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

LINEAR EQUATIONS 209
of the coefficients on the unknown (gulikā). The
quotient will be the value of the unknown (gulikā), if
their possessions be equal.¹
The original verse contains the term "gulikāntara" which
has been here translated as the difference of the coefficients
of the unknowns. We have already stated earlier that Āryabhaṭa
uses the term gulikā or shot for an unknown quantity, (gulikāntara
literally means only the difference of unknowns). This practice
is also followed by other Indian algebraists. Pṛthudaka Svāmī
rightly observed that according to the usual practice in this
country, "the coefficient of the square of the unknown is called
the square (of the unknown) and the coefficient of the (simple)
unknown is called the unknown.²
The rule given by Āryabhaṭa, then, contemplates a problem
of this kind :
Two persons, who are equally rich, possess respectively
a, b times a certain unknown amount together with
c, d units of money in cash. What is that amount ?
Now if x be the amount unknown, then according to the
problem
ax + c = bx + d
Thence
x = (d - c) / (a - b)
Āryabhaṭa has merely expressed this solution in his langu-
age.
Regarding the solution of linear equations, Brahmagupta
says :
In a (linear) equation in one unknown, the difference
of the known terms taken in the reverse order, divided
by the difference of the coefficients of the unknown
(is the value of the unknown).³

1. गुलिकान्तरेण विभजेद् द्वयोः पुरुषयोस्तु रूपकविशेषम् ।
   लब्धं गुलिकामूल्यं यद्यर्थं कृतं भवति तुल्यम् ॥ —Ārya. II. 30
2. BrSpSi, XVIII. 44 (com.)
3. अव्यक्तान्तरभक्तं व्यस्तं रूपान्तरं समेऽन्यवतः ।
   वर्गाव्यक्ताः शोध्या यस्माद् रूपाणि तदधस्तात् ॥ —BrSpSi. XVIII. 43