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CHAPTER IV.

SUBTRACTION.

26. THE simplest cases of Subtraction have already come under the head of addition of like terms, of which some are negative. [Art. 19.] Thus

5a - 3a=

2a, 3a - 7a= - 40, - 3a - ba= -9a.

Also, by the rule for removing brackets [Art. 22.),

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

=lla, and

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

=5a.

SUBTRACTION OF UNLIKE TERMS.
27. The method is shown in the following example.
Example. Subtract 3a —26-c from 4a-36+5c.
The result of subtraction=4a-36+50—(3a —26-c)

=4a-36+50-3a+2b+c
=4a-3a-36 +26+50+0

=Q-6+6c. It is, however, more convenient to arrange the work as follows, the signs of all the terms in the lower line being changed.

4a-36+5c

- 3a+25+ c by addition

a-b+60

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 3yo take 2x2 + 8xy 7y2.

5x2 + xy - 3y2
2.c+ Sxy - 7y?

3.x2 – 7xy +4ya. Example 2. Subtract 2x4 – 3x2 +7x – 8 from x*— 2.03 - 9.C +4.

2c4 – 2003 -- 9x +4
2x4 - 3x2 + 7x – 8
- 24 - 2.2c3 + 3.0" – 16x + 12,

EXAMPLES IV. a. Subtract 1. 4a – 3b+c from 2a - 36 - C. 2. a-3b + 50 from 4a-8b+c. 3. 2.x— 8y +z from 15x + 10y - 18z. 4. 15a - 276+8c from 10a + 3b +4c. 5. – 10x – 14y + 15z from x-,

:- y - 2. 6. -llab +6cd from – 106c+ab-4cd. 7. 4a - 3b + 15c from 25a - 166 - 18c. 8. - 16x – 18y - 15z from –5x+8y + 7z. 9. ab + cd ac bd from ab + cd + ac+bd. 10. - ab + cd - ac+bd from ab- cd+ac bd.

From 11. 3ab + 5cd - 4ac 65d take 3ab + 6cd - 3ac-56d. 12. yz zx + xy take xy + y2 zx. 13. – 2.x3 – x2 – 3x + 2 take 23 – X+1. 14. - 8x2y + 15xy2 + 10xyz take 4.xoy 6xy4 5xyz. 15. }a--b+}c take ja + 36 - c. 16. 2x+y - z take tx 3y }z. 17. -a-36 take za +36- c. 18. 1x-y+joz take - 2x + 4y - 1oz. 19. - 3x – 3y 5z take x-ky-. 20. - *x+3y-} tako }x - Xy-t.

EXAMPLES IV. b.

From 1. 3xy 5yz +8zx take – 4xy + 2yz - 10zx. 2. - 8x2y2 + 15x3y + 13xy: take 4.x+y2 +7cxcoy 8xy3. 3. -8+ 6ab + a262 take 4-3ab - 5a262. 4. aʼbc+b+ca +cab take 3a2bc 5b2ca 4cʻab. 5. - 7a+b+8ab2+ cd take 5a2b 7ab2 +6cd. 6. - 8x2y + 5xy2 x2y2 take 8x'y 5xy2 + x^ya. 7. 10a2b2+ 15ab2 + 8a26 take – 10a2b2 + 15ab2 – 8u2b. 8. 4x2 – 3x + 2 take – 5.22 + 6X – 7. 9. 23 +11.x2+4 take 8.x2 – 5.x – 3. 10. - 8a”x2 +5x2 + 15 take 9a-x2 – 80% – 5.

Subtract 11. 203 – x2 + x +1 from x3 + x2 – X + 1. 12. 3xy2 3x2y + 243 y3 from 23 + 3x2y + 3xy2 + y2. 13. 33 +63 – 2abc from a3 +63 – 3abc. 14. 7xy33 3x+y + 5x3 from 8.43 +7x2y 3xy2 . 15. 24+5+ X – 3.03 from 5.x4 – 8x3 – 2x2 +7. 16. a3 + b3 + c3 – 3abc from 7abc - 323+563 C3. 17. l-x+ 25 – x4 - 203 from 24-1+2 – 3%. 18. 7a4-8a2 + 305 + a from a? - 5a3-7+7a. 19. 10a2b+8ab2 - 82373 – 74 from 5a2b-6ab2 - 7a363. 20. a? 63 +8ab2-7a2b from - 8ab2 + 15a6 + 63.

From 21.1.x2 - xy-Byo take - 2x2 + xy-yo. 22. Ža– 5a-1 take – gaa+a - 3. 23. 3x2 – 3x+take fx-1+3.22. 24.9.2 - şax take } -1.2-2ax. 25.1.23 - }xy2 - y2 take .xy-3y2 - }xy. 26. Ja - 202"- ax take ja-x+1a3 - Sax”.

28. We shall close this chapter with an exercise containing miscellaneous examples of Addition and Subtraction.

EXAMPLES IV. c.

1. To the sum of 2a – 36 – 2c and 26 - Q +70 add the sum of a -4c+76 and c :- 66.

2. From 5x + 3x i take the sum of 2x – 5 + 722 and 3.x2 +4-2.x3 + x.

3. Subtract 3a 7a+5a2 from the sum of 2 +8a2- and 2a– 3a + a - 2.

4. Subtract 5x2 + 3x – 1 from 2.x, and add the result to 3x2 + 3x – 1.

5. Add the sum of 2y 3y2 and 1 -543 to the remainder left when 1 – 2y2 +y is subtracted from 5y?.

6. Take 22 - y2 from 3xy 4y, and add the remainder to the sum of 4xy 22 3y2 and 2x2 +672.

7. Find the sum of 5a - 7b+c and 36 9a, and subtract the result from c - 46.

8. Add together 3x2 – 7.0+5 and 2.2-3 + 5x – 3, and diminish the result by 3x2 +2.

9. What expression must be added to 5x2 – 73+2 to produce 722-1?

10. What expression must be added to 4.x2 – 3x2 +2 to produce 423 + 73 – 6 ?

11. What expression must be subtracted from 3a – 56+c so as to leave 2a-4b+c?

12. What expression must be subtracted from 9x2+11x – 5 so as to leave 6m2 – 17x+3?

13. From what expression must lla?- 5ab 7bc be subtracted so as to give for remainder 5a2+7ab+7bc ?

14. From what expression must 3ab +5bc - 6ca be subtracted so as to leave a remainder 6ca 5bc?

15. To what expression must 7.203 - 6x2 – 5x be added so as to make 9.X2 – 6x – 7x2 ?

16. To what expression must 5ab 11bc 7ca be added so as to produce zero?

17. If 3x2 – 7x0+2 be subtracted from zero, what will be the result ?

18. Subtract 3x3 – 7x + 1 from 2x2 – 5x – 3, then subtract the difference from zero, and add this last result to 22:2 – 2.23 – 4.

19. Subtract 3x2 – 5x+1 from unity, and add 5x2 - 6x to the result.

CHAPTER V.

MULTIPLICATION.

MULTIPLICATION OF SIMPLE EXPRESSIONS.

29. When there is no sign between symbols or expressions, it is understood that the symbols or expressions are to be multiplied together. Thus

ab=axb.

3ab=3x axb.

a (x - y)=ax (x - y). (x+y)(a+b) is the product of x+y and a+b. In Algebra, as in Arithmetic, the product is the same in whatever order the factors are written. (Art. 13.] Example,

20 x 31=2 xa x 3 xb

=2 x 3 xaxo
=6ab.

30. Since, by definition, a:=aaa, and

ab=aaaaa;
α3 χα5 = αααααααα

[ocr errors]

Again,

5a2=5aa,

7@=7aaa;
5a? x 7a3=5 x 7 x aaaaa

=35a5.

31. When the expressions to be multiplied together contain powers of different letters, a similar method is used. Example. 5a3b2 x 8a2b2c3 =5aaabb x Saabsxx

=40a573.03.

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