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4.1. Simultaneous Equations in Two or Three Unknowns^123

(b) Prove that the equations have a root in common if and only if

(cp - c~r)~ = (br - cq)(aq - bp).

5. (a) Find the conditions on a and b that

P-2t+a=o

t2-8t+b=O

have a root in common.

(b) Determine a particular numerical pair (a, b) for which the equa-

tions in (a) have a root in common, and check your result by

showing that the two quadratics have a common factor.

6. Suppose that xyz # 0 and that

p = x - yz/x, q = y - zx/y, P = z - xy/z, a/x + b/y + c/z = 0.

(a) Prove that

O=pxy+qyz+rzx=q/x+r/y+P/z

0 = pxz + qxy + ryz = r/x + p/y + q/z.

(b) Eliminate x, y, z from the given system of equations to obtain

an equation in the remaining variables.

7. Show that the solutions of the system

x+y+z=a

xy+yz+zx=b

xyz = c

are given by the roots of the cubic equation

t3 - at2 + bt - c = 0

taken in some order.

8. Solve the system
   x+y+z=12
   xy+yz+zx=41
   xyz = 42.


9. By expressing x2 + y2 + z2 and x3 + ys + z3 in terms of the elementary
   symmetric functions (Exercise 1.5.8), find a system of the type in
   Exercise 7 equivalent to each of the systems and thence solve it: