ALGEBRA. 
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together with 6 A and 3 a to be subtracted, is 
14 A to be subtracted. 
2. If similar quantities have different signs, 
their sum is found by taking the difference of 
the co-efficients with the sign of the greater, 
and annexing the common letters as before. 
Ex. 5. 7 a -f- 3 b 
— 5a — 9b 
Ans. 2 a — 6 b 
* h 
Ex. 6. 6 a -[- 4 b -J-. 9 c 
9 a — j— 3 b — 16c 
Ans. 
-f-12a — 7 A — 20c 
* 4- 5c 
In the first part of the operation we have 
7 times a to add, and 5 times a to take 
away ; therefore, upon the whole, we have 
2 a to add. In the latter part, we have 
3 times b to add, and 9 times h to take 
away; i. e. we have, upon the whole, 6 times 
b to take away : and thus the sum of all the 
quantities is 2 a — 6 b. 
If several similar .quantities are to be 
added together, some with positive and some 
with negative signs, take the difference be- 
tween the sum of the positive and the sum 
of the negative co-efficients, prefix the sign 
of the greater sum, and annex the common 
letters. 
Ex. 7. 3 a 2 -{- 4 be — - e 2 -f~ 1 0 x — 25 
5 a 2 -\-6 b c 2 c 2 — -15x-j-44 
1 4- 21 a: — 90 
Ans. — 6a 2 -\- be — 9e 2 -f-16.r- 
4ae — :\5bd-\-ex — a.r 
11 a c — j— 7 b 2 - — 1 9 e x -j- 4 ax 
~h 6 h d — 7 d e — 9. ax 
sign of the quantity to be subtracted, and then 
adding it to the other by the rules laid down in 
the last article. 
Ex. 1. From 2 Ax take cy, and the dif- 
ference is properly represented by 2 bx — cy, 
because the — prefixed to cy shews that it is 
to be subtracted from the other; and 2 b x 
cy is the sum of 2 A a: and — cy. 
Ex. 2. Again, from 2 h .r take — cy, and 
the difference, is 2 b cy ; because 2 b x = 
2 A x -f- c y — cy, take away — c y from these 
equal quantities, and the differences will be 
equal ; i. e. the difference between 2 b x and 
cy nib x-\- cy, the quantity which arises 
from adding -\-cy to 2 bx. 
Ex. 3. From a -j- b 
take a — b 
Ans. *-f.2A 
Ex. 4. From 
take ■ 
Ans. 11a — *2 A 
6 a- 
-5 a- 
12 A 
10 A 
Ex. 5. From 5 a 2 4 a b - 
take 1 1 a 2 -f- 6 a b - 
Ans. — 6 a 2 — 2aA- 
-6xy 
- 4 xy 
Ex. 6. From 4 a — 3A-f-6c — 11 
take lOx-f- a — 15 — 2 y 
Ans. 3 a — 3A-j-6c — ] 0 x -j-2 y-|-4 
Ex. 7. From a x 3 — Ax 2 -{-x 
take px 3 — qx 2 -i r rx 
Ans. a — p . x 3 — b — c/ . x 2 -j- 
i — r . x 
are united: 
In t his example the co-efficients 
1 — acc a — p.x 3 is equal to ax 3 — px 3 ; - 
is equal to — Ax 2 -)- q x 1 ; and ~ 
— r x. 
MULTIPLICATION. 
The multiplication of simple algebraical 
quantities must be represented according to 
the notation already pointed out. 
Thus, a X A, or a A, represents the product 
of a multiplied by A ,- a A c, the product of the 
three, quantities a, b, and c. 
It is also indifferent in what order they are 
placed, « x A and A x a being equal. 
lo determine the sign of the product, ob- 
serve the following rule : 
If the multiplier and multiplicand have the 
