THE POPULAR EDUCATOR. 



4. II ne faut pas lui e"crire. 5. A-t-il fallu parler a M. votre pere? 

 6. II a fallu lui parler. 7. Faut-il aller a D. aujourd'hui ? 8. II faut 

 y aller. 9. Me faut-il aller trouver M lle votre sceur ? 10. II vous 

 faut aller la trouver, elle d<5sire vous parler. 11. Combien d'argent 

 faut-il a votre frere ? 12. II lui faut dix francs cinquaute centimes. 

 13. Couibien de livres faut-il a M"e votre soeur ? 14. II lui faut 

 beaucoup do livres, elle lit beaucoup. 15. Que voulez-vous euvoyer 

 au chirurgieu? 16. II nous faut lui envoyer notre cheval; le sieii est 

 malade. 17. Ne lui faut-il pas du papier ? 18. II lui en faut, il a des 

 lettres a e"crire. 19. Lui eu faut-il beaucoup ? 20. II lui en faut vine 

 main. 21. Vous faut-il encore) quelque chose ? 22. II me faut encore 

 quelque chose. 23. II ne me faut plus rien. 24. Vous faut-il cent 

 francs ? 25. II me faut dix dollars. 26. Que faut-il au chirurgien ? 

 27. II lui faut de 1'argent pour payer ses dettes. 28. Le tailleur a-t-il 

 tout ce qu'il lui faut ? 29. II n'a pas tout ce qu'il lui faut. 30. La 

 moclisto a re<;u tout ce qu'il lui faut. 31. Que vous faut-il pour votre 

 peine ? 32. Combien vous faut-il ? 33. Combien nous faut-il ? 34. 

 Que me faut-il faire ? 35. II vous faut e"crire uue lettre. 36. Que lui 

 faut-il dcrire ? 37. II lui faut e"crire quatre pages. 38. II lui faut aller 

 a 1'dglise. 



EXERCISE 93 (Vol. I., page 356). 



1. Does it become you to reproach us with our neglect? 2. It 

 becomes me to reproach you when you deserve it. 3. Does it suit you 

 to go to my brother ? 4. It does not suit me to go to him, I have 

 something else to do. 5. How much may that field be worth ? 6. It 

 may be worth about twenty thousand francs. 7. Are you better than 

 your brother ? 8. My brother is much better than I. 9. Is not thafi 

 knife worth more than yours? 30. Mine is better, it is worth more. 

 11. How much is your watch worth ? 12. It is not worth much, it 

 does not go well. 13. How much is the merchant worth ? 14. I 

 cannot tell you exactly ; he is worth about a hundred thousand francs. 

 15. Is it not better to remain here than to go to market ? 16. It is 

 better to go to market. 17. Is your gold chain worth more thau 

 mine ? 18. It is worth quite as much. 19. It is not worth much, it is 

 broken. 20. Is that worth fifty francs ? 21. It is worth at the most 

 two francs. 22. Have you asked the merchant what that is worth ? 

 23. I have not asked him. 24. He assures me that it is worth about 

 one hundred francs. 



LESSONS IN ALGEBRA. III. 



ADDITION. 



EXAMPLES. (1.) John has x marbles and gains b marbles more. 

 How many marbles has he in all ? 



In this example we wish to add x marbles to 6 marbles. But 

 addition in algebra is denoted by the sign +. Hence x + & is 

 the answer, i.e., John has the sum of * marbles added to b 

 marbles. 



(2.) What is the sum of 35 pounds added to the sum of c 

 pounds and / pounds ? 



By algebraic notation, 3b + c + / pounds is the answer. 



44. The learner may be curious to know Iwiu many marbles 

 there are in x -f- b marbles ; and Iww many pounds in 36 + c + / 

 pounds. This depends upon the number each letter stands for. 

 But the questions do not decide what this number is. It is not 

 the object, in adding them, to ascertain the specific value of x 

 and b, or 36, c, and /; but we find an algebraic expression, which 

 will represent their sum or amount. This process is called 

 addition. Hence 



45. ADDITION in algebra may be defined, the connecting of 

 several quantities with their signs into one expression . 



46. Quantities may be added, by writing them one after another, 

 witlwut altering their signs. 



N.B. A quantity to which no sign is prefixed is always to 

 be considered positive, that is, the sign + is understood [Art. 

 12]. 



EXAMPLE. What is the sum of a + m, b 8, and 2h 3m 

 + d? a + m + b 8 + 2h 3m + d. Ans. 



47. It is immaterial in what order the terms or letters are 

 arranged, If you add 6 and 3 and 9, the amount is the same, 

 whether you put the 6, the 3, or the 9 first namely, 18. But it 

 is frequently more convenient, and therefore customary, to arrange 

 the letters in alphabetical order. 



48. It often happens that the expression denoting the sum or 

 amount may be simplified by reducing several terms to one. 

 Thus, the expression 2a + 7 a -f- 4a may be abridged by uniting 

 the three terms into one. Thus, 2a added to 7 a makes 9a, and 

 4a added to 9a makes 13a, that is, 2a + 7a + 4a = 13a. 



There are two cases in which reductions can be made. 

 49. Case 1. When the quantities are aVJte, and the signs 



alike, as + 46 + 56, or 4y 3y, etc. Find the sum of the 

 co-efficients, annex the common letter or letters, and prefix the 

 common sign. 



EXAMPLES. (1.) What is the sum of 3a, 4a, and 6a? 

 Here, 3a -f 4a + 6a = 13a. Ans. 



(2.) 3xy (3.) 76 + xy (4.) ry + 3abh (5.) cdxy + 3mg 



7xy 86 + 3xy 3ry + abh 2cdxy + mg 



xy 26 + 2xy Gry + 4>abh 5cdxy + 7mg 



2xy 66 + 5xy 2ry + abh Icdxy + 8mg 



13xy 23b + llxy 12ry +9abh IScdxy + 



50. The mode of proceeding is the same when all the signs 

 are. 



EXAMPLES. (1.) What is the sum of - 36c, - be, and 56c? 

 Here 3bc be 5bc = 96c. 

 (2.) - ax (3.) - 2a6 - my (4.) - 3ach - 8bdy 



Sax ab 3my - ach bdy 



2ax 7a6 8my 5ach 7bdy 



Gax I0abI2my 9a,ch IGbdy 



51. Case 2. When the quantities are alike, but the signs 

 unlike, that is, only one of each, as + 96 and 66 ; 



Take tfa less co-efficient from the greater; to the difference 

 annex the common letter or letters, and prefix tlie sign of the 

 greater co-efficient. 



Suppose a man's loss is .500, and his gain .2,000. The 

 algebraic notation is 500 -f 2,000, i.e., .500 is to be sub- 

 tracted from his stock, and .2,000 added to it. But it will be 

 the same in effect, and the expression will be greatly abridged, 

 if we add the difference between .500 and .2,000 viz., .1,500 

 to his stock. 



EXAMPLES. (1.) What is the sum of 16a6 and Tab? 



Ans. 9a&. 



(2.) (3.) (4.) (5.) (6.) 



To +46 56c 2hm dy + 6m 3h dos 

 Add 66 76c 9hm 4dy m 5h + 4dx 



26 26c 7hm 3dy -f 5m 8h + 3dx 



52. If several positive and several negative quantities are to 

 be reduced to one term, first reduce those which are positive, 

 and next those which are negative, to one term, and then pro- 

 ceed as in Art. 51. 



EXAMPLES. (1.) Reduce 136 + 66 + 6 46 56 76, to 

 one term. 



Here, 136 + 66 + 6 = 206 ; and 46 56 - 76 = 166 ; 

 Whence 206 166 = 46. Ans. 



(2.) Add 3xy xy + 2xy Txy + 4xy 9xy + 7xy Gxy. 

 Here, 3xy + 2xy + 4&y + 7xy = IGxy. 

 And, xy 7xy Vxy 6xy 23a;y ; 



Whence, IGxy 23xy = Ixy. Ans. 



(3.) Add 3ad Gad + ad + Tad 2ad + Qad Sad 4ad. 

 Here, 3ad + ad + "lad + Qad = 20acZ ; 

 And, 6ad 2ad Sad 4ad = 20ad ; 



Whence 20ad 20ac? = 0. Ans. 

 (4.) Add 2a6m abm + 7a6m 3a6i + 7a6m. 

 Here, 2a6m + 7a6m + 7a6m = 16a6m ; 

 And, abm 3a6m = 4a6m ; 



Whence, IGabm 4a6m = I2abm. Ans. 

 (5.) Add axy Taxy + 8axy any 8ojxy + 9axy. 

 Here, axy + 8axy + Qaxy = 18axy ; 

 And, 7 'axy axy 8axy = IGaxy ; 



Whence, IQaxy IGaxy = 2axy. Ans. 



53. If two equal quantities have contrary signs, they destroy 

 each other, that is, the results of their addition is 0, and they 

 may be cancelled. Thus + 66 66 = 0. And (3 X 6) 18 

 = 0, so 7&c 76c= 0. 



54. If the letters, or quantities in the several terms to be added, 

 are TINLIKE, they can only be placed after each other, ivith their 

 proper signs (Art. 46). 



EXAMPLES. -(1.) If 46, Gy, 3x, 177i, 5d, and 6, be added, 

 their sum will be 46 6y + 3x + 177& 5d + 6. 



(2.) Add aa, aaa, to xx, xxx, and xxxx. 



Different letters, and different powers of the same letter, can 

 no more be united in the same term, than pounds and guineas 

 can be added, so as to make a single sum. Six guineas and four 

 pounds are neither ten guineas nor ten pounds ; therefore the 

 sum of the above = aa + &&<* + xx + xxx + xxx - 



55. From the foregoing principles we derive the following 



