Equations

Forming Equations

Before proceeding to the actual solution of an equation of any type, certain preliminary operations have necessarily to be carried out in order to prepare it for solution.

Still more preliminary work is that of forming the equation ( samī-karaṇa, samī-kāra or samī-kriyā; from sama, equal and kṛ , to do; hence literally , making equal) from the conditions of the proposed problem. Such preliminary work may require the application of one or more fundamental operations of algebra or arithmetic.

Bhāskara II says: "Let yāvat-tāvat be assumed as the value of the unknown quantity. Then doing precisely as has been specifically told-by subtracting, adding, multiplying or dividing the two equal sides of an equation should be very carefully built.

Algebraic Expressions and Algebraic Equations

What is an algebraic expression ? Let us try to understand this with an example.

Geeta says that she has 10 marbles more than Mala. We do not know exactly how many marbles Mala has. She may have any number of marbles. But we know that

Number of marbles of Geeta = Number of marbles of Mala = 10

We shall denote the ‘number of marbles of Mala' by the letter x. Here x is unknown which could be 1, 2, 3, 4, etc.

Using x, we write,

Number of marbles of Geeta = x+10.

Thus 'x + 10' is an algebraic expression.

Algebra abounds in the usage of symbols. These symbols represent the unknown quantities and operations performed with them. The following table gives the symbols which were used for some basic operations by the ancient Indian Mathematicians.

S. No.	Constituent of an algebraic expression	Samskrit word	Symbol	Examples
1	Unknown	यावत्तावत् कालकः नीलकः , ......	या का नी , ........	या ३ का ४ नी ८	3x 4y 8z
2	Sum	योगः	-	या का या ३ का ४	x + y 3x + 4y
3	Product	भावितम्	भा	याकाभा याकाभा ३	xy 3xy
4	Square	वर्गः	व	याव	x2
5	Cube	घनः	घ	याघ	x3
6	Fourth Power	वर्ग​-वर्गः	वव	यावव	x4
7	Constant Term	रूपम्	रू	रू ३	3
8	Negative	ऋणम्	dot above the quantity (.)	. रू ४	-4

The letter yā (an abbreviation of yāvat-tāvat) was the most popular representation of the unknown quantity. Its square was termed yāva, the abbreviation of yāvat-tāvat-varga (varga means square). While writing equations, the constant term was denoted by the letter rū, an abbreviation of rūpa as seen in the table above. Any negative sign in the equation is denoted by a dot above the term.

If there are three unknown quantities in an expression, the symbols used are yā, kā and nī. These are the abbreviations for yāvat tāvat, kālaka and nīlaka. The product of first two unknown quantities is represented as yākābha where yā and kā are the two unknowns and bha stands for their product.

The following table gives a representation of some of the algebraic expressions used by ancient Indian mathematicians.

S.No.	Modern Notation	Ancient Indian Notation
1	x + 1	या १ रू १
2	3x - 7	या ३ रू ७.
3	2x – 8	या २ रू ८.
4	15x2 + 7x - 2	याव १५ या ७ रू २.
5	1x4 + 6x3 + 25x2 + 48x + 64	यावव १ याघ ६ याव २५ या ४८ रू ६४
6	18x2 + 12xy - 6xz -6x	याव १८ याकाभा १२ यानीभा ६. या ६.

We will see how algebraic expressions are written by ancient Indian mathematicians.

Consider the equation 10 x - 8 = x² +1

This can be written as,

0x² + 10 x - 8 = 1x² + 0x + 1

Observe the positions of x², x¹, x⁰ (constant term). Can you notice any pattern ? The standard way of writing an equation starts from the highest power of x. Then the powers of x were written in the descending order up to its lowest power. This format of writing equation was followed by mathematicians from ancient times.

Brahmagupta called an equation as samakaraṇa or samīkaraṇa. It means 'making equal. The two sides of an equation (LHS and RHS) were written one below the other. The symbol '=' was not used. Both the sides of an equation were made same by finding the appropriate value(s) for the unknown(s).

We shall see how this equation was written by Pṛthūdakasvāmin (864 CE) in his commentary on the Brāhma-sphuṭa-siddhānta. He writes the equation

10x - 8 = x² + 1 as follows:

Devanāgari	Transliteration		Modern notation
याव ०  या १०  रू ८. याव १  या ०    रू १	Yāva 0 yā  10 rū 8. Yāva 1 yā 0 rū 1	⇒	0x2 + 10 x - 8 = 1x2 + 0x + 1

Another example of an equation from Bījagaṇita of Bhāskara II is:

X⁴ - 2x² - 400x = 9999

This is represented as,

यावव १ याव २^.   या  ४^.०० रू ०

यावव ० याव ०   या  ०       रू ९९९९

Operations with Algebraic Expressions

Bhāskara II gives the operations using algebraic terms as follows:

स्याद्रूपवर्णाभिहतौ तु वर्णो द्वित्र्यादिकानां समजातिकानाम् ॥

वधे तु तद्वर्गघनादयः स्युस्तद्भावितं चासमजातिघाते।

भागादिकं रूपवदेव शेषं व्यक्ते यदुक्तं गणिते तदत्र ॥

“The product of a numerical constant and an unknown quantity is an unknown quantity. Products of two or three like terms are their squares or cubes (respectively). Product of unlike terms is bhāvita. Fractions etc. are as in the case of knowns. The other (processes) are same as explained in arithmetic."

Addition and Subtraction of Algebraic Expressions

Bhāskara II gives the rule for addition and subtraction of unknown quantities as follows:

योगोऽन्तरं तेषु समानजात्योर्विभिन्नजात्योश्च पृथक् स्थितिश्च।

“Addition and subtraction are performed amongst like terms. The unlike terms are to be kept separately."

Explanation:

It is well known that the addition and subtraction can be performed only amongst like terms and unlike terms are to be kept separately. Like terms are the terms that contain the same letter variables raised to the same powers. E.g., या ३, या ४, या ५ are like terms. याव २, याव ५, याव ७ are also like terms. का ३, का ७, का १५ are also like terms. Nowadays we say 3x, 4x, 5x are like terms. Similarly 2x², 5x², 7x² are like terms. and 3y, 7y, 15y are also like terms. When we have like terms, the sum and difference can be simplified. E.g. 3x + 5x can be simplified as 8x. 10x² - 4x² can be simplified as 6x².

Unlike terms are those terms having different variables or variables with different powers. E.g. या ३, याव ३, याघ ४, का ५, काव, याकाभा . In modern notation, these are represented as 3x, 3x², 4x³, 5y, y², xy.

Multiplication of Algebraic Expressions

Bījagaṇita gives a rule for multiplication -

गुण्यः पृथग्गुणकखण्डसमो निवेश्यस्तैः खण्डकैः क्रमहतः सहितो यथोक्त्या।

अव्यक्तवर्गकरणीगणनास चिन्त्यो व्यक्तोक्तखण्डगुणनाविधिरेवमत्र॥

“Place the multiplicand at as many places as the terms of the multiplier. Multiply with the terms of the multiplier in order separately and add the results as directed in the problem. This is applicable in the case of squares of unknown numbers and surds also. The method of partial products stated in the case of arithmetic numbers is applicable here also.”

Explanation

Ancient Indian notation	Modern notation
If या ३ रू ५ and या ४ रू ७ are multiplicand and multiplier respectively, their product can be obtained as follows:	If 3x + 5 and 4x + 7 are multiplicand and multiplier respectively, their product can be obtained as follows:
The multiplier has two terms, i.e., या ४ and रू ७	The multiplier has two terms, i.e., 4x and 7
Place the multiplicand at two places. Multiply them with the terms of the multiplier separately as shown. (या ३ रू ५) X या ४ = याव १२ या २० (या ३ रू ५) X रू ७ = या २१ रू ३५	Place the multiplicand at two places. Multiply them with the terms of the multiplier separately as shown. (3x + 5) X 4x = 12x2 + 20x (3x + 5) X 7 = 21x + 35
Add the results. The multiplication result is: याव् १२ या ४१ रू ३५	Add the results. The multiplication result is: 12x2 + 41x + 35

If ax + b and cx + d are multiplicand and multiplier respectively, their product can be obtained as follows:

The multiplier has two terms, i.e., cx and d. Place the multiplicand at two places. Multiply them with the terms of the multiplier separately as shown.

(ax +b) x cx = acx² + bcx

(ax + b) xd = adx + bd

Add the results.

The multiplication result is: acx² + (bc + ad)x + bd

Algebraic Notations

• The symbols used for unknown numbers are the initial syllables yа̄ of yа̄vat-tа̄vat (as much as), kа̄ of kа̄laka (black), nī of nīlaka (blue), pī of pīta (yellow) etc.

• The product of two unknowns is denoted by the initial syllable bhā of bhāvita (product) placed after them. The powers are denoted by the initial letters va of varga (square), gha of ghana (cube); vava stands for vargavarga, the fourth power. Sometimes the initial syllable ghā of ghāta (product) stands for the sum of powers.

• A coefficient is placed next to the symbol. The constant term is denoted by the initial symbol rū of rūpa (form).

• A dot is placed above the negative integer

• The two sides of an equation are placed one below the other. Thus the equation X⁴ - 2X² - 400x = 9999; is written as

यावव​ १ याव २^● या ४००^● रू ०

यावव​ ० याव ० या ० रू ९९९९

which means writing x for या

x⁴ -2x² -400x+0 = 0x⁴ +0x²+0x+9999

If there be several unknowns, those of the same kind are written in the same column with zero coefficients, if necessary. Thus the equation

197x - 1644y - z = 6302 is represented by

yа̄ 197 kа̄ 1644^● ni 1^● ru 0

yа̄ 0 kа̄ 0 ni 0 ru 6302

which means, putting y for kа̄ and z for ni

197x - 1644Y - z + 0 = 0x + 0y + 0z + 6302.

Bhāskara II says:

"Then the unknown on one side of it (the equation) should be subtracted from the unknown on the other side; so also the square and other powers of the unknown;

the known quantities on the other side should be subtracted from the known quantities of another side."

The following illustration is from the Bījagaṇita of Bhāskara II:

"Thus the two sides are

yā va 4 yā 34^● rū 72

yā va 0 yā 0 rū 90

On complete clearance (samaśodhana), the residues of the two sides are

yā va 4 yā 34^● rū 0

yā va 0 yā 0 rū 18

i.e., 4x² - 34x = 18

Classification of Equations

The earliest Hindu classification of equations seems to have been according to their degrees, such as simple (technically called yāvat tāvat ), quadratic (varga), cubic (ghana) and biquadratic (varga-varga). Reference to it is found in a canonical work of circa 300 B.C. But in the absence of further corroborative evidence, we cannot be sure of it. Brahmagupta (628) has classified equations as: (I) equations in one unknown (eka-varna-samīkaraṇa), (2) equations in several unknowns (aneka-varna-samīkaraṇa), and (3) equations involving products of unknowns (bhaivita).

The first class is again divided into two sub classes, viz.,(i) linear equations, and (ii) quadratic equations (avyakta-varga-samīkaraṇa). Here then we have the beginning of our present method of classifying equations according to their degrees. The method of classification adopted by Pṛthūdakasvāmī (860) is slightly different. His four classes are: (1) linear equations with one unknown, (2) linear equations with more unknowns, (3) equations with one, two or more unknowns in their second and higher powers, and (4) equations involving products of unknowns. As the method of solution of an equation of the third class is based upon the principle of the elimination of the middle term, that class is called by the name madhyamāharaṇa (from madhyama, "middle", aharana "elimination", hence meaning "elimination of the middle term"). For the other classes, the old names given by Brahmagupta have been retained. This method of classification has been followed by subsequent writers.

Bhāskara II distinguishes two types in the third class, viZ" (i) equations in one unknown in its second and higher powers and (ii) equations having two or more unknowns in their second and higher powers.' According to Krsna (1580) equations are primarily of two classes: (1) equations in one unknown and (z) equations in two or more unknowns. The class (1), again, comprises two subclasses: (i) simple equations and (ii) quadratic and higher equations. The class (2) has three subclasses: (i) simultaneous linear equations, (ii) equations involving the second and higher powers of unknowns, and (iii) equations involving products of unknowns. He then observes that these five classes can be reduced to four by including the second subclasses of classes (1) and (2) into one class as madhyamāharaṇa.

Linear Equations in One Unknown

A Linear equation is an equation containing only the first power of the variables, coefficients and constants. For example, the equation 2x + 4 = 5 is a linear equation in one variable. This is called first-order equation since the power of the variable (x) is one. If the equation has highest power of x as two, i.e. x² , then it will be a quadratic (second order) equation.

Early Solutions:

As already stated, the geometrical solution of a linear equation in one unknown is found in the śulba , the earliest of which is not later than 800 B.C. There is a reference in the Sthānāṅga-Sūtra (c. 300 B.C.) to a linear equation by its name (yāvat -tāvat ) which is suggestive of the method of solution! followed at that time.We have, however, no further evidence about it. The earliest Hindu record of doubtless value of problems involving simple algebraic equations and of a method for their solution occurs in the Bakhshālī treatise, which was written very probably about the beginning of the Christian Era.

One problem is "The amount given to the first is not known. The second is given twice as much as the first; the third thrice as much as the second; and the fourth four times as much as the third. The total amount distributed is 132. What is the amount of the first?"

If x be the amount given to the first, then according to the problem,

x + 2X + 6x + 24X = 132.

Rule of False Position:

The solution of this equation is given as follows:

" 'Putting any desired quantity in the vacant place' ; any desired quantity is 1 ; 'then construct the series.

1	2	2 3	6 4
1	1	1 1	1 1

'multiplied'

1

2

6

24

added' 33. "Divide the visible quantity'

132 33

(which) on reduction becomes

4 1

(This is) the amount given (to the first)."

The solution of another set of problems in the Bakhshālī treatise, leads ultimately to an equation of the type ax+ b=p. The method given for its solution is to put any arbitrary value g for x, so that

ag+ b =p' say.

Then the correct value will be

${\displaystyle x={\frac {(p-p')}{a}}+g}$

Solution of Linear Equations

āryabhaṭaI(499) says:

"The difference of the known "amounts" relating to the two persons should be divided by the difference of the coefficients of the unknown. The quotient will be the value of the unknown, if their possessions be equal."

This rule contemplates a problem of this kind: Two persons, who are equally rich, possess respectively a, b times a certain unknown amount together with c, d

units of money in cash. What is that amount?

If x be the unknown amount, then by the problem

ax+ c= bx+ d.

Therefore ${\displaystyle x={\frac {(d-c)}{(a-b)}}}$

Hence the rule.

Rule for solving the linear equation of the form bx + c = dx + e where b, c, d and e are given numbers is given by Brahmagupta as follows.

अव्यक्तान्तरभक्तं व्यस्तं रूपान्तरं समेऽव्यक्तः।

वर्गाव्यक्ताः शोध्या यस्माद्रूपाणि तदधस्तात् ॥

"The difference of absolute numbers, inverted and divided by the difference of unknown, is the [value of the) unknown in an equation.”

Explanation: Consider the equation, bx + c = dx + e

Here x is the unknown quantity whose value is to be found. The letters b and d are its coefficients. The remaining letters c and e are numerical constants.

Difference of absolute numbers = c-e

Difference of absolute numbers inverted = e-c

Difference of coefficients of unknown = b - d

x is found as

${\displaystyle x={\frac {(e-c)}{(b-d)}}}$

Bhāskara II explains how the above formula is obtained.

यावत्तावत् कल्प्यमव्यक्तराशेर्मानं तस्मिन् कुर्वतोद्दिष्टमेव ।

तुल्यौ पक्षौ साधनीयौ प्रयत्नात्त्यक्त्वा क्षिप्त्वा वाऽपि संगुण्य भक्त्वा ॥

एकाव्यक्तं शोधयेदन्यपक्षाद्रूपाण्यन्यस्येतरस्माच्च पक्षात्

शेषाव्यक्तेनोद्धरेद्रूपशेषं व्यक्तं मानं जायतेऽव्यक्तराशेः॥

“Assume the unknown quantity (x). Perform the desired process by transposing the factors involving unknown terms to one side and constant terms to the other side after cancelling or reducing or multiplying or dividing. Divide the terms by the coefficient of the unknown and calculate the value of the unknown factor."

Explanation: For instance, let us consider the following equation:

6x - 5 = 2x + 3

(i) Transposing the factors involving unknown terms to one side and the constants to the other side, we get,

6x - 2x = 3 + 5

Hence, 4x = 8

ii) Dividing the terms by the coefficient of the unknown, we get

x = 2

Sripati writes :

"First remove the unknown from anyone of the sides (of the equation) leaving the known term; the reverse (should be done) on the other side. The difference of the absolute terms taken in the reverse order divided by the difference of the coefficients of the unknown will be the value of the unknown.

Naraya writes:

"From one side clear off the' unknown and from the other the known quantities; then divide the residual known by the residual coefficient of the unknown. Thus will certainly become known the value of the unknown. "

For illustration we take a problem proposed by Brahmagupta :

"Tell the number of elapsed days for the time when four times the twelfth part of the residual degrees increased by one, plus eight will be equal to the residual

degrees plus one."

It has been solved by Pṛthūdakasvāmī as follows:

"Here the residual degrees are (put as) yāvat -tāvat ,

ya increased by one, ya 1 ru 1; twelfth part of it, (ya 1 ru 1) / 12

four times this, (ya 1 ru 1) / 3 ; plus the absolute quantity eight, (ya 1 ru 25) / 3 . This is equal to the residual degrees plus unity. The statement of both sides

tripled is

ya 1 ru 25

ya 3 ru 3

The difference between the coefficients of the unknown is 2. By this the difference of the absolute terms, namely 22, being divided, is produced the residual of the degrees of the sun 11. These residual degrees should be known to be irreducible. The elapsed days can be deduced then, (proceeding) as before."

In other words, we have to solve the equation

${\displaystyle {{\frac {4}{12}}(x+1)+8=x+1}}$

which gives x + 25 = 3x + 3

2x = 22

Therefore x= 11

The following problem and its solution are from the Bijaganita of Bhāskara II :

"One person has three hundred coins and six horses. Another has ten horses (each) of similar value and he has further a debt of hundred coins. But they

are of equal worth. What is the price of a horse?

"Here the statement for equi-clearance is :

6x + 300 = 10x - 100.

Now, by the rule, 'Subtract the unknown on one side from that on the other etc.,' unknown on the first side being subtracted from the unknown on the other side,

the remainder is 4x. The absolute term on the second side being subtracted from the absolute term on the first side, the remainder is 400. The residual known

number 400 being divided by the coefficient of the residual unknown 4x, the quotient is recognized to be the value of x, (namely) 100."

Linear Equations with Two Unknowns

Rule of Concurrence

One topic commonly discussed by almost all Hindu writers goes by the special name of sankramana (concurrence). According to Nārāyana(1350), it is also called sankrama and sankraama. Brahmagupta (628) includes it in algebra while others consider it as falling within the scope of arithmetic. As explained by the commentator Ganga-,dhara (1420), the subject of discussion here is "the investigation of two quantities concurrent or grown together in the form of their sum and difference."

In other words sankramana is the solution of the simultaneous equations

x+ y= a, x-y= b.

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 unknown quantities): (this is) concurrence. The same rule is restated by him on a different occasion 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.

Linear Equations

Mahāvīra gives the following examples leading to simultaneous linear equations together with rules for the solution of each.

Example. "The price of 9 citrons and 7 fragrant wood-apples taken together is 107; again the price of 7 citrons and 9 fragrant wood-apples taken together

is 101. O mathematician, tell me quickly the price of a citron and of a fragrant wood-apple quite separately."

If x, y be the prices of a citron and of a fragrant wood-apple respectively, then

9x+7y= 107,

7x+9y = 101.

Or, in general,

ax+ by = m

bx + ay = n

Solution. "From the larger amount of price multiplied by the (corresponding) bigger number of things subtract the smaller amount of price multiplied by the (corresponding) smaller number of things. (The remainder) divided by the difference of the squares of the numbers of things will be the price of each of the bigger number of things. The price of the other will be obtained by reversing the multipliers.

Thus ${\displaystyle x={\frac {(am-bn)}{(a^{2}-b^{2})}}}$ , ${\displaystyle y={\frac {(an-bm)}{(a^{2}-b^{2})}}}$

The following example with its solution is taken from the BfjagatJita of Bhāskara II :

Example. "One says, 'Give me a hundred, friend, I shall then become twice as rich as you.' The other replies, 'If you give me ten, I shall be six times as rich

as you.' Tell me what is the amount of their (respective) capitals ?"

The equations are

x + 100 = 2(y - 100), (I)

y + 10 = 6(x - 10). (2)

Bhāskara II indicates two methods of solving these equations. They are substantially as follows:

First Method: Assume x = 2z.- 100, y = z + 100,

so that equation (I) is identically satisfied. Substituting

these values in the other equation, we get

z + 110 = 12z- 660;

Hence z =.70 Therefore, x = 40 , y = 170 .

Second Method: From equation (I), we get

x =2y - 300,

and from equation (2)

${\displaystyle x={\frac {1}{6}}(y+70)}$

Equating these two values of x, we have

${\displaystyle 2y-300={\frac {1}{6}}(y+70)}$

${\displaystyle 12y-1800=y+70}$

Hence y= 170. Substituting this value of y in any of the two expressions for x, we get x = 40.

Linear Equations With Several Unknowns

A Type of Linear Equations

The earliest Hindu treatment of systems of linear equations involving several unknowns is found in the Bakhshālī 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."

If x1, x2, x3 be the wealths of the three merchants respectively, then x1 + x2 = 13, x2 + x3 = 14, x3 + x1 = 15.

Another problem is "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. x₁ x₂ x₃ x₄ x₅

x₁ + x₂ = 16, x₂ + x₃ = 17, x₃+ x₄ = 18, x₄ + x₅ = 19, x₅ + x₁= 20.

There are in the work a few other similar problems. 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 n being odd.

Solution by False Position

A system of linear equations of this type is solved in the Bakhshālī treatise substantially as follows:

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

$x_1=p+\frac{1}{2}(a_n-<!--\; -\; -->$