154 4. Equations
- Given that a 2 1 and b are real numbers, prove that the system
y=x3+ax+b
z=y3+ay+b
x=z3+az+b
has exactly one real solution.
- Solve
x(x - y - z) = a
Y(Y - z-x)=b
I(% - x - y) = c.
- If x + x-l = a, y + y-l = b and a2 + b2 + z2 = 4 + abz, determine z
in terms of x and y. - If
and
ux vy W%
show that (x/u + y/v + .z/w)~ = a2/u2 + b2/v2 + c2/w2.
- Let a, b, c be positive numbers such that fi + 4 + fi = a/2.
Prove that the system
Jy--a+JTT=l
Jzi;+&-S=1
JKT+Jy-c=1
has exactly one real solution.
- Solve for 2, y, z:
yz = a(y + z)
xz = b(x + z)
xy = c(x + y).