SUBTRACTION. ] 



MATHEMATICS. ALGEBE A. 



455 



and the terms then added. We need scarcely say that 

 the results of subtraction, like those of addition, are 

 written from left to right. 



2. From lax 3by fez 

 Take Sax + 46y 9cz 2 



Rem. ax 7&y + 2cz + 2 



3. From 8mz + bx Gay 2 

 Take 2mz 36x -f 4a y 3 



Rem. 



10uy + l 



As subtraction is thus converted into addition, all that 

 has been said about addition applies of course here. 

 Terms that have no like are to be brought down in the 

 remainder with their proper signs ; those that are in the 

 subtractive row with changed signs. 



1. 



EXAMPLES FOB F.XEHCISE. 



2. 



2or+4iy 7cz 



Zax 2l/y-\-icz 



3. 



4. 5. G. 



2.1<7y l-Uj+G \\ubc-\-3de tyff 2b:Zax n 

 _18 fl y 94x 'Jabc 7 ' 



7. 



86+217 

 3 + 59+24 



9. 



~ayz <3bx-\-6c 2e 

 4ay:+Sl>f-\-2c id 



8. 

 NWy llftr 12/; 3m 



10. 



Zcejc-\-aby 2djc 3z 



In the following examples the suitable arrangement 

 of the terms, so as to bring bike quantities under one 

 another, is left for you to manage yourself. 



11. From 5ox 3Jy + "c: take Uy 3ax + 2ez. 



12. From 44z 2cyax take 3oj + 4cy Sliz. 



1 3. From "ay + 54-r 1 6 take 3 bx + 4ay + 3ez. 



14. From 4lj? 2y + 13z 4 take 5j + 6y $x. 



15. From 8?.ory 31* + 4m take G.JAz 4<wy + 2n. 



16. From Glyr 5>jr 2J take 4ay Sjax + 4yr + f . 



If you have correctly worked the examples now given, 

 you will have acquired a pretty good knowledge of 

 '>raical Addition and Subtraction, and have become 

 familiar with the meaning and use of the plus and minus 

 . We shall now explain a few further particulars, 

 and shall then give a short specimen of the application of 

 Algebra to what are called Simple Equations. These 

 will throw some light upon the practical utility of the 

 science ; they will afford an insight into the value of 

 algebraic symbols in matters of calculation, and enable 

 you to see the great advantage in such matters of com- 

 bining Utters with figures. 



You already know that the sign minus prefixed to a 

 quantity indicates the subtraction of that quantity. 

 Hitherto the sign has been prefixed to simple quantities 

 only ; but it may be prefixed to a compound quantity, so 

 as to indicate that nil the simple terms of which it is 

 composed are to be subtracted. In order to this it is 

 only necessary to unite all the terms by some link, so as 

 to imply that when a sign is put before the compound 

 quantity, that sign is to affect the tc/i" v, or 



every individual term of which it is composed : whatever 

 be the link employed, it is called a vinculum. Suppose 

 we have the compound quantity 4a 26 -f- 3c to subtract 

 from 9a + 66 5c, as in Exercise 1, page 454. We 

 might indicate the subtraction thus : 



9a + C6 5c (4<i 26 + 3c); or 



9a + C6 oc J4a 26 + 3c } ; or 

 Oa + 66 6c [4a 26 + 3c]. 



Such vincula as these are also called brackets. It is 

 plain, from the rule of subtraction, that a bracketed 

 quantity may be freed from the brackets, when a minus 

 sign is prefixed to it, by simply changing the signs of all 

 the terms which compose it. Thus the preceding expres- 

 sion, without brackets, is, 



9a + 6b 5c 4(1 + 26 3c = 5a + 86 8e. 

 And the subtraction of a compound quantity may in 

 general be performed in this way without taking the 

 trouble of writing it under the quantity it is to be taken 

 from. When the terms to be subtracted are all con- 

 nected, with changed signs, as above, to the other quan- 

 tity, we shall merely have a row- of terms to be united, 

 like with like, as in addition; just the same as in the 

 example now given. 



If the sign plus appear before a bracketed quantity, 

 then, since addition implies no change of sign, the signs 

 must remain undisturbed, though the brackets be re- 

 moved : thus 

 9a+C6 5c+(4a 26+ 3c) = 9a+66 5c+4a 26+3c 



= 13a+46 2c. 



Now, that yon may never find yourself puzzled, as to 

 si^ns, when you have to free an expression from brackets, 

 always be careful to notice the sign, whether -f- or , 

 which precedes the bracket ; fancy this sign rubbed out 

 along with the brackets ; if it be -{-, the terms thus set 

 free present their proper signs, without any change being 

 necessary. Of course the leading term within the 

 brackets, if itself a phis term, as the 4a, above, will not 

 have its sign actually inserted ; so that when the -f- 

 before the bracket is rubbed out with the brackets them- 

 selves, there will be a gap as between the 5c and the 4a 

 here ; you need scarcely be told that in this gap the -f- 

 belonging to the 4a must be inserted, because 4a is now 

 not a leading quantity. 



But if the sign before the bracket be , then having 

 rubbed out this sign, with the brackets, or having fan- 

 cied it rubbed out, write all the terms, thus set free, 

 with changed signs. For example 

 8oz + by + ( oax Sby) = 8ox + by Sax Zby 



= 3ax 26y. 

 Sax + by ( Sax 36y) = Sax + by + Sax + 36y 



= 13ax -f 4% 



You see that in the first of these expressions you have 

 only to fancy that, with the tip of your finger, you rub 

 out the marks + ( , between the by aud the 5ax ; and 

 in the second expression, that you in like manner rub 

 out ( , between the by and the 5ax, and then that 

 the signs of the terms thus set free are changed. By 

 attending to these hints, you will not be likely "to stick " 

 at bracketed expressions. 



Coefficient* are frequently found before bracketed quan- 

 tities : you are aware that coefficients are multipliers or 

 factors. When the brackets are removed, the factor is 

 to be introduced into each simple term, as in the in- 

 stances following : 

 3 ( 4a 26 + 3c ) = 12a 66 + 9c ; 6 ( Sax 3by) 



_ 4 ( _ Zz 5ay + 2) = 12* + 20ay 8 ; 



2a (toy 2te) = 6a.n/ 4a6z. 



As already stated, brackets are not the only kind of 

 vincula used to bind a set of simple quantities into one 

 compound whole ; a bar or line, put over the row of 

 quantities, is sometimes, though less frequently, em- 

 ployed ; thus, a -)- 6 + a 6, and a + 6 a 6, are the 

 same as a + 6 + (a 6), and a -f 6 (<i 6) ; but this kind 

 of vinculum is getting out of use, though the bar or line, 

 which separates the numerator from the denominator of 

 a fraction, still performs the office of a vinculum when 

 there are several terms in the numerator; thus, in the 



fraction ^ y ' c , the minus sign, before the bar of 

 separation, operates as it would do if the compound 

 numerator were enclosed in brackets ; thus, 



ax+by 6 

 that is, the fraction is the same as -^p- 



We shall not detain you further with these explana- 



