Examples of Substitution of Unknown Quantities.

238. Corollary. When there are two unknown quantities which enter symmetrically into the given equation, the solution is often simplified by substituting for them two other unknown quantities, one of which is their product and the other their sum.

239. EXAMPLES.

1. Find two numbers whose sum is 5, and the sum of whose fifth powers is 275.

Solution. Let the numbers be x and y, represent their product by p, and we have

But we also have

x + y = 5,

x5y5=275.

(x+y)5=x5+5x1y+10x3 y2+10x2 y3+5x y2+y5 =x5+y5+5x y (x3 + y3)+10 x2 y2 (x+y);

(x+y)5=275+5p (125 — 15p) +10 p2 × 5=55;

=

y=3, or=2, or (551).

2. Solve the two equations

(xy) (x2- y2) = 7,
(x + y) (x2 + y2) = 175.

Examples of Substitution of Unknown Quantities.

Solution. These equations become, by development,

x3 — x2 y — x y2+y3=7,

3. Solve the two equations

x + y = x y

x + y + x2+ y2 = 12..

:

8. Find two numbers such, that their sum and product may together be 34, and the sum of their squares may exceed the sum of the numbers themselves by 42.

Ans. 4 and 6;

or 1 (−11+✔―59,) and 1 (—11—✔—59).

9. What two numbers are they, whose sum is 3, and the sum of whose fourth powers is 17?

Ans. 2 and 1;

or (355), and (3—✓—55).

10. What two numbers are they, whose product is 3, and the sum of whose fourth powers is 82?

240. Corollary. In many cases, in which two unknown quantities enter into the given equations symmetrically except in regard to their signs, the solution is simplified by substituting for them two other unknown quantities, one of which is their difference, and the other is their sum or their product.

Examples of Equations of the Second Degree.

and

4

$2401 ± 7, or = ±7√−1;
24017,

x y (x3-y3) — 2x2 y2 (x − y) + (x — y)2= 157.

Ans. x 4, or=-3, or =( 1±√-51);

or x=

=

y=3, or 4, or =(-1±√-51).

=

-157?±2√(624+1574)) — 781, y=±}v(−1572±2√(624+1574))+784.

6. What two numbers are they, whose difference is 1, and the difference of whose third powers is 7?

7. What two numbers are they, whose difference is 3, and the sum of whose fourth powers is 257?

or (±√(—79)+3) and † (±√ (—79) —3).