Equations

Equations
File:Algebraic equation notation.svg

Forming Equations

We need to carry out certain preliminary operations on equations before moving into actual solution.

We need to form the equation ( samī-karaṇa, samī-kāra or samī-kriyā; from sama, equal and kṛ , to do; hence literally , making equal) from the given conditions of the proposed problem. This 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

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Algebraic Equation

Algebraic expression can be understood with the following example.

Ram says that he has 10 coins more than Shyam. We do not know exactly how many coins Ram has. Ram may have any number of coins. With the given information

Number of coins held by Ram = Number of coins held by Shyam + 10

We will denote the ‘number of coins held by Shyam by the letter x. Here x is unknown which could be 1, 2, 3, 4, etc.

Using x, we write,

Number of coin held by Ram = x+10.

Thus 'x + 10' is an algebraic expression.

Algebra utilizes 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.	Components of an algebraic expression	Samskrit word	Symbol	Examples
1	Unknown	यावत्तावत् कालकः नीलकः , ......	या का नी , ........	या ३५ का १४ नी ८२	35x 14y 82z
2	Sum	योगः	-	या का या ३५ का १४	x + y 35x + 14y
3	Product	भावितम्	भा	याकाभा याकाभा ३२	xy 32xy
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 ३२	32
8	Negative	ऋणम्	dot above the quantity (.)	. रू ४३२	-432

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). constant term was denoted by the letter rū, an abbreviation of rūpa as shown in the above table. 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 + 17	या १ रू १७
2	7x - 17	या ७ रू १७.
3	18x – 8	या १८ रू ८.
4	15x2 + 17x - 2	याव १५ या १७ रू २.
5	1x4 + 16x3 + 25x2 + 8x + 6	यावव १ याघ १६ याव २५ या ८ रू ६
6	8x2 + 12xy - 6xz -16x	याव ८ याकाभा १२ यानीभा ६. या १६.

How algebraic expressions are written by Ancient Indian mathematicians.

Consider the equation 10 x - 18 = x² +14

This can be written as,

0x² + 10 x - 18 = 1x² + 0x + 14

By looking at the positions of x², x¹, x⁰ (constant term), there is some 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).

Pṛthūdakasvāmin (864 CE) in his commentary on the Brāhma-sphuṭa-siddhānta writes the equation 40x - 48 = x² + 51 as below

Devanāgari	Transliteration		Modern notation
याव ०  या ४०  रू ४८. याव १  या ०    रू ५१	Yāva 0 yā  40 rū 48. Yāva 1 yā 0 rū 51	⇒	0x2 + 40 x - 48 = 1x2 + 0x + 51

Here is an 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:

Addition and subtraction can be performed with like terms and unlike terms are to be kept separately. Same letter variables raised to the same powers are treated as like terms. E.g., या ४, या ५, या ६ are like terms. याव ७, याव ८, याव ९ are also like terms. का ३, का ७, का १५ are also like terms. Presently we say 4x, 5x, 6x are like terms. Similarly 7x², 8x², 9x² 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. 4x + 6x can be simplified as 10x. 9x² - 7x² can be simplified as 2x².

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 . Presently, 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 २ रू ४ and या ३ रू ५ are multiplicand and multiplier respectively, we can get their product as mentioned below	If 2x + 4 and 3x + 5 are multiplicand and multiplier respectively, we can get their product as mentioned below
The multiplier contains two terms, i.e., या ३ and रू ५	The multiplier contains two terms, i.e., 3x and 5
Multiply the multiplicand with the two terms of the multiplier separately as mentioned below. (या २ रू ४) X या ३ = याव ६ या १२ (या २ रू ४) X रू ५ = या १० रू २०	Multiply the multiplicand with the two terms of the multiplier separately as mentioned below. (2x + 4) X 3x = 6x2 + 12x (2x + 4) X 5 = 10x + 20
Add the results. The result of the multiplication is : याव् ६ या २२ रू २०	Add the results. The result of the multiplication is : 6x2 + 22x + 20

If $ax+b$ and $cx+d$ are multiplicand and multiplier respectively, we get their product as mentioned below.

The multiplier has two terms, i.e., cx and d. Multiply the multiplicand with the two terms of the multiplier separately as mentioned below.

$(ax+b)cx=acx^{2}+bcx$

$(ax+b)d=adx+bd$

Add the results.

The multiplication result is : $acx^{2}+(bc+ad)x+bd$

Classification of Equations

In the canonical work of circa 300 B.C. found that Hindu classification of 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).

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).

Equations in one unknown (eka-varna-samīkaraṇa) 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. He classified as : (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, namely " (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 first classification again, comprises of two subclasses: (i) simple equations and (ii) quadratic and higher equations. The second classification 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 having only the first power of the variables, coefficients and constants. For example, the equation 4x + 7 = 8 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:

In śulba geometrical solution of a linear equation in one unknown is found , the earliest of which is not later than 800 B.C.

Sthānāṅga-Sūtra (c. 300 B.C.) has a reference to a linear equation by its name (yāvat -tāvat ) which is suggestive of the method of solution followed at that time.

Bakhshālī treatise has problems involving simple algebraic equations and solution method, probably written in 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

2*3=6

6*4 =24

1

2

6

24

added

1 + 2 + 6 + 24 = 33

. "Divide the visible quantity'

132 33

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

āryabhaṭa I(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 considers 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?

Let x be the unknown amount, with the given information

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.

Narayana 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 Gangadhara (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

File:Linear equation for y=3x+1.png

Linear Equation

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

$$