ADDITION .] 



MATHEMATICS. ALGEBRA. 



453 



letters into numbers. But in many cases you can actually 

 add and subtract, though you may remain in utter igno- 

 rance as to the meaning of the letters: there is such a thing 

 as addition of algebra as well as addition of common num- 

 bers ; and subtraction of algebra as well as arithmetical 

 subtraction. AVe shall now introduce you to these 

 operations ; you will at once see how it happens that 

 you can actually perform, the operations indicated by -f- 

 and , even upon quantities whose values are unknown, 

 by examining a simple instance or two, before proceeding 

 to RULES. Suppose we have such an expression as oax 

 + 3ax fax -j- 2ax ax ; you would surely not re- 

 quire to know what ax stands for, before you could 

 perform the operations indicated by the signs ; you 

 would say to yourself whatever thing ax may represent, 

 5 plus 3 of them must make 8 of those things : 8 of them 

 minus 7 of them leaves one ; this, with the two followin _<-, 

 make 3 of them, which diminished by one, gives 2 of the 

 things, whatever they be, as the amount of the whole 

 row ; you would thus be sure that the result is 2ax ; you 

 would thus state with confidence that 



5ax + Sax lax + 2ax ax 2ax. 

 The compound expression on the left of the sign of equa- 

 lity, is thus reduced to the simple expression on the 

 right, solely because the several quantities in the com- 



i'l expression are like quantities; that is, they di tier 

 in nothing except in coefficient and in siyn ; the letters 



: 'ie same in all. It is only when the terms of a 

 mnd expression are utdike quantities that the opera- 

 tions indicated by -f- and cannot be performed til] 

 the values of the letters are stated. You will no doubt 

 be able of yourself to put the proper simple value, on 

 the right of the sign of equality in each of the following 



1. iry 4xy + I3xy + 3fy = 



2. axz "taz -f- 1 laxz -f- axz = 



3. 9mr -f- bmnx 13mn,r == 



4 4 4 44 



ADDITIOX. 



Addition of algebra U the finding the amount of a set 

 ot quantities, of which some are additive, and the others 

 subtractive. It differs, therefore, from addition in com- 

 mon arithmetic in tlii.s ; that, in the latter operation all 

 the numbers are additive. Addition of algebra 'therefore 

 combines the two operations which, in arithmetic, are 

 called addition and subtraction. 



CASE I. IVJien the quantities to be added are all like 

 quantities. 



RULE 1. Find the sum of the positive coefficients. 



2. Find the sum of the negative coefficients. 



3. Take the difference of these two sums, and 



prefix to that difference the sign belonging 

 to the greater sum. 



4. Annex to the difference the letters common 



to all the quantities, and the correct sum 



will be obtained. 



1. 2. 3. " 



Za Jx Say 



So 3x lay 



2a 2x bay 

 ~a 13.r ay 



4a x 9ay 



9a 



\4x 



4. 



2ax -\- Zly 

 3ax 2iy 

 ba.T -)- 4//y 

 4a.r by 

 Cox -f ~,t',y 



10or+ Illy 



Say 



5. 

 baxz 3bcy 



(>axz 2bcy 

 lax: -|- blicy 



Zajcz bey 



4axz -f- 4bcy 



6<urz -(- 3l/ey 



The fourth and fifth of these examples each consists of 

 two vertical rows of /< iei ; the first row is always 



thnt which is first computed ; so that iu algebra we be^in 

 with the column at which, in arithmetic, we end. Tliu 



reason why the columns are added up in this order is, 

 that it is more convenient to write the results of the 

 several columns, witli the proper signs, from loft to 

 right, than from right to left. 



From carefully looking over these examples, you will 

 see that the only work performed is the adding up of all 

 the positive coefficients in each column, and all the nega- 

 tive coefficients in separate totals, and then writing the 

 difference of these sums for the coefficient of the result 

 prefixing to it the sign belonging to the greater of th 

 two sums, and then writing against it the Utters common 

 to all the terms in the column. The work, therefore, i 

 purely arithmetical ; the letters in, the finished result are 

 simply copied from tho terms above. 



EXAMPLES TOR EXEKOISE. 



1. 



24 



74 



94 



114 



34 



4. 



3a.ry 



a ty 



2axy 



12ajy 



7. 

 -\- 3mx n 



4mx-{-2n 



2mx Sn 



SIM* 3n 



9. 



- alz Zx -\- a 3 



- 2o6* -{- x 2a 



- 4al/z If +2 

 babz 13* -)- 4a 



\\abz -|-5a 



II. 



3z -\- ate 1 Imp 



5a4e -)- 4mji 



Is 2abe Zmp 



5z + 9m/; 



1 8a4c Imp 



4abc -\- mp 



3. 



Gly 

 Zby 



44y 



Siy 

 'by 



G. 



Zax Zbz 

 box + 36: 



lax 84* 



lax -\- Gbz 

 lax 4bz 



8. 

 2a4 



7a4 6 



2xyz 3a4 + 7 

 bxyz o4 13 



8xy: + 4o4 + 9 



10. 



9ac.r -)- 2liey tg 



2acx Tbey + Zky 



4acx 3l/ey 1kg 



acx -f- Sky 



lacx 54?y 



4 bey iy 



12. 



3mz -|- 



1 6ty -f- 9 8a4c 



bmz 



4ty + <jmz 3abc 

 bky -\- labc 



2ky \\rnz abc 



Add together the following quantities : 



13. 7a4z 3cey-\-1mx, -f-3a4z ~mx, 



14. 2ar+6pgy 13, +Spyy+ll, +7 ox 2, 3ox-pqy, 



15. 3axy, 7aoc, 3ary-|-2a4-(-5c, axy 5a4 2c. 



16. bjyz 2an4-34n, 7am 74, Gxyz bn, 4am bbn. 



17. 6bcx-\-3ny, "imz, Sny, 2bc.rny, bny 3mz, 2m:. 



18. 9exz 13, 7bcy-\-4, GM; 44cy+l, \3bcy~ 3, \exz-bcy. 



19. Impy 2ax-\-3, .'iar 7, 3mpy 2, mj>y-\~Sax-\-6. 



20. cyx 4, 6ai-)-2, 2cyx ab 5, 17, 4cgx-\-7a&, 13. 



CASK II. Wlten the quantities (o be added are not all 

 like quantities. 



When rows of quantities are arranged one under 

 another, so as to present a set of vertical columns, as in 

 Examples 1 to 12 above, if the vertical rows arc not rows 

 of lilx quantities, then you will have carefully to examine 

 all the rows, aud to pick out from among them the dif- 

 ferent sets of like quantities, and add them together as 

 Before ; those quantities that have none like them must 

 jo merely connected with their signs to the sums thus 



