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124 4. Equations

(4

x+y+z = 15

x2+y2 +z2 = 83

x3 + y3 + z3 = 495

(b)

x+y+z = 2

x2 +y2 + z2 = 14

xyz = -6.

10. By factoring the left side of the first equation, or otherwise, solve the
    system
    x2+3xy-4y2 = 0
    x2 +xy+y2 = 7y3 -4.


11. (a) Suppose that r, oi, bi (i = 1,2,... , n) are numbers for which
    ai/bi = r for each i. Let a = Cai, b = Cbi. Show that a/b = r.

(b) If X

-=-z-z Y Z

11-y 6-z 7-x

2

use (a) to determine x -+ y + z, and solve this equation for x, y,

Z.

4.2 Surd Equations

In solving an equation in a single variable x, we begin with the assumption
that x satisfies the equation and deduce that x must be one of a number of
possibilities. If a polynomial equation is involved, we are content to accept
all of these as valid solutions. However, strictly speaking, the solution of
the equation is not properly complete until the possible solutions have been
checked by substitution into the equation.
While, for polynomial equations, all of the putative solutions turn out
to be valid, for surd equations, more care is needed. The manipulations for
solving surd equations often lead to more general equations, not equivalent
to their predecessors, so that in effect information about the solution is lost.
The result is that only some of the values turned up by the analysis may
satisfy the original equation. The remaining values which do not satisfy the
equation are said to be ettraneous. This phenomenon will be illustrated in
the exercises.
For surds involving real numbers, e denotes the unique real number y
for which yk = x when k is an odd integer. However, when k is an even
integer, fi is defined only for z > 0 and denotes the unique nonnegative
real number y for which y” = x.