kramana is the solution of the simultaneous equations of the type : RULE OF CONCURRENCE 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 (Samkra mana).¹ Therefore x+y=a x=y=b 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 Hence Lincar Equations with Several Unknowns The first mention of a solution of the problem with more than one unknown is found in the Bakhasali Manuscript, and a system of linear equations of this type is solved in the Bakhaśali treatise substantially by the False Position Rule. A generalised system of linear equations will be b₁xcixi-ar b₂x-caxe-A2........ box-cox an X Σ(a/c) 2(b/c)-1 211 Xx= >(alc) Σ(b/c)-1 ar Cr X 1 2 3 S One particular case, where bi-b₂=b3=......-bn =1 and c₁= C₂=CB=......=Cnc has been treated by Brahmagupta at one place. He gives the rule as follows: 1. योगोऽन्तरयुक्तहीनो द्विहृतः संक्रमणमन्तरविभक्तं वा । वर्गान्तरमन्तरयुतहीनं द्विहृतं विषमकर्म ।। 2. भागकला विकलैक्यं दृष्ट्वा विकलान्तरं च के शेषे । ऐक्यं द्विवाउन्तराधिक होनं च विभाजित शेषे || - BrSpSi. XVIII. 36 - BrSpSi. XVIII. 96
