That is, x = bt + cq and y = at + cp.
Positive integral solutions, unlimited in number, are obtained by giving t any positive integral value, and any negative integral value, so long as it is numerically less than the smaller of the quantities cq/b, cp/a; t may also be zero.
(β) If aq − bp = −1, we obtain x = bt − cq, y = at − cp, from which positive integral solutions, again unlimited in number, are obtained by giving t any positive integral value which exceeds the greater of the two quantities cq/b, cp/a.
If a or b is unity, a/b cannot be converted into a continued fraction with unit numerators, and the above method fails. In this case the solutions can be derived directly, for if b is unity, the equation may be written y = ax − c, and solutions are obtained by giving x positive integral values greater than c/a.
4. To solve ax + by = c in positive integers. Converting a b into a continued fraction and proceeding as before, we obtain, in the case of aq − bp = 1,
x = cq − bt, y = at − cp.
Positive integral solutions are obtained by giving t positive integral values not less than cp/a and not greater than cq/b.
In this case the number of solutions is limited. If aq − bp = −1 we obtain the general solution x = bt − cq, y = cp − at, which is of the same form as in the preceding case. For the determination of the number of solutions the reader is referred to H.S. Hall and S.R. Knight’s Higher Algebra, G. Chrystal’s Algebra, and other text-books.
5. If an equation were proposed involving three unknown quantities, as ax + by + cz = d, by transposition we have ax + by = d − cz, and, putting d − cz = c′, ax + by = c′. From this last equation we may find values of x and y of this form,
x = mr + nc′, y = mr + n′c′,
or x = mr + n (d − cz), y = m′r + n′ (d − cz);