CHAPTER 1V

INDETERMINATE EQUATIONS OF THE FIRST DEGREE IN
TWO UNKNOWNS

397. In the next few pages we wish to consider equations
of the form ax + by = ¢, where a, b, and ¢ are given inte-
gers, neither ¢ nor b being zero, and x and y are unknowns,
as, for example, the equations 4x + 5y = 7, 2x — 3y = 6,
6x +4y =5,3x— 7y = 0.

398. If x has any finite value and y = (¢ — ax) : b, the
equation is evidently satisfied.

Def. Thus the equation has an infinit number of solutions
and is on this account called indeterminate.

399. We will be interested in integral solutions, that is,
pairs of values of x and y that are integers and that satisfy
the equation.

Def. If there exists an integral solution, we will call the
equation possible (in integers), otherwise impossible.

400. Th. If a and b have any common factor and the equation
s possible, ¢ must have the same factor (1).
This follows evidently from §§ 61, 65, 68.

401. Th. If a and b have any common factor that ¢ does not
have, the equation is impossible.

This follows immediately from the preceding theorem.

For example, 2x 4+ 2y = 1 is impossible.

402. We will now assume that c is divisible by any common
factor of @ and b.

(1) See Charles Smith, A Treatise on Algebra, 5th ed., p. 503, London,
Macmillan and Co., 1913.

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