View source for Indeterminate Equations of the First Degree

Āryabhaṭa I (476) <ref>{{Cite book|last=Datta|first=Bibhutibhusan|title=History of Hindu Mathematics|last2=Narayan Singh|first2=Avadhesh|publisher=Asia Publishing House|year=1962|location=Mumbai}}</ref>was the earliest Hindu Algebraist worked on the Indeterminate Equations of the First Degree. He provided a method for solving the simple indeterminate equation

<math>by-ax=c</math> 

where a, b and c are integers.He also provided how to extend this to solve Simultaneous Indeterminate Equations of the first degree.

Bhāskara I (522) disciple of Āryabhaṭa I has displayed that the same method might be applied to solve the equation

<math>by-ax=-c</math>

and further that the solution of this equation would follow from that of

<math>by-ax=-1</math>

Brahmagupta and others followed the methods of Āryabhaṭa I and Bhāskara I

== Importance ==
The subject of indeterminate analysis of the first degree was considered so important by ancient Hindu Algebraists that the whole science of algebra was once named after it. Āryabhaṭa II , Bhāskara II  and others mentions precisely along with the sciences of arithmetic, algebra and astronomy. 

On account of its special importance exclusive work on this entitled kuṭṭākāra śiromaṇi by Devarāja  a commentator of of Āryabhaṭa I.

== Varieties of Problems ==
There are three varieties of problems pertaining to Indeterminate equations of the first degree.

Variety 1 : Find a number (N) which when divided by two given numbers (a, b) will leave two remainders (R<sub>1</sub>,R<sub>2</sub>).

Now we have <math>N=ax+R_1=by+R_2
</math>

<math>by-ax = R_1-R_2
</math>

Putting <math>c=R_1 \thicksim R_2
</math>

<math>by-ax=  \pm c
</math>


the upper or lower sign being taken according as Rl

> or < R2• In a pro blem of the second kind we are

required to find a number (x) such that its product with

a given number (a) being increased or decreased .by

another given number .(v) and then divided by a 'third

given number (~) will leave no remainder. In other

words we shall have to solve

ax±- v =Y

~

in positive integers. The third variety of problems

similarly leads to equations of the form

by + ax = ± c.

== References ==