LESSONS IN ALGEBRA. 



347 



le i-ifhth iiinl tin- ninth, we hare 

 4s 20, or = 5 ; 

 i, wo have 



usposing in tho third, wo have 



a>= 12 y * = 3; 

 Transposing iu tho second, wo have 



w = 9 x y = 2. 

 EXAMPLE (2). Given w+ 50 = x, 

 a + 120 = 3y, 

 y + 120 = 2, 

 s + 195 = 3io ; 



to find tho values of w, x, y, and *. Ant. to = 100, x = 150, I 

 y = 90, and = 105. 



itciss 40. 



1. There is a certain fraction, such, that if 3 be added to the 

 numerator, the value of tho fraction will be J ; but if 1 be sub- 

 tracted from the denominator, tho valuo will bo J. What is tho 

 fraction t 



2. i'l.i.!.' tins number 90 into four such parts, that if the first is incrtated 

 by 2, tho second dim ''nuhcd by 2, tho third vnuUiplud by 2, and 

 irth divided by 2, shall all be c.|>i >!. 



3. Find three numbers, such that the .first, with half tho sum of tho 



Mcond and third, shall bo 120 ; tho <xid, with J the difference 

 of the third audyir.it, shall be 70 ; and half tho sum of th<> 

 numbers shall be 95. 



4. What two numbers are those whose difference, sum, and product 



are as tho numbers 2, 3, and 5 ? 



5. A vintner sold at one time 20 dozen of port wine, and 30 dozen of 



sherry; and for the whole received 120 guineas. At another 

 time, he sold 30 dozen of port and 25 dozen of sherry, at the 

 same prices as before, and for the whole received 140 guineas. 

 What was the price per dozen of each sort of wine ? 



6. A merchant having mixed a certain number of gallons of brandy 



and water, found that, if ho had mixed 18 gallons more of each, 

 he would have put into the mixture 8 gallons of brandy for 

 every 7 of water. But if ho had mixed 18 less of each, ho 

 would have put in 5 gallons of brandy for every 4 of water. 

 How many gallons of each did he mix ? 



7. What fraction is that, whose numerator being doubled, and the 



denominator increased by 7, the value becomes j ; but the 

 denominator being doubled, and tho numerator increased by 2, 

 the value becomes j ? 



If in tho algebraic statement of tho conditions of a problem, 

 the original equations are more numerous than tho unknown 

 quantities, these equations will either bo contradictory, or one or 

 more of them will be superfluous. 



Thus, the equations 3x = 60, and x = 20, are contradictory. 

 For, by tho first, x = 20 ; while, by the second, x = 40. 



But if the latter equation be altered, so as to give to x tho 

 same valuo as in the former, it will be useless, in the statement 

 of a problem. For nothing can be determined from the one 

 which cannot be from tho other. 



Thus, in the equations 3x = 60, and fa 10, one is super- 

 fluous. 



But if the number of independent equations pix>duced from 

 the conditions of a problem bo less than tho number of unknown 

 quantities, tho subject is not sufficiently limited to admit of a 

 definite answer. If, for instance, in the equation a:-|-y = 100, 

 x and y are required, there may be fifty different answers. The 

 values of x and y may be either 99 and 1, or 98 and 2, or 97 and 

 3, etc. For the sum of each pair of these numbers is equal to 

 100. But if there be a second equation which determines one 

 of these quantities, the other may then be found from tho 

 equation already given. As x + y = 100, if x = 46, y must bo 

 such a number as added to 46 will mako 100, that is, it must bo 

 54 ; and no other number will answer this condition. 



In most cases, also, the solution of a problem which contains 

 many unknown quantities may bo abridged by particular artifices 

 in tii'lixlidititig a single letter for several. 



EXAMPLE (3). -Suppose four numbers, u, x, y, and , arc' 

 required, of which the sum of the first three is 13, tho sura of 

 the first two and the last is 17, tho sum of the first and the last 

 two is 18, and the sum of the last three is 21. 

 Here, u -f + y = 13, 

 u 4- x -f- z = 17, 

 u n-y + * = 18, and 

 x 4- y + * = 21, by the question. 



No\r, lot 8 be substituted for the mm at the four number*, 

 that i*. u + * + y -f *. It will then be Men that of tbeee four 



."OS, 



The first contains all tho letters except * , that is, 



8-= 1 

 Tbe second contains all except y, that u, 



8-y = 17; 

 Tho third contains all except x, that is, 



S-*= 18, and 

 The fourth contains all except u, that Ls, 



8-u = 21. 



Adding all these latter equations together, we have, 

 4S-r-y-x-u = 09, or 



10, 



But 8= (3 + y+x+u) by substitution. 

 Therefore, 48 - 8 = 69, that is, 38 - 69, and 8 - 23. 

 Now, putting 23 for S, in tho four equations in which it is 

 first introduced, wo have, 



OO 1 Q / . _ ft*l _ 1 Q 



W- 28-17 



23-y=17, 



23-* = 18, 



and 23-u=21. 



Therefore, 



1 a =23 -18 = 5, and 



(u=23-21 





Contrivances of this sort for facilitating the solution of par- 

 ticular problems, must be discovered by the student's own 

 ingenuity and skill. They are of a nature not to be taught by 

 a system of rules, but by practice and plodding industry, which 

 is genius. 



KEY TO EXERCISES IN LESSON'S IN ALGEBRA. XXI. 



EXEKCISE 35. 

 . x* + 3**y + 3jr./ a + y*. 



2. a 4 + 4a s i) + 6a*b* + 4ab 3 + b 4 . 



3. o - 6a s b + 15ab* - 20a 3 b 1 + 15aV - Cab* + b*. 



4. * + S* 4 !/ + 10xV + 10*V + 5r/ + y*. 



6. m 7 + 7m'n + 21m s n* + 35m*u 3 + 35m*n* + 21m*ii l 



7. a + 9ab + 36aV + 84ab 3 + 126o s b + 126o6 + 



9ab + b. 



8. a" 



13)*V 



9. * ls IS-e^y + 78* Il y* 2ti6a M> j/s + 715**jr* 1287r'y + 1716*'y* 

 1716*V + 1287*V - 71*Hy + 286iV- 78jc*y n + 13xj u - y" 



10. a' 7a*b + 21a B l>* 35aV< 3 + 35o*i>* 21a*b* + 7ab* b*. 



11. a* + 8a'b + 28ab + SSa'b 1 + 70at< 4 + 50ob* + 28ab + 8ab T -I- b. 



12. 32 + 80* + 80x* + 40a + lOa.* + *. 



13. a 3 -3a z br + 3a 1 o+3ob m T_6a*cat + Sac* - I V + 3b*e* - <* + A 



14. a s + 9a'bc + 27abV + 27b*c. 



15. 16a*b* 32a s b 3 * + 24**l>** 



16. 16a*b + 40abc* + 25c 4 . 



17. 27**- 162* 2 ;; - 324jry - 2 



18. 125a 3 + 225a*d + ISSad 1 4 



KEY TO EXERCISES IN LESSONS IN ALGEBRA. 2CXIL 

 EXEUCISE 36. 



1. x = 5, and y = C. 



2. * = 10, and y = 3. 



3. * = 6, and y <= 4. 



15, and y 

 11, aud v ' 



9. 



1. x= 5, and y ' 



2. x = 2, and y 



3. x = 4, and y = 



4. * = 8, and y 



10. 

 20. 

 LB. 



EXERCISE 37. 



5. ICO and 140 miles. 



6. A 49 years, and B =* 21 years. 



7. 15 the (TMter, and 10 the less. 



1. x = 6, and y = 4. 



2. * S, and y 6. 

 ::. r = 12, and y = 2. 



4. * = 7, and y =3. 



5. 11,111 = greater,anny, sad 9,9K> 



= smaller army. 



6. 120 the greater, and 100 the less. 



7. soft, the lower portion, 48ft. 



tho upper portion, 108 ft. the 

 total height 



EXKBCISE 38. 



8. 10 and 2. 



9. S and 2. 



10. 20 and 12. 



11. Bud J'. 



!.. m, 



13. 63. 



14. 78 galloos of brandy, and 



gallons of gia. 



