and bears that meaning only; but in Algebra it is also taken to mean the sum of the two quantities a and b without any regard to the relative magnitudes of a and b.

Example 1. Find the sum of 3a - 5b+2c, 2a+3b - d,

= 3a+2a - 4a - 5b+3b+2b+2c - d
= a + 2c - d,

by collecting like terms.

-4a+2b.

The addition is however more conveniently effected by the following rule:

Rule. Arrange the expressions in lines so that the like terms may be in the same vertical columns: then add each column beginning with that on the left.

3. 4p+3q+5r;

4. 7a-5b+3c;

-2p+3q-8r; p-q+r.

lla+2b-c; 16a+5b-2c.

5. 81-2m+5n; - 67+7m+4n; -1-4m-8n.
6. 5a-7b+3c-4d; 6b-5c+3d; b+2c - d.
7. 2a+4b-5x ; 2b-5x; - 3a + 2y ;
8. 7x-5y-7%; 4x+y; 5z; 5x-3y+27.

- Cb + Sx+y.

9. a-2b+7c+3; 2b-3c+5; 3c+2a; a-8-7c.
10. 5-x-y; 7+2x; 3y-22; - 4+x-2y.
11. 25a-15b+c; 4c-10b+13a; a-c+20b.
12. 2a-3b-2c+2x; 5x+3b-7c; 9c - 6x-2a.
13. 3a-5c+2b-2d; b+2d -a; 5c+3f+ 3e −2a - 3b.
14. p-q+7r; 6q+r−p; q-3p-r; 6q-7p.
15. 17ab - 13kl - 5xy; 7xy; 12kl-5ab; 3xy - 4kl – ab.
2ax-3by - 2c; 2by-ax+7cz; ax-4cz+7by; cz-6by.
3ax + cz-4by; 7by - 8ax - cz; -3by+9ax.
18. 3+5cd; 2fg-3st; 1-5cd;

16.

17.

- 4+2st-fg.

19. 5cx+3fy-2+28 ; - 2fy+6-98; 20. -3ab+7cd -5qr; 2ry+Sqr- cd;

-38-4+2cx-fy. 2cd-3qr+ab - 2ry.

29. Different powers of the same letter are unlike terms; thus the result of adding together 2.3 and 3 cannot be expressed by a single term, but must be left in the form 23 +3x2.

Similarly the algebraical sum of 5a2b2, -3ab3, and -b1 is 5a2b2 - 3ab3-b1. This expression is in its simplest form and cannot be abridged.

Example. Find the sum of 6x3- 5x, 2x2, 5x, - 2x3, -3x2, 2. 6x3-5x+2x2 + 5x − 2x3 – 3x2 + 2

The sum =

30. In adding together several algebraical expressions containing terms with different powers of the same letter, it will be found convenient to arrange all expressions in descending or ascending powers of that letter. This will be made clear by the following example.

Example 1. Add together 4x-2x2+3x2; 3x3-9x-x2;

3x35x2+6x+7 2x2-9x-8

- 2x3+3x2+4x 3x3x2-9x x3- x2+x+4 3x32x27x+3

323 +7 +6x x − x2 - x3 +4.

5x2; 2x2 - 8 – 9x;

In writing down the first expression we put in the first term the highest power of x, in the second term the next highest power, and so on till the last term in which a does not appear. The other expressions are arranged in the same way, so that in each column we have like powers of the same letter.

Example 2. Add together

3ab2-2b3+ a3; 5a2b-ab2-3a3; 8a3+5b3; 9a2b-2a3 + ab2.

Find the sum of the following expressions:

1. x2+3xy-3y2; -3x2+xy+2y2; 2x2 - 3xy + y2. 2. 2x2-2x+3; -2x2+5x+4; x2-2x-6.

3. 5x3-x2+x-1; 2x2 - 2x+5; - 5x + 5x – 4.

4. a3 - a2b+5ab2+b3 ; - a3-10ab2+b3; 2a2b+5ab2 - b3. 5. 3x3-922-11x+7; 2x3-5x2+2; 5x3+ 15×2 – 7x ; 8x-9. 6. 2-5x+8x; 7x+4x3+5x; 8x3-9x ; 2x3-7x3-4x. 7. 4m3 +2m2 - 5m+7; 3m3 + 6m2 − 2 ;

-5m2+3m; 2m - 6.

8. ax3-4bx2+ cx; 3bx2 - 2cx-d; bx2+2d; 2ax3+d. 9. py2-9qy+7r; -2py2+3qy-6r; 7qy-4r; 3py2.

10. 5y3 +20y2+3y-1; - 2y +5 −7y2; 11. 2-a+Sa2-a3; 2a3 - 3a2 + 2a − 2 ;

12. 1+2y-3y2-5y3;

-3y2-4+2y3- y.

-3a+7a3-5a2.

-1+2y2-y; 5y3+3y2+4.

13. a2x3-3a3x2+x; 5x+7a3x2; 4a3x2 - a2x3 – 5x.

14. 25-4xy-5x3y3; 3x4y+2x3y3 -- 6xy1; 3x3y3 +6xy1 — y3.

15. a3-4a2b+6abc; a2b-10abc+c3; b3+3a2b+abc.

16. ap5-6bp3+7cp; 5-6cp+5lp3; 3-2ap5; 2cp--7.
17. c2-2c5+11c6; - 2c7 - 3c6+5c5; 4c6 - 10c5; 4c7-c6.
18. 4h3-7+3h1 -- 2h; 7h -3h3+2-h1; 2h4+2h3 - 5.
19. 3x3+2y2 – 5x+2; 7x3-5y2+7x-5;

6xy2-18x3- 7.

9x3+11-8x+4y2;

20. x2+2xy + 3y2; 3z2+2yz + y2; x2+3z2+2xz; z2 - 3xy - 3yz;

xy+xz+yz-6z2 - 4y2 - 2x2.

CHAPTER IV.

SUBTRACTION.

31. THE simplest cases of Subtraction have already come under the head of addition of like terms, of which some are negative. [Art. 20.]

Now subtract -b from the left-hand side and erase -b on the right; we thus get

a-(-b)=a+b.

This also follows directly from the rule for removing brackets.

Subtraction of Unlike Terms.

32. We may proceed as in the following example. Example. Subtract 3a-2b-c from 4a-3b+5c.

The difference

=4a-3b+5c-(3a-2b-c)
=4a-3b+5c-3a+2b+c
=4a-3a-3b+2b+5c+c

=a-b+6c.

The expression to be subtracted is first enclosed in brackets with a minus sign prefixed, then on removal of the brackets the like terms are combined by the rules already exclaimed in Art. 20.

It is, however, more convenient to arrange the work as follows, the signs of all the terms in the lower line being changed.

by addition,

4a-3b+5c -3a+2b+c

a- b+6c

The like terms are written in the same vertical column, and each column is treated separately.

Rule. Change the sign of every term in the expression to be subtracted, and add to the other expression.

Note. It is not necessary that in the expression to be subtracted the signs should be actually changed; the operation of changing signs ought to be performed mentally.

Example 1. From 5x2+xy take 2x2+8xy - 7y2.

5 + xy 2x2 + Sxy -- 7y2 3x2-7xy + 7y2

Example 2.

In the first column we combine mentally 5x2 In and -2x2, the algebraic sum of which is 3.. the last column the sign of the term - 72 has to be changed before it is put down in the result. Subtract 32 - 2x from 1 − x3.

Terms containing different powers of the same letter being unlike must stand in different columns.

-2-3

3x2 - 2x

+1

- x3-3x2+ 2x + 1

In the first and last columns, as there is nothing to be subtracted, the terms are put down without change of sign. In the second and third columns each sign has to be changed.

The re-arrangement of terms in the first line is not necessary, but it is convenient, because it gives the result of subtraction in descending powers of x.