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A LG E B R A.

Part I.
Of Equa- Cor. The figns of all the terms of an equation may

Or
7*=84

Of Equa-
tion.

cion.

84
be changed into the contrary figns, and it will continue

R. 2.
to be true.
Rule 2. Any quantity by which the unknown quan.

III.
tity is multiplied may be taken away, by dividing all

9
the other quantities of the equation by it.

1+2=16
Thus, if ax=1
b

R. 3. 20+9x=64*
Also, if mx+nb-am

R. I. 2055*
nb

R. 2.
xtra

55
For if equal quantities are divided by the same
quantity, the quotients are equal.

2. Solution of Questions producing fimple Equations.
Rule 3. If a term of an equation is fractional, its de From the resolution of equations we obtain the re-

nominator may be taken away, by multiplying all solultion of a variety of useful problems, both in pure
the other terms by it.

mathematics and phyfics, and also in the practical arts

founded upon these sciences. In this place, we conThus, if =670 Also, if a

sider the application of it to those questions where the

quantities are expressed by numbers, and their magnix=ab + ac

ax-b-cx

tude alone is to be considered. And by trans. ax-x=b

When an equation, containing only one unknown

b
· And by div. x=

quantity, is deduced from the question by the follow
ing rules, it is sometimes called a final equation. If it

be fimple, it may be resolved by the preceding rules ;
For if all the terms of the equation are multiplied by but if it be of a superior order, it must

be resolved by the same quanity, it remains a true propofition.

the rules afterwards to be explained. The examples in

this chapter are so contrived, that the final equation
Corollary to the three last Rules.

may be fimple.
If any quantity be found on both sides of the equa- The rules given in this fe&tion for the solution of
tion, with the same sign, it may be taken away from questions, though they contain a reference to simple
both. (Rule 1.)

equations only, are to be considered as general, and as
Also, if all the terms in the equation are multiplied applicable to questions which produce equations of any
or divided by the same quantity, it may be taken out order.
of them all. (Rule 2. and 3.)

General Rule. The unknown quantities in the question
Ex. If 3xta=a+b, then 3x=b.

proposed must be expressed by letters, and the rela-
If 2ax + 3ab=ma+a*, then 2x+36=mta.

3b

tions of the known and unknown quantities con

tained in it, or the conditions of it, as they are call. If

then x-4=16.
3

ed, must be expressed by equations. These equa-
3 -3
3

tions being resolved by the rules of this science, will
Any fimple equation may be resolved by these rules

give the answer of the queftion.
in the following manner. 1/1, Any fraction may be
taken away by R. 3. 2dly, All the terms including

For example, if the question is concerning two num-
the unknown quantity, may be brought to one side of bers, they may be called x and y, and the conditions
the equation, and the known terms to the other, by from which they are to be investigated must be expref-
R. 1. Lastly, If the unknown quantity is multiplied fible by equations.
by any known quantity, it may be made to stand alone Thus, if it be required that the
by Ri 2. and the equation will then be resolved. Def. 3.

sum of two numbers fought

x+y=360

be 6o, that condition is exExamples of fimple Equations resolved by these Rules.

prefled thus :
I.

If their difference must be 24, then xy=24
If 3x+5=x+9

If their product is 1640, then xy=1640
R. 1. 2x=4

4
R. 2. X=
=.

2,

=
II.

2therefore
4x
If x*+12=*+26
~
3

These are some of the relations which are moft ea.

fily expreffed. Many others occur which are less ob-
5x__

4*
=14

vious ; but as they cannot be described in particular
3
R. 3• 30x154–8x=84.

rules, the algebraical expression of them is best explain

ed by examples, and must be acquired by experience. No it,

A

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If their quotient mult be 6, then =6
If their ratio is as 3. tor Forthen}2x=34.

5x5

2

R. 1. 5*

2

Equa

00.

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of Equa- A diftinct conception of the nature of the question, added a half, a third part, and a fourth part of itself, of Equa. tions.

tions,
and of the relations of the several quantities to which the sum will be 50.
it refers, will generally lead to the proper method of
stating it, which in effect may be considered only as a Let it be z: then half of it is a third of it &c.

3'
translation from common language into that of alge-
bra.

Therefore, zt;tit. 50
, =

3
Cafe I. When there is only one unknown quantity to

242 +122 +82 +6z=1200 be found.

502= 1200 Rule. An equation involving the unknown quantity

224 must be deduced from the question (by the general

If the operation be more complicated, it may be
rule). This equation being resolved by the rules of useful to register the several steps of it, as in the fol-
the last fe&tion, will give the answer.

lowing
It is obvious, that, when there is only one unknown Example. 2. A trader allows L. 100 per annum for the
quantity, there must be only one independent equation expences of his family, and augments yearly that
contained in the question ; for any other would be un-

part of his stock which is not so expended by a
necessary, and might be contradictory to the former. third part of it; at the end of three years his original
Example. 1. To find a number, to which if there be stock was doubled. What had he at fire ?

Let his firft ftock be
Of which he spends the first year L. 100, and

there remains
This remainder is increased by a third of it-

47400
felf

3

3 42-400

42-700 The fecond year be spends L. 100, and there } 41

3

3 He increases the remainder by one-third of 42-700, 42-700

162-2800 5

}
it


3
9

9
The third year he spends L. 100, and there

-2800 remains

9

9

162—3700_643–14800 He increases it by one-third

7

+
9
27

27
But at the end of the third year his stock is 642–14800

8 doubled; therefore

27 By R. 3.

9642--14800=542 By R. i.

10 1OZ=14800
By R. 2.

112=1480
Therefore his stock was L. 1480; which being tried, tion, under two different forms, from which no solu-
answers the conditions of the question.

tion can be derived.

Example. 3. Two persons, A and B, were talking of
Cafe II. When there are two unknown quantities.

their
ages:

: says A to B, Seven years ago I was just
Rule. Two independent equations involving the two three times as old as you were, and seven years
unknown quantities, must be derived from the que-

hence I shall be just twice as old as you will be. I ftion. A value of one of the unknown quantities

demand their present ages. must be derived from each of the equations: and

Let the ages

of A
these two values being put equal to each other, a and B be refpec- 1 x and y
new equation will arise, involving only one unknown
quantity, and may therefore be resolved by the pre- Seven years ago they

tively
ceding rule.

2 x7 and y-7
Two equations must be deduced from the question :

Seven years hence
they will be

3 x+7 and y+7
for, from one including two unknown quantities, it is
plain, a known value of either of them cannot be ob. Therefore by Quest.
tained, more than two equations would be unnecessary;

1. and 2.

4 x–73Xy-y=34-21 and if any third condition were assumed at pleasure,

Also by Queft. 2.

5x+7= 2Xy+7=27+14
most probably it would be inconsistent with the other
two, and a question containing three such conditions By 4. and tranfp. 6x=3-14
would be absurd.

By 5. and transp. 7 *= 2y+7
It is to be observed, however, that the two condi- By 6. and 7. 839-14=2y+7
tions, and hence the two equations exprefling them, Transp. and 6.

9 y=21
must be independent; that is, the one must not be de- By 9. and 6. or 7.

101 x=49
ducible from the other by any algebraical reasoning : The
for, otherwise, there would in effect be only one equa- swer the conditions.

of A and B then are 49 and 21, which an

ages Vol. I. Part I.

The

were

and 3.

3 F

410

2

2/+*+2

= 34

68-y

2

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tions. Of Eqau.

The operation might have been a little shortened by independent equations may be derived from a question of Equations. fubtracting the 4th from 5th, and thus 14 =-,+35; as there are unknown quantities in it, these quantities

and thence g=21. therefore (by 6tb) x= (39–14) may be found by the resolution of equations.
=49.

Examp. 6. To find three numbers, so that the first,

with half the other two, the second with one-third Example. 4. A gentleman distributing money among

of the other two, and the third with one-fourth of
fome poor people, found he wanted 105. to be able

the other two, may each be equal to 34.
to give 53. to each ; therefore he gives each 48. on-
ly, and finds he has 5 s. left. - To find the number of

Let the numbers be x, y, z, and the equations are
Thillings and poor people.

ytz
**

= 34
If any question such as this, in which there are two
quantities fought, can be resolved by means of one let-

= 34
ter, the solution is in general more fimple than when

3 two are employed. There must be, however, two in

32+

ty
dependent conditions ; one of which is used in the no.

4
tation of one of the unknown quantities, and the other
gives an equation.

From the ift

4 x
Let the number of poor be

From the 2d 51x=102-3--Z
The number of shillings will be 2.52-10

From the 3d 6x=136-42-y
The number of thillings is also 1342+5

68-y-z
By 2. and 3.
1452-JO=42+5. From 4th and 5th 7

=102-34-2
Transp.

1512=15.
The number of poor therefore is 15, and the num-

136-Z

8
ber of shillings is (42+s=) 65, which answers the 7th reduced
)

5
conditions.

5=6, and reduced 91y=

32-34 Examp. 5. A courier sets out from a certain place, and

32-34_136-Z

136— travels at the rate of 7 miles in 5 hours ; and 8 hours 8 and 9

5 after; another sets out from the same place, and tra

Toth reduced

11|152-170=272-22
vels the same road, at the rate of s miles in 3 hours :
I demand how long and how far the first must travel

172=442 or z=26

By 8 and 5 131 before he is overtaken by the second ?

y=22 and x=10 Let the number of hours

y

Examp. 7. To find a number confisting of three places, Then the fecond travelled

whose digits are in arithmetical proportion; if this 23-8 The first travelled seven

number be didvided by the sum of its digits, the quo

7y
miles in 5 hours, and 3 (5:7::y:)”-miles

5

tient will be 48 ; and if from the number be subtherefore in y hours

tracted 198, the digits will be inverted. In like manner the second

59-40 4 (3:5::y--8:

Let the 3 digits be 1 x, y, z travelled in y-8 hours

miles

3

Then the number
But they both travelled the

is

2100x+10y + same number of miles; 5

54 — 40

If the digits be 2 therefore by 3. and 4.

3 5

inverted, it is

3 10oz +10y +* Mult.

6257-200= 219 The digits are Transp.

4y= 200

4*+z=2y
Divid.

8
y= 50

therfore
The first then travelled 50 hours, the second By question

100x+10y+z

5 (y-8=) 42 hours.

=48

x+y+z

By question 6100x+10y+2-198=100Z
The miles travelled by each

= .
5 3

+10y+x

From 6 and tranf. 7.99x=99z+198 Case III. When there are three or more unknown Divid. by 99 81x=2+2 quantities.

|9f*= 2y--% Rule. When there are three unknown quantities, there Transp.

10|2y-2=2+2 must be three independent equations arifing from the Mult. 5.

Ily=zt! queflion; and from each of these a value of one of Transp.

12100xti@y+z=48x+48y+487" 13

52x=38y+472 paring these three values, two equations will arise, for x and y

14 522+104=38z+38+472 . involving only two unknown quantities, which may Transp. therefore be resolved by the rule for Case 2.

115

332= 66 In like manner may the rule be extended to such

Divid.

16

By=(z+1=)3 questions as contain four or more unknown quantities;

(x=(2+2=)4. and hence it may be inferred, That, when just as many The number then is 432, which succeeds upon trial.

5

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of Equa- It sometimes happens, that all the unknown quan- part, the given quantities (being numbers) disappear Of Equa

tions. tions. tities, when there are more than two, are not in all in the last conclusion, so that no general rules for like

the equations expressing the conditions, and therefore cases can be deduced from them. But if letters are
the preceding rule cannot be literally followed. The used to denote the known quantities, as well as the
solution, however, will be obtained by such substitu- unknown, a general solution may be obtained, because,
tions, which need not be particularly described. during the whole course of the operation, they retain

their original form. Hence also the connection of the
Corollary to the preceding Rules.

quantities will appear in such a manner as to discover
It appears, that in every question there must be as the necessary limitations of the data, when there are
many independent equations as unknown quantities; any, which is necessary to the perfect solution of a pro-
if there are not, then the question is called indetermi- blem. From this method, too, it is easy to derive a
nate, because it may admit of an infinite number of synthetical demonstration of the solution.
answers; since the equations wanting may be assumed When letters, or any other such symbols, are em-
at pleasure. There may be other circumstances, how. ployed to express all the quantities, the algebra is some-
ever, to limit the answers to one, or a precise number, times called specious or literal.
and which, at the same time, cannot be directly ex-
pressed by equations. Such are these; that the num. Examp. 8. To find two numbers, of which the fum
bers must be integers, squares, cubes, and many others.

and difference are given.
The solution of such problems, which are also called
diophantine, shall be confidered afterwards.

Let s be the sum given, and d the given difference.

Also, let x and y be the two numbers fought.
Scholium.

x+y=s
On many occasions, by particular contrivances, the

メープd operations by the preceding rules may be much abrid

Whence

Sx=y ged. This, however, must be left to the skill and practice

2x=dty of the learner. A few examples are the following:

d+y=smy 1. It is often easy to employ fewer letters than there

2y=smd
are unknown quantities, by expressing some of them

sand
from a simple relation to others contained in the condi-
tions of the question. Thus the solution becomes more
easy and elegant. (See Ex. 4. 5.)

2. Sometimes it is convenient to express by letters, Thus, let the given sum be 100, and the difference 24.
not the unknown quantities themselves, but some o-

76
ther quantities connected with them, as their sum, Then x=

&y=
difference, &c. from which they may be easily derived.
(See Ex. I. of chap. 5.)

In the same manner may the canon be applied to a-
3. In the operation, also, circumstances will suggest a ny other values of , and d. By reversing the steps in
more eafy road than that pointed out by the general the operation, it is easy to show, that if x=

std

and
rules. Two of the original equations may be added
together, or may be subtracted ; sometimes they must
be previously multiplied by some quantity, to render y=7, the sum of x and y must be s, and their dif-

.
such addition or subtraction effe&ual, in exterminating ference d.
one of the unknown quantities, or otherwise promo-
ting the solution. Substitutions may be made of the Examp. 9 If A and B together can perform a piece
values of quantities, in place of quantities themselves, of work in the time a, A and C together in the
and various other such contrivances may be used, which time b, and B and C together in the time c, in what
will render the solution much less complicated. (See time will each of them perform it alone?
Ex. 3. 7. and 9.)

Let A perform the work in the time x, B in y, and
Sect. II. General Solution of Problems. C in z; then as the work is the same in all cases, it
In the solutions of the questions in the preceding' may be represented by unity.

3 F 2

By

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Part I. A L G E B R A.

If particular values be inserted for these letters, a Of Involu1: :2:) A in a days particular folution will be obtained for that cafe. Lettion and

Evolution. them denote the numbers in Example 5. 2 y:1::a:) Bin a days

gra
Then x=

4
3(*:1
::be)
A in 1 days

Here it is obvious, that gr must be greater than ps,
b
41(2:1:::)

=
C in b days

else the problem is impoffible ; for then the value of

x would either be infinite or negative. This limita-
B in c days

tion appears also from the nature of the question, as
the second courier muft travel at a greater rate than.

the first, in order to overtake him. For the rate of
6(2:1::6: C in c days

p the first courier is to the rate of the second as

9 += ; and ay tax=xy

that is, as ps to qr; and therefore or must be greater
6
8+
8+-=j and bz+bx=xz
Ej

Scholium.
+-- I and cz+cy=zy
9

Sometimes when there are many known quantities.

in a general solution, it may fimplify the operation to bc

labe abc

+ abc Mult. 7th by

express certain combinations of them by new letters,
y
xy
labc abc

ftill to be confidered as known.
+
Mult. 8th by

labe abc
ab

CHAP. IV. t=ab

.
Mult. gth by Ý

29
Add roth, 2 abc 2abc 2abc

Of Involution and Evolution.
u Ith, 12th, 13

=batactal

у From 13th subt. 2 abc

2abc In order to resolve equations of the higher orders, twice ioth

141Z
=actab-bc, & z=actab-bc it is necessary to premite the rules of Involution and

-
From 13th fubt. Izabc

2 abc

Evolution.
twice 11th
15

LEMMA.
From 1 3th subt. 2abc

2abc
twice 12th =bctacab, & *= betac ab expressed by that quantity, with negative responents

, 116

The reciprocals of the powers of a quantity may be. , x

. Example in numbers. Let a=8 days, b=9 days, of the fame denomination. That is, the series a, 1,

34

23 =177 49

31

m, &c.
It appears likewise that a, b, c, mult be such, that the a–, a-, a-",
product of any two of them must be less than the fum For the rule for dividing the powers of the fame
of these two multiplied by the third. This is necef- root was to subtract the exponents; if then the index
sary to give positive values of x, y, and z, which alone of the divisor be greater than that of the dividend, the.
can take place in this question. Besides, if x, y, and z, index of the quotient mult be negative.
be assumed as any known numbers whatever, and if.
values of a, b, and c, be deduced from fteps 7th, 8th, Thus, =a?-a?=a-'. Also,

a3
and gth, of the preceding operation, it will appear,
that a, b, and c, will have the property required in the

=a. Also, =1. and so on of others. limitation here mentioned.

If a, b, and c, were such, that any of the quantities *, y, or z, became equal to o, it implies that one of the numerator or denominator of a fraction, may be

Cor. 1. Hence any quantity which multiplies either : the agents did nothing in the work. "If the values of transposed from the one to the other, by changing the any of these quairtities be negative, the only suppor- fign of its index.

. tion which could give them any meaning would be,

43 xy-2.

&c. work, either obstructed it, or undid it to a certain exe

3x~ tent.

Cor. 2. From this notation, it is evident that thefe

negative powers, as they are called, are multiplied by
Examp. 10. In question 5th, let the first courier travel adding, and divided by subtracting their exponents.
p miles in q hours; the second r miles in s hours; let.
the interval between their setting out be a;

Thus, a_axa_3=1-5.
Then, by working as formerly,

qra gr--ps

1. Of

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