51. When dissimilar terms have a common literal part, this may be taken as the unit of addition. The sum of the terms will then be expressed by inclosing the sum of the coefficients in a parenthesis, and prefixing it to the common unit.

Ans. (4a+4)y+(3c+6)x.

8. Add 3x+2xy, bx+exy, and (a+b)x+2cdxy.

Ans. (a+2b+3)x+(2cd+c+2)xy.

9. Add ax+7y, 7ax-3y, and -2x+4y.

Ans. (8a-2)x+8y. 10. Add (b-a)/x, and (c+2a-b)/x. Ans. (c+a)√x. 11. Add (a+2b) m—c√m, (2a—6c) m—3a√m, (5c—4a) m— bm, and (2a-36)m+4a√m. Ans. (a-b-c) (m+√m). 12. Add ax+y+z, x+ay+z, and x+y+az.

Ans. (a+2)(x+y+*).

SUBTRACTION.

52. Subtraction, in Algebra, is the process of finding the difference between two quantities.

53. It is evident that 5 units of any kind or quality subtracted from 8 units of the same kind or quality, must leave 3 units of the same kind or quality. That is,

But these remainders are the same as we shall obtain by changing the signs of the subtrahends and then adding the results, algebraically, to the minuends. Thus,

+8a-(+5a)=+8a-5a=+3a
-8a-(-5a)=-8a+5a=-3a

Hence, in Algebra,

Subtracting any quantity consists in adding the same quantity with its sign changed.

54. This principle may be established in a more general manner as follows:

Let it be required to subtract the quantity b-c from a.

OPERATION.

Minuend,
Subtrahend,

Difference,

α

b-c

a-b+c

We first subtract b from a, indicating the operation, and obtain for a result, a-b. But the true subtrahend is not b, but b-c; and, as we have subtracted a quantity too great by c, the remainder thus obtained must be too small by c; we therefore add c to the first result, and obtain the true remainder, a-b+c. But this result is the same as would be obtained by adding-b+c to u. 55. It follows from the principle enunciated above, that any quantity is subtracted from nothing or zero, by simply changing its sign or signs. Thus,

56. From these principles and illustrations we deduce the following

RULE. I. Write the subtrahend underneath the minuend, placing the similar terms together in the same column.

II. Conceive the signs of the subtrahend to be changed, unite the similar terms as in addition, and bring down all the remaining terms with their proper signs.

Ans. 8a-6c.

11. From 7a-5c+2 subtract -a+c+2.
12. From 8x-3xy+2y+c subtract x2-6xy+3y2-2c.

Ans. 7x+3xy-y'+3c.

13. From a+b subtract ab.

Ans. 2b.

14. From xy subtract x-y.

Ans. y.

15. From a+b+c subtract —a—b—c. Ans. 2a+2b+2c. 16. From 3a-b-2x+7 take 8-3b+a+4x.

Ans. 2a+2b-6x-1.

17. From 6y'-2y-5 take -8y'—5y+12.

Ans. 14y+3y-17.

18. From 3p+g+r-3s take q—8r+2s—8.

Ans. 3p+9r-58+8.

19. From 13a2-2ax+9x2 take 5a2-7ax-x3.

Ans. 8a'+5ax+10x1.

20. From x-3x+5x-7x+12 take x-4x+2x2-6x+15. Ans. x+3x-x-3.

21. From a 3a c+5a c-2a'c'+4ac-c take a'-4a'c+ 2a'c-5a'c'+3ac-c°. Ans. a'c+3a'c'+3a2c3+ac*. 22. From 2x+28x+134x3-252x+144 take 2x+21x+ 67x2-63x+84. Ans. 7x+67x-189x+60. 23. From x2+5x*y+10x3y2+10x3y3+5xy*+y° take xo—5x1y+ . 10x'y'-10x'y3+5xy*—y°. Ans. 10x y+20x3y3+2y°

24. From the sum of 6x'y-11ax' and 8x'y+3ax3, take 4x3y— 4ax3+a. Ans. 10x'y-4ax3—a. 25. From the sum of 8cdx+15a2b-3 and 2cdx-8a3b+24 take the sum of 12a3b-3cdx-8 and cdx-4a2b+16.

Ans. 12cdx-a2b+13.

57. The difference of two dissimilar terms may often be conveniently expressed in a single term, as in (51), by taking some common letter or letters as the unit of subtraction.

Ans. (a-m)x+(bn)y+(c—p)z.
Ans. (c-1)x.

6. From ax+bx+cx take x+ax+bx.
7. From (a+2b+c) √xy take (2b—c)√xy.

Ans. (a+2c)√xy.

8. From (3a-2m)x3+(5a+2m)x2+(4a—m)x take (a—m)x3

-(2a+m)x2+(2a−3m)x.

Ans. (2a-m)x+(7a+3m)x2+(2a+2m)x.

9. From 1+2az* +·3a*z*+4a3z®+5a1z take z+2az*+3a'z°+ 4a3z®.

Ans. 1+(2a-1)**+(3a2—-2a)z*+(4a3—3a*)≈°+(5a*—4a3)z°.

USE OF THE PARENTHESIS.

58. The term, parenthesis, will be employed hereafter as a general name to designate the various signs of aggregation employed in algebraic operations. The following rules respecting the use of the parenthesis should be thoroughly considered by the learner, if he would acquire facility in algebraic transformations.

59. From the definition of the signs of aggregation, (17), we understand that if the plus sign occurs before a parenthesis, all the terms inclosed are to be added, which does not require that the signs of the terms be changed; but if the minus sign occurs before a parenthesis, all the terms inclosed are to be subtracted, which requires that the signs of all the terms be changed. Hence,

1. A parenthesis preceded by the plus sign may be removed, and the inclosed terms written with their proper signs. Thus,

ab+(c―d+e)=a—b+c-d+e

2. Conversely: Any number of terms, with their proper signs, may be inclosed by a parenthesis, and the plus sign written before the whole. Thus,

a—b+c―d+e=a+(—b+c−d+e)

3. A parenthesis preceded by the minus sign may be removed, provided the signs of all the inclosed terms be changed. Thus,

a-(b-c+d-e)=a—b+c―d+e

4. Conversely: Any number of terms may be inclosed by a parenthesis, preceded by the minus sign, provided the signs of all the given terms be changed. Thus,

ab+c=d+e—a—b+c—(d—e)

60. When two or more parentheses are used in the same expression, they may be removed successively by the above rules. Thus, a—{b—c―(d—e)}=a—{b—c―d+e}=a—b+c+d—e

Or, in a different order,

a―{b—c―(d—e)}=a—b+c+(d—e)—a—b+c+d—e