Question

MEDAL!!! PLZ HELP!!!

Answer 1

This is a test of algebra. The question is well structured, so that you have to have just one insight
at each stage. Note that the equations are symmetric: if you replace b by a, c by b and a by c, and
perform the same operation with x, y and z, the equations are unchanged. This leads to a saving
of work, but can also be used to check for algebraic errors.
The method of solution along which you are steered is a bit convoluted; it is therefore necessary
to satisfy yourself that the method does actually lead to solutions of the original equations and
that you have not missed any on the way. This is best done by considering the relationship of the
equations you actually solved to the original equations.
Note that the sign of 4 is not determined. If x, y and z satisfy the equations, then so also do −x,
−y and −z, and for this solution 4 will have the same magnitude but the opposite sign. It is not
surprising that there are two solutions for given a, b and c: the equations are quadratic.
Geometrically, equations (1) – (3) represent hyperboloids: for example, equation (1) can be written
x
2 + (y − z)
2/4 − (y + z)
2/4 = a, which looks something like an eggtimer if a > 0 and like a pair of
radar dishes if a < 0. Three eggtimers intersect in two points, corresponding to the solutions you
are about to find, but the six radar dishes do not intersect at all; in this case, a + b + c < 0 which
means that 4 is imaginary.
There are neater ways of solving these equations (for example, using matrices, but not in an obvious
way). When the question was set (without the intermediate steps) in a Cambridge University
examination for undergraduates in January of 1860, the examiners provided a model solution.
Their idea was to square equation (1) and subtract the product of equations (2) and (3); a good
plan, but not one that many people would think of under examination conditions.