LESSONS IN ENGLISH. 



86 



GENERAL ROLE rou ADDITION. 



Write down the quantities to la added without altering their 

 tigns, placing tho$e that are alike under each other ; and uniic 

 tuch terms at are simii<n: 



( )t licrwiae. Write the quantities to be added one after another, 

 jmttiny the sign + between them, and then simplify the expression 

 l>\j incorporating like quantities. 



1 . If any of tho quantities bo in brockets and tho sign 

 -f be before the brackets, tho brackets may bo removed without 

 altering tho result. 



Hv brockets is meant tho vinculnm or parenthesis, already 

 explained [Art. 21], This is ono of tho most important things 

 in tho study of Algebra ; its use is unlimited. If quantities 

 be included in any manner between brackets or parentheses, 

 they must be treated as a single quontity, that is, tho result of 

 the operation of the signs within tho brackets is to bo used 

 instead of tho quantities themselves, as a general rale. If tho 

 signs of tho quan ties within tho brackets be either plus or 

 minus, or a combination of both, and if a factor be outside tho 

 bracket, each of tho quantities within may bo multiplied by 

 that factor, preserving their signs, and the product will be the 

 Biimo as if the result were multiplied by that factor. Thus, 

 x (a -\- b c) = ax -\- bx ex ; or, if a. + 6 c = e ; then 

 a (a -\- b c) = ex. Conversely, if tho result of the quantities 

 within tho brackets bo multiplied by any factor, tho result will 

 b3 tho same as if each of the quantities were multiplied by that 

 factor. Thus, if a + 6 c = e; then, ex = (a + b c) x = 

 ax -|- bx ex. If several factors be employed, the same results 

 will tako place. Thus, axy -f- bxy cxy = xy ( a -J- b c) = 

 (a -f- b c) xy ; and mbcd nbcd -f- pbcd = bed (m n -f- p) = 

 (m n + p) bed ; and pxyz -\- qxyz rxyz = xyz (p + q r) = 

 (p + q r) xyz. Expressions of this kind may be varied in- 

 definitely. 



Note 2. If tho sign be before the brackets, they cannot be 

 removed without vitiating the result, until the signs of all the 

 terms within the brackets be changed, viz., + into , and con- 

 Tersely. 



EXAMPLE. To She 6d + 26 3y, add 36c -f x 3d 

 + bg, and 2d + y +3x + b. 



These may be arranged thus : 3&c 6eZ + 26 3y 



3bc 3d + x + bg 



2d+ b+ y+3x 



And the sum will be 



7d -|-36 2y +4x + bg 



EXERCISE 4. 

 Add together the following quantities-^- 



1. ob + 8, to cd - 3, and Sab - 4m + 2. 



2. x + 3y - dx, to 7 - * - 8 + Jim,. 



3. abtrv 3* + bm, to y x + 7, and 5x 6y + 9. 



4. 3am + 6 7xy - 8, to lOxy 9 + Sam. 



5. 6ahy + 7d 1 + inxy, to Sahy 7d + 17 nury. 

 6 7ad - 7i + 8ory - ad, to 5ad + h - 7*y. 



7. 2by - Sax + 2a, to 36* - by + a. 



8. ax + by - xy, to - by + 2ry + So*. 



9. 41cd/ - 10-ry - 18b, to 7xy + 24b + 3cd/. 



10. 862?- 17xy + 18a, to 4ax - Sbx + 63cx. 



11. 3at> - 6be + 4cd - 7xy, to 17mn -f 18/g - 2a*. 



12. - 42abc + lOabd, to SOabc + 15abd + 5xyz. 



13. ax - y + 6 - d/ + 44, to 4d/ - 20 +-3a* + 75y. 



14. 45<z - lOb + 4cd/, to 82b - 4cdf + lOa - 46. 



15. 12 (a + b) + 3 (a + b), to 2 (a + b) - 10 (a + b). 



L6. xy (a + b) + 3xy (a + b), to 2xy (a + b) - 4ry (a + b). 



17. ax + aa, * + xxx, 4aa + 2z + ax, and SJ.T.C. 



18. y - yy + xy, 2xx + lOyy, to 4ry + 6y - 8*r. 



19. oaa + 4aoa, to 1- aa -.Uaaa + 8aaa. 



20. 12yyyy - 10i-jr, to 20xjc - Syyyy + 2xx + 3yyyy. 



21. 4 (x - y) - 13, to (a + b) - 16 (z - y) - 7 ( a + b). 



22. a (x + y) - 6y, to 40 (a - b) + 8a (* + y) - 36 (a - b). 



23. lOazy + ITbcd - aa-y, to Gary - Ubcd. 



24. - x + y + 6x (a - b) - 7x, to IGy - 15* (a - b) + 25*. 



25. - 4 (x + y) + 16 (x + y), to 15nbc - 10 (* + y). 



26. Sabo - 6*y + run, a + 6abc + 14y - llo + 6mw, to 15ry - 17abo 

 loa -abo + xy - 3mn + abc. 



5S7. a (* + y) - 3b (* + y)-4a (x + y) - 4 (*+ y)- (x + y),to 4b (* + y) 

 + 7a(x + y)+5(x + y) + 6b(x + y). 



Note. As the expressions x a (square x), y 3 (cube y), etc., are 

 nsed for the first time in the following exercises, the learner is 

 referred to Art. 28 : 



2o - So* - 



- 2* - * - 1, and 



83. Jc*, a'x, y', */, and 3b/. 



29. a* - 2a'6 - 3a* + 2t, 3d 

 * b. and 5<i 3 - 4o6 - at 1 + 3W. 



80. *-** - a + 1, Z^ 

 - ifl + Z* - 8. 



31. -a-fb+c + d, a-b + c + d, a + b-c-fd, tmda+b+o-d, 

 - 2b, 2b - 3c, 3o - 4d, 4d - 5, and 5 - Of. 



83. v* + 2yz - 3yj, 2y + 2/z + Syt*, and 3-j* - 4'j*z - 2y**. 



34. or* + bx\ b* - ex 1 , and c* 3 + d*. 



35. m** nz, tu* pz, and 2* *. 



KEY TO EXEBCISES IN LESSONS IN ALOEBBA. 

 EXERCISE 1. 



i. 



1. (a 7i) x (b + c + d) = 37m 



2. a + b : - : : ac : 12/L 



c 

 _ a + b + c 



^ = 4 (a + I + c) d. 



Sake 



EXERCISE 2. 



1. The product of a and b increased by the quotient of 3 time* Ik 

 minus c, divided by the sum of x and y, is equal to the product of d b/ 

 a increased by the sum of b and c, and diminished by the quotient ut 

 h divided by the sum of 6 and b. 



2. If a be added to 7 times the sum of h and *, and from this sum, 

 the quotient of c less 6 times d, divided by the sum of twice a and 4, 

 be subtracted, the remainder will be equal to the sum of a and It, 

 multiplied by the difference of b and c. 



3. The difference of a and b, is to the product of a and c, as the 

 difference of d and 4, divided by m, is to 3 multiplied by the sum of 

 h, d, and y. 



4. If the quotient of the difference between a and h, divided by the 

 sum of 3, and b less c, be addod to the quotient of the sum of d and 

 the product of a and b, divided by twice m, the whole will be equal to 

 the quotient of b times a multiplied by tho sum of d and /, divided by 

 a times m, lessened by the quotient of c times d divided by h increased 

 by d times m. 



EXERCISE 3. 



1. 

 2. 



4 + 80 

 2x6 



; + 8 x 10 = 



(4 x 2) + 10 _ 

 3x0 ~ 



80 = 92. 



81 



a 4+ J 3x6) +(4x2x8) ^4+(4x2xlO) :=iJ - 

 i o x 4, J 



= f + 64-7 = 68. 



44x8+ ( 3 x 4) + (3 x 6) (3 x 4 x 10) -(6x2) _ 



12 + 18 



5. (3 x 4 x 8) + !-2-f + (2 x 10) = 96 + ? + 20 = 118. 



o 4 4 



6. (3 + 2) x (10 8) + - 3 = 5 x 2 + * 3 = 



7. 



3 x (6 



10-6 



- TT = 24. 



10 (4 x 2) 



(3 x 2) + (5 x 8) (4 x 6-4) x (3-2) 6 + 40 + 



' 72 x 10J + 3 ~~^ = i*^"- 



(24 4) x 1 _ 46 20 _ 



* ;nr T + i~ 4, 



10 



' 20+ 3 



I 8 + 



10 



LESSONS 



IN ENGLISH. XXVIII. 



LATIN STEMS. 



WE are about to lay before the student a large portion of the 

 roots of tho Latin language. In the study of them, he may 

 become acquainted with the treasures of the Roman literature, 

 and the tone and strength of the Roman mind. These lessons do 

 not indeed, lie on the surface. Nevertheless, they are to be learnt 

 by care and diligence. For this purpose, the learner should 

 impress on his mind the preceding remarks, and remembering 

 that a language is the mirror of a nation's mind, accustom 

 himself to see and contemplate the Romans in their worda 

 those unerring tokens of thought, those mental miniatures. 



Of course it is only so much of the Latin vocabulary as exists 

 in English that I shall set forth in these pages. The Latin 



