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History of Hindu Mathematics: A Source Book v2.c17.s12: 14. One Linear Equation in More Than Two Unknowns

History of Hindu Mathematics: A Source Book v2.c17.s12

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14. One Linear Equation in More Than Two Unknowns

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To solve a linear equation involving more than two unknowns the usual Hindu method is to assume arbitrary values for all the unknowns except two and then to apply the method of the pulveriser. Thus Brahmagupta remarks, "The method of the pulveriser (should be employed), if there be present many unknowns (in an equation)."² Similar directions have been given by Bhâskara II and others.³

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² *BrSpŚi*, xviii. 51.

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³ *BBi*, p. 76.

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One of the astronomical problems proposed by Brahmagupta⁴ leads to the equation: $197x - 1644y - z = 6302.$ Hence $x = (1644y + z + 6302)/197.$ The commentator assumes z = 131. Then $x = (1644y + 6433)/197,$ hence by the method of the pulveriser $x = 41, y = 1.$

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⁴ *BrSpŚi*, xviii. 55.

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The following example with its solution is from the algebra of Bhâskara II :

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“The numbers of flawless rubies, sapphires, and pearls with one person are respectively 5, 8 and 7 ; and O friend, another has 7, 9 and 6 respectively of the same gems. In addition they have coins to the extent of 90 and 62. They are thus equally rich. Tell quickly, O intelligent algebraist, the price of each gem.”

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If x, y, z represent the prices of a ruby, sapphire and pearl respectively, then by the question $5x + 8y + 7z + 90 = 7x + 9y + 6z + 62.$ Therefore $x = (-y + z + 28)/2.$ Assume z = 1; then $x = (-y + 29)/2,$ whence by the method of the pulveriser, we get $x = 14 - m, y = 2m + 1,$ where m is an arbitrary integer. Putting m = 0, 1, 2, 3, we get the values of (x, y, z) as (14, 1, 1), (13, 3, 1), (12, 5, 1), (11, 7, 1), etc. Bhâskara II then observes, “By virtue of a variety of assumptions multiplicity of values may thus be obtained.”

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Sometimes the values of most of the unknowns present in an equation are assumed arbitrarily or in terms of any one of them, so as to reduce the equation to a simple determinate one. Thus Bhâskara II says :

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increased or decreased by them, or in some cases (simply) any known values may be assumed for the other unknowns according to one’s own sagacity. Knowing these (the rest is an equation in one unknown).”¹

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¹ *BBi*, p. 44.

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The above example has been solved again by Bhâskara II in accordance with this rule thus:²

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² *BBi*, p. 46.

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(1) Assume x = 3z, y = 2z. Then the equation reduces to $38z + 90 = 45z + 62.$ Therefore z = 4. Hence x = 12, y = 8.

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(2) Or assume y = 5, z = 3. Then the equation becomes $5x + 151 = 7x + 125.$ Whence x = 13.

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