha don gian.
Cach 2:Ta cong phuang trinh (1) vai k Ian phuang trinh (2). +2y^+2x + 8y + 6 + k x^+xy + y + 4x + l =0
<=>(l + k)x^ +(2 + 4k + ky)x + 2y^ +8y + ky + k + 6 = 0 Ta C O
A = (2 + 4k + ky)2-4(k + l)(2y2+8y + ky + k + 6)
= (k^ - 8k - 8)y2 + (4k2 - 32k - 32)y + Uk^ - 12k - 20 .
Ta mong muon A c6 d^ng (Ay -hB)^ o A = 0 c6 nghi^m kep:
o (4k2 - 32k - 32)^ - 4(k2 - 8k - 8)(l2k2 - 12k - 2o) = 0 o k = - | Tu do ta C O each giai nhu sau:
Lay 2 Ian phuang trinh (1) tru 3 Ian phuang trinh (2) cua h? ta c6: 2(x2+2y2+2x + 8y + 6)-3(x2+xy + y + 4x + l J = 0
u = V u = -4v u = -4v
<=> -x^ - 3 x y - 8 x + 4y^ +13y + 9 = 0<:> x^ + (3y + 8)x - (4y2 + 13y + 9J = 0 Ta C O A = (3y + 8)^ + 4(4y^ + 13y + 9) = 25y^ + lOOy +100 = (5y + lO)^
Tu do tinh du^c: 3y + 8-(5y + 10) x = ^ = - V -1 2 3y + 8 + (5y + 10) x = ỵ = 4y + 9
Phan vỉc con lai la kha don gian. n , u