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

240 BRAHMAGUPTA AS AN ALGEBRAIST
If the multiplier be negative, it must be made positive;
and the additive must be made negative : and then the
method of the pulveriser should be employed.
Pṛthūdaka Svāmī, however, does not indicate how to derive
the solution of the equation.
by = -ax+c ...(1)
from that of the equation
by = ax-c ...(2)
The method, however, seems to have been this :
Let x=α, y=β be the minimum solution of (2). Then
we get
bβ = a α-c
or b(a-β) = -a(a-b)+c
Hence x=a-b, y=a-β is the minimum solution of (1). This
rule is very clearly indicated by Bhāskara II and others.
We shall give two examples from Bhāskara II (Bījagaṇita)
to illustrate the rule :
Example I.
13y = -60x + 3
By the method described before, we find that the minimum
solution of
13y = 60x+3
is x=11, y=51. Subtracting these values from their respective
abraders, namely 13 and 60, we get 2 and 9. Then by the
maxim : "In the case of the dividend and divisor being of differ-
ent signs, the results from the operation of division should be
known to be so", making the quotient negative we get the solu-
tion of
13y = -60x+3
as x=2, y=-9. Subtracting these values again from their respec-
tive abraders (13, 60), we get the solution of
13y = -60x-3
as x=11, y=-51.
Example II.
11y = 18x+10