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

RULE OF CONCURRENCE 211
kramaṇa is the solution of the simultaneous equations of the
type :
x + y = a
x - y = b
Brahmagupta's rule for solution is:
The sum is increased and diminished by the difference
and divided by two; (the result will be the two un-
known quantities) : (this is) concurrence (Saṁkra-
maṇa).¹
Brahmagupta restates this rule in the form of a problem
and its solution :
The sum and difference of the residues of two (heavenly
bodies) are known in degrees and minutes. What are
the residues? The difference is both added to and
subtracted from the sum, and halved; (the results are)
the residues.²
Linear Equations with Several Unknowns
The first mention of a solution of the problem with more
than one unknown is found in the Bakhshālī Manuscript, and a
system of linear equations of this type is solved in the Bakhshālī
treatise substantially by the False Position Rule.
A generalised system of linear equations will be
b₁Σx - c₁x₁ = a₁, b₂Σx - c₂x₂ = a₂,......,
bₙ Σx - cₙxₙ = aₙ
Therefore
Σx = Σ(a/c) / (Σ(b/c) - 1)
Hence
xᵣ = (bᵣ / cᵣ) × [Σ(a/c) / (Σ(b/c) - 1)] - aᵣ / cᵣ
r = 1, 2, 3......, n
One particular case, where b₁ = b₂ = b₃ = ...... = bₙ = 1 and c₁ =
c₂ = c₃ = ...... = cₙ = c has been treated by Brahmagupta at one
place. He gives the rule as follows :

1. योगोऽन्तरयुतहीनो द्विहतः संक्रमणमन्तरविभक्तं वा ।
   वर्गान्तरमन्तरयुतहीनं द्विहृतं विषमकर्म ॥ — BrSpSi. XVIII. 36
2. भागकला विकलैक्यं दृष्ट्वा विकलान्तरं च के शेषे ।
   ऐक्यं द्विधाऽन्तराधिक हीनं च विभाजितं शेषे ॥ — BrSpSi. XVIII. 96