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