Hints 1578.1. Begin by eliminating the terms with coefficient 9/8 and determine
simple possible relations between t and y.8.2. The equations lead to relations of the form u + v = u-l + V-‘; study
this first.(^8). (^5). Let u = a2 - x2 =... , and write zy + yz + zt in terms of U.
8.6. One strategy is to find a homogeneous equation of the fourth degree
in z and y which might be factored. Multiply the second equation
by 14xy. Note also that the second equation can be used to give an
expression for 64.
8.7. Factor the left sides as differences of squares. Let u = -z + y+ z, etc.
8.9. Solve first for zy, yz and zx.
8.10. (a) 1 = (U + v - U-‘)(u + v - V-‘).
(b) Equate the four members in pairs. Eliminate denominators and
manipulate terms to find factors which might be cancelled out.
8.11. Solve for u = x + y, v = xy.
8.13. If o, b, c can not all be the same, the first equation and (y - z) + (z -
x) + (CC - y) = 0 lead to a determination of (y - z) : (z-x) : (x - y).
8.14. The last two equations can be used to determine (Y+.z)~ = (Y-z)~+
4yz in terms of 2.
8.15. Note that (x+~i)~ = (x2-y2)+2xyi. What is (x+yi)[(a+b)+(a-b)i]?
Be careful; x and y need not be real.
8.16. Express the difference of two expressions for b2c2 as a sum of squares.
8.20. Dispose of the case a = b first. Make a change of variable x =
(1 + a)/( 1 - u), y = (1 + v)/( 1 - V) ; u and v each satisfy a quartic
equation whose roots are obvious.
8.25. For any quadratic polynomial p(t), ap(a) = Chp(k2)xk.
8.28. If (21, yr, zi) and (22,y2,z2) satisfy the system, then (zr + x2,.. .)
should satisfy the linear equation. Observe that (x, y, z) = (16,9,8)
satisfies the linear equation.
8.33. Let T be an equilateral triangle of side 1. There is a one-one corre-
spondence between points P inside the triangle and positive reals a,
b, c for which &+A+&= 34, and the distances from P to the
respective sides are fi, fi, 4. Also, Jy(l is the side length of a
right triangle with hypotenuse fi and other side 4.