History of Hindu Mathematics: A Source Book v2.c17.s7: 9. Linear Equations with Several Unknowns

History of Hindu Mathematics: A Source Book v2.c17.s7

Language: English

9. Linear Equations with Several Unknowns

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**A Type of Linear Equations.** The earliest Hindu treatment of systems of linear equations involving several unknowns is found in the Bakhshâlî treatise. One problem in it runs as follows:

"[Three persons possess a certain amount of riches each.] The riches of the first and the second taken together amount to 13; the riches of the second and the third taken together are 14; and the riches of the first and the third mixed are known to be 15. Tell me the riches of each."¹

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¹ *BMs*, Folio 29, recto. The portions within [ ] in this and the following illustration have been restored.

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If x₁, x₂, x₃ be the wealths of the three merchants respectively, then $x₁ + x₂ = 13, x₂ + x₃ = 14, x₃ + x₁ = 15. (1)$ Another problem is

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“[Five persons possess a certain amount of riches each. The riches of the first] and the second mixed together amount to 16 ; the riches of the second and the third taken together are known to be 17 ; the riches of the third and the fourth taken together are known to be 18 ; the riches of the fourth and the fifth mixed together are 19 ; and the riches of the first and the fifth together amount to 20. Tell me what is the amount of each.”
*i.e.*, x₁ + x₂ = 16, x₂ + x₃ = 17, x₃ + x₄ = 18,
x₄ + x₅ = 19, x₅ + x₁ = 20. (2)

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There are in the work a few other similar problems.² Every one of them belongs to a system of linear equations of the type $x₁ + x₂ = a₁, x₂ + x₃ = a₂, , x_n + x₁ = a_n, (1)$ n being odd.

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² *BMs*, Folio 30, recto ; also see Kaye’s Introduction, p. 40.

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**Solution by False Position.** A system of linear equations of this type is solved in the Bakhshâlî treatise substantially as follows :

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Assume an arbitrary value p for x₁ and then calculate the values of x₂, x₃, ... corresponding to it. Finally let the calculated value of x_n + x₁ be equal to b (say). Then the true value of x₁ is obtained by the formula $x₁ = p + ½(a_n - b).$

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In the particular case (1) the author³ assumes the arbitrary value 5 for x₁; then are successively calculated the values x'₂ = 8, x'₃ = 6 and x'₃ + x'₁ = 11. The correct values are, therefore, $x₁ = 5 + (15 - 11)/2 = 7, x₂ = 6, x₃ = 8.$

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”1 *BMs*, Folios 27 and 29, verso.

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³ *BMs*, Folio, 29, recto.

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*Rationale.* By the process of elimination we get from equations (I) $(a₂ - a₁) + (a₄ - a₃) + + (a_n-1 - a_n-2) + 2x₁ = a_n.$ Assume x₁ = p; so that $(a₂ - a₁) + (a₄ - a₃) + + (a_n-1 - a_n-2) + 2p = b, say.$ Subtracting 2(x₁ - p) = a_n - b. Therefore x₁ = p + ½(a_n - b).

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**Second Type.** A particular case of the type of equations (I) for which n = 3, may also be looked upon as belonging to a different type of systems of linear equations, *viz.*, $Σ x - x₁ = a₁, Σ x - x₂ = a₂, , Σ x - x_n = a_n, II$ where Σ x stands for x₁ + x₂ + + x_n. But it will not be proper to say that equations of this type have been treated in the Bakhshâlî treatise.¹ They have, however, been solved by Âryabhaṭa (499) and Mahâvîra (850). The former says:

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¹ The example cited by Kaye (*BMs*, Introd., p. 40, Ex. vi) which conforms to this type of equations is based upon a misapprehension of the text.

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“The (given) sums of certain (unknown) numbers, leaving out one number in succession, are added together separately and divided by the number of terms less one; that (quotient) will be the value of the whole.”²

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² *Â*, ii. 29.

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*i.e.*, $Σ x = _r=1n a_r / (n - 1).$

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Mahâvîra states the solution thus:

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“The stated amounts of the commodities added together should be divided by the number of men less

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one. The quotient will be the total value (of all the commodities). Each of the stated amounts being subtracted from that, (the value) in the hands (of each will be found).”¹

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¹ *GSS*, vi. 159.

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In formulating his rule Mahâvîra had in view the following example:

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“Four merchants were each asked separately by the customs officer about the total value of their commodities. The first merchant, leaving out his own investment, stated it to be 22; the second stated it to be 23, the third 24 and the fourth 27; each of them deducted his own amount in the investment. O friend, tell me separately the value of (the share of) the commodity owned by each.”²

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² *GSS*, vi. 160-2.

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$Here x₁ + x₂ + x₃ + x₄ = (22+23+24+27)/(4-1) = 32.$

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$Therefore x₁ = 10, x₂ = 9, x₃ = 8, x₄ = 5.$

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Nârâyaṇa says:

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“The sum of the depleted amounts divided by the number of persons less one, is the total amount. On subtracting from it the stated amounts severally will be found the different amounts.”³

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³ *GK*, ii. 28.

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The above type of equations is supposed by some modern historians of mathematics⁴ to be a modification of the type considered by the Greek Thymaridas and solved by his well known rule *Epanthema*, namely,⁵

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⁴ Cantor, *Vorlesungen über Geschichte der Mathematik* (referred to hereafter as Cantor, *Geschichte*), 1, p. 624; Kaye, *Ind. Math.*, p. 13; *JASB*, 1908, p. 135.

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⁵ Heath, *Greek Math.*, I, p. 94.

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$x + x₁ + x₂ + + x_n-1 = s,$ $x + x₁ = a₁, x + x₂ = a₂, , x + x_n-1 = a_n-1.$

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The solution given is $x = ((a₁ + a₂ + + a_n-1) - s)/(n - 2).$

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But that supposition has been disputed by others.¹ Sarada Kanta Ganguly has shown that it is based upon a misapprehension. It will the *citra-kuṭṭaka-miśra*,¹ tell me quickly what was the amount in the hand of each."²

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That is, we get the equations x + 4 + 5 &= 2(y + z - 4 - 5), / y + 4 + 6 &= 3(z + x - 4 - 6), / z + 5 + 6 &= 5(x + y - 5 - 6); or 2(x + y + z) - 3x &= 27, / 3(x + y + z) - 4y &= 40, / 5(x + y + z) - 6z &= 66; a particular case of the system (III). Substituting in (1), we get x = 7, y = 8, z = 9.

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In general, suppose a_r,¹, a_r,², a_r,r-1, a_r,r+1 a_r,n to be the amounts begged by the rth merchant from the others; and x_r the amount that he had initially. Then x_1 + \Sigma' a_{1,m} &= b_1 (\Sigma x - x_1 - \Sigma' a_{1,m}), / x_2 + \Sigma' a_{2,m} &= b_2 (\Sigma x - x_2 - \Sigma' a_{2,m}), / \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots / x_n + \Sigma' a_{n,m} &= b_n (\Sigma x - x_n - \Sigma' a_{n,m}), where Σ' a_r,m denotes summation from m = 1 to m = n excluding m = r. Therefore \Sigma x + (b_1 + 1)\Sigma' a_{1,m} &= (b_1 + 1)(\Sigma x - x_1), / \Sigma x + (b_2 + 1)\Sigma' a_{2,m} &= (b_2 + 1)(\Sigma x - x_2), / \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots / \Sigma x + (b_n + 1)\Sigma' a_{n,m} &= (b_n + 1)(\Sigma x - x_n). Let k_r = (b_r + 1)Σ' a_r,m, r = 1, 2, 3,, n.

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¹ This is the name given by Mahâvîra to problems involving equations of type (III).

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² *GSS*, vi. 253½-5½.

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Then dividing the foregoing equations by b₁ + 1, b₂ + 1,, respectively, and adding together, we get Σ x (1/(b₁ + 1) + 1/(b₂ + 1) + + 1/(b_n + 1)) + (k₁/(b₁ + 1) + k₂/(b₂ + 1) + + k_n/(b_n + 1)) = (n - 1)Σ x. \therefore \Sigma x = & (k_1/(b_1 + 1) + k_2/(b_2 + 1) + \dots + k_n/(b_n + 1)) / & \div (b_1/(b_1 + 1) + b_2/(b_2 + 1) + \dots + b_n/(b_n + 1) - 1). Whence x_r = & { (k_r + b_r k_1)/(b_1 + 1) + (k_r + b_r k_2)/(b_2 + 1) + \dots + (k_r + b_r k_{r-1})/(b_{r-1} + 1) / & + (k_r + b_r k_{r+1})/(b_{r+1} + 1) + \dots + (k_r + b_r k_n)/(b_n + 1) - (n - 2)k_r } / & \div (b_r + 1)(b_1/(b_1 + 1) + b_2/(b_2 + 1) + \dots + b_n/(b_n + 1) - 1).

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Mahâvîra describes the solution thus:

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“The sum of the amounts begged by each person is multiplied by the multiple number relating to him as increased by unity. With these (products), the amounts at hand are determined according to the rule *Iṣṭaguṇa-ghna* etc.¹ They are reduced to a common denominator, and then divided by the sum diminished by unity of the multiple numbers divided by themselves as increased by unity. (The quotients) should be known to be the amounts in the hands of the persons.”²

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¹ The reference is to rule vi. 241.

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² *GSS*, vi. 251½-252½.

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Problems of the above kind have been treated also by Nârâyaṇa (1357). He says:

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“Multiply the sum of the monies received by each person by his multiple number plus unity. Then proceed as in the method for “the purse of discord.” Divide the multiple number related to each by the same as increased by unity. By the sum diminished by unity of these quotients, divide the results just obtained. The quotients will be the several amounts in their possession.”¹

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¹ *GK*, ii. 33f.

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One particular case, where b₁ = b₂ = b_n = 1 and c₁ = c₂ = = c_n = c, was treated by Brahmagupta (628). He gave the rule:

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“The total value (of the unknown quantities) plus or minus the individual values (of the unknowns) multiplied by an optional number being severally (given), the sum (of the given quantities) divided by the number of unknowns increased or decreased by the multiplier will be the total value; thence the rest (can be determined).”²

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² *BrSpSi*, xiii. 47.

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$Σ x ± cx₁ = a₁, Σ x ± cx₂ = a₂, , Σ x ± cx_n = a_n.$

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$Therefore Σ x = (a₁ + a₂ + + a_n)/(n ± c).$

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$Hence x₁ = 1/c ( ± a₁ (a₁ + a₂ + + a_n)/(n ± c) );$ and so on.

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**Brahmagupta's Rule.** Brahmagupta (628) states the following rule for solving linear equations involving several unknowns:

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“Removing the other unknowns from (the side of) the first unknown and dividing by the coefficient of the first unknown, the value of the first unknown (is obtained). In the case of more (values of the first unknown),

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two and two (of them) should be considered after reducing them to common denominators. And (so on) repeatedly. If more unknowns remain (in the final equation), the method of the pulveriser (should be employed). (Then proceeding) reversely (the values of other unknowns can be found).”¹

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¹ *BrSpSi*, xviii. 51.

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Pṛthûdakasvâmî (860) has explained it thus:

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“In an example in which there are two or more unknown quantities, colours such as *yâvat-tâvat*, etc., should be assumed for their values. Upon them should be performed all operations conformably to the statement of the example and thus should be carefully framed two or more sides and also equations. Equi-clearance should be made first between two and two of them and so on to the last: from one side one unknown should be cleared, other unknowns reduced to a common denominator and also the absolute numbers should be cleared from the side opposite. The residue of other unknowns being divided by the residual coefficient of the first unknown will give the value of the first unknown. If there be obtained several such values, then with two and two of them, equations should be formed after reduction to common denominators. Proceeding in this way to the end find out the value of one unknown. If that value be (in terms of) another unknown then the coefficients of those two will be reciprocally the values of the two unknowns. If, however, there be present more unknowns in that value, the method of the pulveriser should be employed. Arbitrary values may then be assumed for some of the unknowns.”

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It will be noted that the above rule embraces the indeterminate as well as the determinate equations. In fact, all the examples given by Brahmagupta in illustration of the rule are of indeterminate character. We shall mention some of them subsequently at their proper places. So far as the determinate simultaneous equations are concerned, Brahmagupta's method for solving them will be easily recognised to be the same as our present one.

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**Mahâvîra's Rules.** Certain interest problems treated by Mahâvîra lead to simple simultaneous equations involving several unknowns. In these problems certain capital amounts (c₁, c₂, c₃, . . .) are stated to have been lent out at the same rate of interest (r) for different periods of time (t₁, t₂, t₃, . . .). If (i₁, i₂, i₃, . . .) be the interests accrued on the several capitals, $i₁ = rt₁c₁/100, i₂ = rt₂c₂/100, i₃ = rt₃c₃/100, . . . .$

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(i) If i₁ + i₂ + i₃ + . . . . = I, c_r and t_r be known (for r = 1, 2, . . .), we have $i₁ = Ic₁t₁/(c₁t₁ + c₂t₂ + c₃t₃ + . . . .),$ with similar values for i₂, i₃, . . . .

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(ii) Or, if c₁ + c₂ + c₃ + . . . = C, i_r and t_r be known (for r = 1, 2, . . .), we have $c₁ = (Ci₁/t₁)/(i₁/t₁ + i₂/t₂ + . . . . .),$ and so on.

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(iii) Or, if we are given the sum of the periods t₁ + t₂ + . . . = T, c_r and i_r, then $t₁ = (Ti₁/c₁)/(i₁/c₁ + i₂/c₂ + . . . . .),$ with similar values for t₂, t₃, . . . .

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Mahâvîra has given separate rules for the solution of problems of each of the above three kinds.¹

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¹ *GSS*, vi. 37, 39, 42.

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**Bhâskara's Rule.** Bhâskara II has given practically the same rule as that of Brahmagupta for the solution of simultaneous linear equations involving several unknowns. We take the following illustrations from his works.

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*Example 1.* "Eight rubies, ten emeralds and a hundred pearls which are in thy ear-ring were purchased by me for thee at an equal amount; the sum of the price rates of the three sorts of gems is three less than the half of a hundred. Tell me, O dear auspicious lady, if thou be skilled in mathematics, the price of each."²

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² *BBi*, p. 47.

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If x, y, z be the prices of a ruby, emerald and pearl respectively, then $8x = 10y = 100z,$ $x + y + z = 47.$

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Assuming the equal amount to be w, says Bhâskara II, we shall get $x = w/8, y = w/10, z = w/100.$

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Substituting in the remaining equation, we easily get w = 200. Therefore $x = 25, y = 20, z = 2.$

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*Example 2.* "Tell the three numbers which become equal when added with their half, one-fifth and one-ninth parts, and each of which, when diminished by those parts of the other two, leaves sixty as remainder."³

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³ *BBi*, p. 52.

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Here we have the equations $1 x + x/2 = y + y/5 = z + z/9,$ $2 x - y/5 - z/9 = y - z/9 - x/2 = z - x/2 - y/5 = 60.$

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Bhâskara puts *w* for each of the equal quantities in (1), so that $x = ⅔w, y = ⅚w, z = 9/10w.$ Substituting these values in (2), any one of them will give $⅖w = 60;$ whence w = 150. Therefore $x = 100, y = 125, z = 135.$

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It should be noted that the equations (2) are sufficient for the determination of the unknowns.

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*Example 3*. Another type of problem is: Three portions (x, y, z) of a sum of money (c) were lent out at three different rates of interest (r₁, r₂, r₃ per cent per month) for three different periods (t₁, t₂, t₃ months). The interests accrued on them severally were the same. What were those portions?¹

Linked texts (6)

Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Bījagaṇita — Bhāskara II, Sanskrit
Bakhshālī Manuscript — Sanskrit
Gaṇitakaumudī — Nārāyaṇa Paṇḍita (mathematician), Sanskrit