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312 Answers to Exercises and Solutions to Problems

Suppose that u = VW. Substituting for w in (2) leads to v2 = 1, and we

get the solutions (u, v, w) = (-1, 1, -l), (-1, -1,1) already noted.

Suppose that u = -VW. Substituting for w in u-l + w + uw-l = v-l +

w-l + VW leads to

(v - u)(l + v + u - UV) = 0.

If v = u, then w = -1. Expressing everything in terms of u, we find that

the given system is satisfied iff

0 = u3 + u2 + u - 3 = (u - l)(u2 + 2u + 3)

and

O=3u3+u2-u-1=(3u-5)(u2+2u+3)+14.

But these are inconsistent and there is no solution with v = u. If uv =

1 + u + v (and u = -VW), then the first member of the given system is

equal to the fourth and the second is equal to the third. Thus, the system

is satisfied iff (4) holds:

-u(u + v)(-uv-l - u) = u(v - 1)(-uv-l - 1)

or
u2(u + v)(l + v) = -u(v - l)(u + v)
or u(1 + v) = -(v - 1) or u+v+uv = 1 or u+v =O. Hence (u,v,w) =
(i, -i, 1) or (-;, i, l), which has already been noted.
8.11. Let u = x + y, v = xy. Then the two equations become u2 - 2v = 13,
~(13 - v) = 35, whence 0 = u3 - 39u + 70 = (u - 5)(u - 2)(u + 7). Hence
x and y are the zeros of any one of the quadratics t2 - 5t + 6, 2t2 - 4t - 9,
t2 + 7t + 18.

8.12. x, y, .z are the zeros of the cubic

t3-3ut2+(2u2-a-7)t+(4u2+10a-6) = (t+2)[t-(2u-l)][t-(u+3)].

8.13. Suppose a, b, c are all distinct. From the first equation and (y - Z) +

(z - x) + (x - y) = 0, it f o 11 ows that x - y = t(a - b), y - t = t(b - c),

z - x = t(c - a) for some t. The second equation then determines t. Finally,

writing two of the variables in terms of a third, we find from the third

equation that

3x = e + t(2a - b - c), 3y = e + t(2b - a - c), 32 = e + t(2c - a - b).

If exactly two of a, b, c are equal, say a = b # c, then x = y and the
second equation is consistent iff d = 0. If d = 0, any value oft will work
and the solution can be completed using the third equation as before.
If a = b = c, then the first equation imposes no restriction. We can
substitute z = e - (x + y) into the second equation, choose an arbitrary
value of x and solve a cubic equation in y and thence determine Z.