LESSONS IN MUSIC. 





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HERIVALK FRERES. 



LESSONS IN ALGEBRA XV. 



SIMPLE EQUATIONS (continual). 

 REDUCTION BY MULTIPLICATION. 



ir,o. WHEN the unknown quantity is connected with a known 

 quantity by the sign of division, the reduction is effected by 

 multiplying both members of the equation by the latter, if it bo 

 the divisor ; and by the former, if it be the divisor. 



In this case, it will bo particularly useful to remember a rule 

 formerly given, namely, that a fraction is multiplied by its do- 

 nominator, by removing the denominator ; or, in other words, 

 putting down the numerator as the product. Also, that after this 

 process has been performed, transposition is still to be employed 

 as in the preceding examples. x 



EXAMPLE. Reduce the equation --f- a = 5 -j- d. 



Here, multiplying both sides by c, we have, for the product, 

 x + ac = be cd; and, by transposition, x = be -f ctl ac. 



161. Though it is not always necessary, yet it is often conve- 

 nient, to remove the denominators from fractions consisting of 

 known quantities only. This is done in the same manner as in 

 the preceding rule. , , 



EXAMPLE. Eeduce the equation - = - -I 



a be 



Here, multiplying by a, we have x = h i 



dbh b c 



again, multi- 





plying by b, we have bx = ad + - ; lastly, multiplying by c, 



acd + abh 



we have bcx = oca + abh. Whence x = , - 



oc 



Ans. 



162. An equation may be cleared of fractions by multiplying 

 both members by all the denominators. 



163. In clearing an equation of fractions, it often happens 

 that a numerator becomes a multiple of its denominator (i.e., 

 can be divided by it without a remainder), or that some of the 

 fractions can be reduced to lower terms. When this occurs, the 

 operation may be shortened by performing the division indicated, 

 and by reducing the fractions to their lowest terms. 



164. In clearing an equation of fractions, it will be necessary 

 to observe, that the sign prefixed to any fraction, denotes 

 that the whole value is to be subtracted, which is done by 

 changing the signs of ail the terms in the numerator. 



EXAMPLE. Reduce - - = c - SLUl 



(a d)r x 



36 + 2/im -f- 6n' 



EXERCISE 26. 



x-4, 

 1. Reduce the equation g + 5 = 20. 



x 

 1. Reduce the equation fr^ + d = h. 



6 

 3. Reduce the equation ..._- +7=8. 



4. Reduce the equation - = -, + 



a d <? m 



x 2 4 6 

 5. Reduce the equation ;, ; = - + - + -. 



?. Ans.x = 



- 



. Reduce * - = 6. 

 3 4 



4r 3 3* 

 - Seduce + 



8 

 i6' 





8. Reduce 2* -^ = . + = 

 o H> a 



3x 

 4 



10 



REDUCTION BT DIVISION. 



165. When the unknown quantity containt any known qn 



at a factor, the equation it reduced by dividing every term, on 

 both members by thit known quantity. 



\ M I-I.K. Reduce the equation ox -f- b 8h = d. 

 Here, by transposition, we have ax d -f 3h b ; and 



dividing by a, we hare x = -^ . Ant. 



a 



166. If the unknown quantity haa co-efficients in several 

 terms, the equation must be divided by the aum of all theae co- 

 efficient*. 



EXAMPLE. Reduce the equation 3x bx = a d. 

 Here, 3x bx = (3 b) x ; and (3 6) X * = a d. 



Whence, dividing by 3 6, we have x = H . 



3 6 



Am. 



167. If any quantity, either known or unknown, is found aa a 

 factor in every term, both members of the equation may be 

 divided by it. On the other hand, if any quantity is a divisor 

 in every term, both members of the equation may be multiplied 

 by it. In this way, the factor or divisor will be removed, and 

 the reduction may be effected as before. 



EXAMPLES. (1.) Reduce the equation aar-f- Sab = 6ad + a. 

 Here, dividing by a, we have x + 36 = 6d + 1 ; and, Dy 

 transposition, x = Gd + 1 36. Ans. 



(2.) Reduce the equation J = . 



xxx 



Here, multiplying by x, we have x+ 1 b = h d; and, by 

 transposition, x=h d-\-b 1. Ans. 



168. A proportion is converted into an equation by making the 

 product of the extremes, one member of the equation; and the 

 product of the means, the othei' member. 



EXAMPLE. Reduce to an equation ax-.b : : ch: d. 

 Here the product of the extremes is adx, and the product of 

 the means bch ; the equation is, therefore, adx = bch. When'^ 



bch 



x= . Ans. 

 ad 



169. An equation may be converted into a proportion, by resolv- 

 ing one side of the equation into two factors, for the middle terms 

 of the proportion ; and the other side into two factors, for the 

 extremes. 



EXAMPLE. Convert the equation adx = bch into a proportion. 



Here the first member may be divided into the two factors ax 

 and d ; the second into ch and b. From these factors we may 

 form the proportion ax-.b -.: ch:d. 



EXERCISE 27. 



a d 



1. Reduce the equation 2* = - - , + 4b. 



c ti 



2. Reduce the equation ax + x = h 4. 



x b_ * + d 



3. Reduce the equation x r ~r 



Reduce the equation * * (a + b) - a - b = d x (a + b). 



Reduce to an equation a + b : c : : 7i m : y. 



Reduce to a proportion the equation ay + by = ch cm. 



Reduce the equation 16* + 2 = 34. 



Reduce the equation 4z 8 = 3x + 13. 



Reduce the equation 10* 19 = 7* + 17. 



Reduce the equation 8* - 3 + 9 = - 7.c + 9 + 27. 



KEY TO EXERCISES IN LESSONS IN ALGEBRA. 

 EXERCISE 25. 



1. x = b + 4. 



2. v = 2ob 2Jim - a. 

 3 x = b-7h-d-22. 



4. x = 8bh -H 9. 



5. x = 15. 



6. 

 7. 

 8. 

 9. 

 10. 



= 11. 

 = 20. 

 = 31. 

 = 8 

 = 6. 



11. * 



12. * 



13. v 



14. * 



24. 



LESSONS IN MUSIC. XXI. 



EXERCISES " HONEST FELLOW " AULD LANG SYNE." 



IN our last Lesson in Music (Vol. in., p. 398) we gave the 

 learner a great deal of necessary information on the different 

 kinds of voices of men, women, and boys, proper enunciation, 

 and singing in parts. We now propose, in accordance with 

 our promise, to set before our pupils some exercises in port- 

 singing; but before any student commences to practise theae, it 

 will be as well for him or her, aa the case may be, to read orer 



