ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

274 BRAHMAGUPTA AS AN ALGEBRAIST
y = 1/a ((ad + bc)/m + b)
if b > c and m > (ad + bc)/m. If these conditions be reversed then x
and y will have their values interchanged.
Datta and Singh have given the following rationale of
these solutions :
axy = bx + cy + d,
or a²xy − abx − acy = ad,
or (ax − c) (ay − b) = ad + bc.
Suppose ax − c = m, a rational number;
then ay − b = (ad + bc)/m.
Therefore
x = 1/a (m + c)
y = 1/a ((ad + bc)/m + b)
Or, we may put ay − b = m;
in that case, we shall have ax − c = (ad + bc)/m;
whence x = 1/a ((ad + bc)/m + c),
y = 1/a (m + b).
Brahmagupta’s own rule.
Whilst the rule given above is ascribed to an unknown
author, Brahmagupta’s own rule for the solution of a quadratic
indeterminate equation involving a factum is as follows :
With the exception of an optional unknown, assume
arbitrary values for the rest of the unknowns, the
product of which forms the factum. The sum of the
product of these (assumed values) and the (respective)
coefficients of the unknowns will be absolute quantities.
The continued products of the assumed values and of
the coefficient of the factum will be the coefficient of
the optionally (left out) unknown. Thus the solution