MATHEMATICS. ALGEBB \. 



[BCBTltAtTIO.X. 



1 found ; for we cannot ad*alty add or subtract when tho 

 I quantities are unlike. The f. -the rule. 



lit sots of lilct quan- 



titi n each neparat \actlyasin 



> the last case; then, to tin- sum thus f..in. 

 I by the signs belonging to them, tho rooudning tin/i/.c 

 ! quantities. 



Although it matters not in what order a row of 

 ' algebraical terms is written, yet in addition, it is usual to 

 commeuce with the quantity at the top of tho left-hand 

 column, and to select from among all tho columns tho 

 several quantities like it ; then passing to the quantity 

 at the top of the next column, to add together the 

 quantities like it, and so on. 



1. 2- 



\r 7y + S* 5aj + 24y 7 



4y + G * 34y + 18 4j 



_ 3r 2y + a ** 9 *. 



4x4-3j 26 +3a* 24y 



2ax + 24y + 28 4z 



3. 



7xy _ 4a* + 24e 

 (lor -\- 5m 3p 

 24f 3xy + 8a * 

 _ jy Ac _ or 



6xy + 5az 



4. 



2yi 3a4 

 5cr + 2y* 4- 5a4 

 Ja4 3fjf -f y* 

 4f* 6yj CJT 

 2cj + yr 8 



_- 4yj + 9o4 



8 



EXAMPLES FOE EXERCTSB. 

 1. 2. 



2<tr -4- 34y + 4c 5jry 3ar -(- 7 



54y -f 3c ax 6xy + 44x 3 



7e + tax 24y 4aj 2.ry + e 



3ez bar+M 19 + <u J-y 



3. 



9y.* + 8<tr 34e 



. ten 44c + 2yr 



rys + 3ar 4y* 



5. 

 . Try 34; + 4io 



54z + 4ry 3c./ 

 2ic 64i -f- 9jy 



4jy 4- 3w b: 



1C -fSxy + w 



4. 



4n4 



Ccd + t/ 1 ft 

 2ff Zed + 14 

 3* + a4 rd 



6. 



2am: + 2iu. 

 Grip 84 



am: 24t> 



fin* 14 



From comparing tho examples in this second case of 

 .vMition with those given in the first, you will see that, 

 although tli<< placing of tho several expressions to be 

 a<lded one uudi r aiintlu-r was of considerable assistance 

 there, because the lib: quantities all appeared in vertical 

 rows, yut tho arrangement is of no advantage here, since 

 we have to pick out tho like quantities after a careful 

 search for th>>m nmong the entire set of expri-s 



inld -1" thin just as well if the expressions were all 

 written side by xiilo, as in the next examples, without 

 t ikirr.; tin; i,"iil'l" nf first arranging them one iin.l-r 



If you think it easier, however, you may 

 first one set of like quantities out of the express*!.. 

 each of tho fnllowin;; examples, and arrange them ai in 

 Case I. ; then a aecutid set, and so on. \ou m.ty thus 

 change this Case II. into Case I., taking care, however, 



that the unlike quantities be connected to the sum of tho 

 .utities in the final result. 



7. 2a.ry 34?+. 7//.--C. J-2A: Jary, 44r 3c+2. 



8. Ita4r 3rf< !/ V~~ 24c+rf*. 4J 3a4c-f-S, aticf. 



9. 4iz+2y 7^+9, 3x ^y+J, 8y 5j+2r, 2y 



10. 8^4- a. 54, 64 1- 3<u. 2? 3?, 474. 



y y * V 



11. 7xyr 3 24e, 5- tm-\-p, \2lic-\-3xy: 



a c 



12. -ax 5y+7* 8, ly 2.-, 4; 6or+2y, 3y V:+ax. 

 3 * 



8UBTRACT1OX. 



Before giving you the rule for subtraction of alg.-' 

 we must explain tho principle upon which tho rule is 

 founded; for whatever part "f mathematics be stii.li".!, 

 you must never confide in a "Kule" till conviricc.l i.f 

 its correctness. 



Now, let us take any two numbers at random, say 9 

 and 4 ; and let us endeavour to subtract the latter from 

 the former, when the algebraic signs +, are, the one 

 or the other, prefixed to the 9 and the 4, and try to put 

 down what we are certain must be the correct remainder. 

 You know tliat to ivbtraot means to take away ; if, 

 therefore, we can actually take away the 4 with its sign, 

 from the 9 with its sign, wo shall be sure of the remainder 

 sought. In order to this, let us write 9 in this form, 

 namely, 9 + 4 4, which you see, although it takes up 

 more room is only 9, for 4 4 is nothing. From the 9 

 thus expressed, take away tho + 4, that is, fancy it 

 actually taken up in your fingers and removed, or which 

 will answer as well, take it away by rubbing it out; what 

 remains ia evidently 9 4 or 5. 



Again, from the same expression for the 9, now take 

 the 4 away; that is, rub it out: wliat remains is 

 evidently 9 + 4 or 13. 



You thus see that if from a positive number, as 9, you 

 have to subtract a positive number, as 4, the true 

 remainder will bo got by clumgitig the sirjn, of tho 

 nuiiilier to be subtracted, and then ailJinj, as in 

 the margin. And that if, from a positive ii'imher, 

 ynii have to subtract ;i number, the true 



remainder will bo got, in like manner, by cl 

 the tign of the number to be subtracted, ai. 

 /; as in tho mar 



Hitherto, the has l.eeii supposed positive: let 

 it be negative, and, in imitation of the plan above, 

 let us write this 9 in the form 9 + 4 4. 



From the 9, thus written, take away, or re- 



move, the + 4 ; the remainder is 9 4, or I ::. 



From the same expression for the 9, take away the 



4; tho remainder is 9 + 4, or 5. 



You see, therefore, that if from a negative number, as 



9, you have to subtract a positive number, as 4, the 

 true remainder will be got by rhunyiinj the sign of the 

 nunilior to be subtracted, and tlfu <tilil!ng, as in 



tho margin. And that if from a negative number 

 you have to subtract a negative number, tho true 

 remainder will, in like manner, be got by r'. 



i of the number to be subtracted, and ihrn 

 i, as in th 



This reasoning, whicli of course applies to any 

 pair of niimliei.s, as well as to 9 and 4, at once 

 suggests the following rule for subtraction of 

 algebra. 



UUI/B. Change the signs of all tho (..run in the 



rmtity to bo subtracted, or con- to be 



: th.-ii proceed as if it wore ad'liti/i 



'. ,iet ion : the result will be the remainder. 

 1. From9a + 66-Bc { -' '$ a ' j To Oa + 06-5e 

 Take 4a 26 + 3c Add 4a + 2i 3c 



9 

 4 



9 

 4 



13 



9 



13 



9 

 4 



5 



Rcm. 5a + 86 8c 



In the operation on the left, the signs of the subtract.! ve 

 terms are only a I to be i<l then ad'i 



performed. On the right tho change is actually n 



