ALGEBRA. 
55 
If the highest power } 1st, ) The e- C Simple, 
of the unknown quan- > 2d, r quation < Quadrat, 
titv in any term be the ) 3d, ) is called ( Cubic, 
: &c. &c. 
But the exponents of the unknown quantities 
are supposed to be integers, and the equation is 
supposed to be cleared of fractions, in which the 
unknown quantity, or any of its powers, enter 
i 3.v — b . 
the denominators, xhus, .*• -j- a = - — is 
a simple equation ; 3 a- = 12 , when clear- 
ed of the fraction by multiplying both sides by 
2a;, becomes 6.x 1 — 5 = 24a- a quadratic ; v 3 — 
2 d — at 6 — 20 is an equation of the sixth order, 
&c. 
To resolve an equation is to find the value of 
the unknown term in known terms. 
Of Simple Equations, and their Resolu- 
tions. 
Rule 1. Any quantity may be transposed 
from one side of an equation to the other, by 
changing its sign. 
Thus, if 3x — 10 — 2 a- -{- 5 
Then, 3a- — 2x — 10 -j- 5 or r — 15 
Thus also, 5a- -j - h = a -j- 2a- 
By transp. 3at = a — b. 
For equal quantities are thus addAl to or sub- 
tracted from both sides. 
Carol. The signs of all the terms of an equa- 
tion may he changed into the contrary signs, 
and it will continue to be true. 
Ru le 2. Any quantity by which the unknown 
quantity is multiplied may be taken away, by 
dividing all the other quantities of the equation 
by if. 
Thus, if ax — b 
_ b 
a 
Also, if mx -{- nb = am 
nb 
x — — = a. 
m 
For if equal quantities are divided by the 
same quantity, the quotients are equal. 
Rule 3. If a term of an equation is fractional, 
its denominator may be taken away by multi- 
plying all the other terms by it. 
Thus, if — — L c Also, if a — = 
a x 
x = ab -J- ac ax — b — c 
And by trans. ax — cx = b 
And by div. x = ■ . 
a — c 
For, if all the terms of the equation are mul- 
tiplied by the same quantity, the quantities on 
each side will be equal. 
Corol. If any quantity be found on both sides 
of the equation, with the same sign, it may be 
taken away from both. 
Also, if all the terms iifthe equation are mul- 
tiplied or divided by the same quantity, it may 
be taken out of them all. 
EXAMPLE. 
If 3x -j- a — a -j- b, then 3x = b. 
If 2 ax -(- 3 ab = ma -f- a 2 , then 2x 3b = m a. 
re x 4 16 , 
If 3 3 = -p Aen x- 4=1 6. 
By these rules the unknown term may be se- 
parated from the known, and the equation is 
resolved. 
Examples of Simple Equations resolved by these Rules. 
If 3x 5 ~ x - j- 9 
If 5x - 
,5x , 4x 
---4-12= -■ 
2 1 3 
5x - 
5x 4.v 
- : = 26 
2 3 
30a - 
— I5x ■ — 8.v = 84. 
Or 7a- = 84 
84 
-f 26 
12 = 14 
~r 
16 
-f 9 = 64 
Solution oi 
20 -j- = 64. r 
20 = 64a' — 9x = 55x 
__ 20 4 
'' = 55 ~ lT 
Questions producing Simple 
Equations. 
General Rule. The unknown quantities in 
the question proposed must be expressed by let- 
ters, and the relations of the known and un- 
known quantities contained in it, or the condi- 
tions of it, as they are called, must be expressed 
by equations. These equations being resolved, 
give the answer to the question. 
For example, if the question is concerning 
two numbers, they may be called x and y, and 
the conditions from which they are to be inves- 
tigated must be expressible by equations. 
Thus, if it be required that the 4 
sum of two numbers sought f , 
be 6’0, that condition is ex- f ' Y T D 
pressed thus: j 
If their difference must be 24, then x — y = 24 
If their product is 1640, then xy = 1640 
If their quotient must be 6 , then — = 6 . 
y 
Case I. When there is only one unknown quan- 
tity to be found. 
Rule. An equation involving the unknown 
quantity must be deduced from the question ; 
and it is obvious, that, when there is only one 
unknown quantity, there must be only one in- 
dependent equation contained in the question ; 
for any other would be unnecessary, and might 
be contradictory to the former. 
Ex. 1 . To find a number, to which if there be 
added a half, a third part, and a fourth part of 
Itself, the sum will be 50. 
Let it be z : then half of it is 
it is — , &c. 
a third of 
2a- = 4 
4 
*= 2 
— = 2 . 
therefore = 
x -J- 7 = y -J- 7 X 2 = 2y -J- 14 
x = 3y — 21 -J- -7 = 3y — 14 
x = 2y -J- 1 4 — 7 = 2 _y -j- 7 
Therefore Sj — 14 = 2 y + 7 
y = 21 , and 
a = 49. 
Ex. 3. A gentleman distributing money among 
some poor people, found he wanted 10 ,. to be 
able to give 5s. to each ; therefore he gives each 
4,. only, and finds lie has 5s. left. — To find the 
number of shillings and poor people. 
If any question such as this, in which there 
are two quantities sought, can be resolved by 
means of one letter, the solution is in general 
more simple than when two are employed. 
There must be, however, two independent con- 
ditions; one of which is used in the notation of 
one of the unknown quantities, and the other 
gives an equation. 
Let tiie number of poor lie z, then the num- 
ber of shillings will be 5z — 10, and also 4z-j-5; 
therefore, 5z — 10 = 4z -j- 5 
s = 15, equal the number of poor* 
of course the number of shill ligs is 05. 
Case III. When there are three or more un- 
known quantities. 
Rule. When there are three unknown quan- 
tities, there must be three independent equa- 
tions arising from the question; and from, each 
of these a value of one of the unknown quanti- 
ties must be obtained. By comparingthese three 
values, two equations will arise, involving only 
two unknown quantities; and in like manner 
may the rule be extended to such questions as 
contain four or more unknown quantities. — 
Hence it may be inferred, that when just as 
many independent equations may be derived 
from a question as there are unknown quantities 
in it, these quantities may be found by the re- 
solution of equations. 
Ex. 4. To find three numbers, so that the 
first with half the other two, the second with 
one-third of the other two, and the third with 
one-fourth of the other two, may each be equal 
to 34. 
Let the numbers be x,y, z , and the equations 
will be 
Therefore, * + — + - -f- -J- = 50 
24z + 12 2 -f 82 -(- 6z = 1200 
50z = 1 200 
z = 24. 
Case II. When there are two unknown quan- 
tities. 
Rule. Two independent equations involving 
the two unknown quantities, must be derived 
from the question. A value of one of the un- 
known quantities must be derived from each of 
the equations ; and these two values being put 
equal to each other, a new equation will arise, 
involving only one unknown quantity. 
Ex. 2 . Two persons, A and B, were talking 
of their ages ; says A to B, Seven years ago I 
was just three times as old as you were, and se- 
ven years hence I shall be just twice as old as 
you will be. I demand their present ages. 
Let the ages of A and B be x andy, then seven 
years ago their ages were x — 7 and y — 7 ; and 
seven years hence they will be a 4- j and y 4- 7 ; 
^ 2 
y + 
_+f 
3 
- -} -y 
34 
= 34 
= 34 ; then, by the first equation. 
1 4 
x = - ; and by the second 
x = 102 — 3y — z ; and by the third 
x = 136 — 4z — y ; therefore 
68 — y 
136 — 
y = 
= 102 — 3y — • s 
; and by the two latter equations 
34 
J 2 
3z — 34 
therefore. 
136 
2 5 
15z — 170 = 272 — 2z 
17z = 442, or z = 26 
y = 22 , and x = 10. 
On many occasions, by particular contriv- 
ances, the operations by the preceding rules 
may be much abridged. This, however, must 
be left to the skill and practice of the learner; 
A few examples are the following. 
' 1 . It is often easy to employ fewer letters 
than there are unknown quantities, bv express- 
ing some of them from a simple relation to 
others contained in the conditions of tha ques- 
tion. 
