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

LINEAR EQUATIONS 241
Proceeding as before, we find the minimum solution of
11y=18x+10
to be x=8, y=14. These will also be the values of x and y in
the case of the negative divisor but the quotient for the reasons
stated before should be made negative. So the solution of
-11y=18x+10
is x=8, y=-14. Subtracting these (i.e., their numerical values)
from their respective abraders, we get the solution of
-11y=18x-10
as x=3, y=-4,
"When the divisor is positive or negative the numeri-
cal values of the quotient and multiplier remain the
same : when either the divisor or the dividend is
negative, the quotient must always be known to be
negative"¹.
One Linear Equation in
More Than Two Unknowns
Whenever a linear equation involves more than two unkno-
wn's the Indian algebraists used to assume arbitrary values for all
the unknowns except two and then to apply the method of kuṭ-
ṭaka or "pulveriser". In this connection, Brahmagupta says :
The method of the pulveriser (should be employed if
there be present many unknowns (in any equation)²,

1. Bhāskara II gives the following rule :
   "Those(the multiplier and quotient)cbtained for a positive divi-
   dend being treated in the same manner give the results corres-
   ponding to a negative dividend."
   The treatment alluded to in this rule is that of subtraction
   from the respective abraders. He has further elaborated it
   thus :
   The multiplier and quotient should be determined by
   taking the dividend, divisor and interpolator as positive. They
   will be the quantities for the additive interpolator.
   Subtracting them from their respective abraders, the quantities
   for a negative interpolator are found. If the dividend or divi-
   sor, be negative, the quotient should be stated as negative, the
   quotient should be stated as negative.
   —Bījagaṇita
2. आद्याद्वर्णादन्यान् वर्णान् प्रोह्याद्यमानमाद्यहृतम् ।
   सदृशच्छेदावसकृद् द्वौ व्यस्तौ कुट्टको बहुषु ॥ — BrSpSi. XVIII. 51