4.8. Problems on Systems of Equations^153
- Prove that (x,y) = (1,2) is the unique real solution of
x(x + y)2 = 9“(Y3 - x3) = 7.- Given that none of Ial, lb(, ICI is equal to 1, and that
22 = y2 + %2 - Say%y2 = .z2 + x2 - 26%x
%2 = x2 + y2 - acxy,
show that
-=-= x2 y2^22
1 - a2 1 - b2 sCa.
What happens if Ial = l?- x” + 2y2 + 3%’ = 36
3x2 + 2y2 + z2 = 84
xy+xz+yz = -7. - Find all real a for which there exist nonnegative reals xi for which
c kxk =^0
kc1
5
c
k3xk = a2
k=l
5
c k5 xk = a3.
k=l- Determine all real p for which the system x+y+% = 2, y%+%x+xy =
1, xyz = p has a real solution. - xy = 2
(3 + Syl(x - Y))~ + (3 - Syl(x + Y>)~ = 82. - Show that the real solutions (xi, yi, %i) of
(x - 5)2 + (y - 2)2 + (z - 6)2 = 49(x - 11)2 + (y - 7)s + (% - 2)2 = 49
38x - 56y - 132 = 0
satisfy xl + x2 = 16, yr + y2 = 9, zr + z2 = 8.