ALGEBRA. 



Fx 1 Let -* =13? To find x 

 1 ' Let ?2a>fry8l5 and y. 



Multiply the terms of the first equation 

 by 2, and the terms of the other by 3, 

 then 6 a 10 #=26 

 6 oH-21 #=243 

 By subtraction, 31 y 217 



217 

 and v= =7; 







13, 



also, 3 x 5 y = 13, or 3 x 35 

 therefore 3 a; = 13 + 35 = 48 



and x = = 16 



Fx 2 T^fr $ax+t>y=c-> To find x 



-r***. t_mxny=d$ and y. 

 From the first, m a x + * ^ y = * c 

 from the other, max n a y = ad 

 by subtraction, m b y -\- n a y=m c ad, 



m c a d 



therefore, y = r-r -- . 

 m b-\-n a 



Again, n a x-\~n b y=n c 

 m b x n b y=b d 



by addition, n a-\-m b .x = n c + b d, 

 *P n c-\-b d 



therefore x = - ; -.. 

 n a-\-m o 



If there be three independent simple 

 equations, and three unknown quantities, 

 reduce two of the equations to one, con- 

 taining 1 only two of the unknown quanti- 

 ties, by the preceding rules ; then reduce 

 the third equation and either of the form- 

 er to one, containing the same two un- 

 known quantities ; and from the two 

 equations thus obtained, the unknown 

 quantities which they involve may be 

 found. The third quantity may be found 

 by substituting their values in any of the 

 proposed equations. 



C2 x+3 #+4 z=16} To find.r, 

 Ex. Let ^3oH-2# 5z=8 C #, and 



CSx 6 #+3 z=9 ) z. 

 From the 21st equa. 6 x-^-n #-4-12 z=48 

 6 x+4 #10 z=16 

 by subtr.5# 22z=32 

 from the lt and 3rd 10 x + 15 #+20 2=80 

 10 j 12 y +6z=12 

 bysubtr. 27#-f-14z=68 

 and 5 #+22 -=,32 

 hence 135 y + 70 2=340 

 andl35#+594z=864 

 bysubtr. 524 z= 524 

 z = 1 



5 y -f 22 z = 32 

 that is, 5 y -f- 22 =32 



5# = 32 22 = 10 

 10 



that is, 2 x + 6 + 4 == 16 

 2# = 16 -6 4 



The same method may be applied to 

 any number of simple equations. 



That the unknown quantities may have 

 definite values, there must be as many 

 independent equations as unknown quan- 

 tities. 



Thus, if a: -}- y = a, x = a y ; and 

 assuming y at pleasure, we obtain a value 

 of x, such that x-\- y = a. 



These equations must also be inde- 

 pendent, that is, not deducible one from 

 another. 



Let x -f- y = a, and 2a:+2# = 2a; 

 the latter equation being deducible from 

 the former, it involves no different sup- 

 position, nor requires any thing more for 

 its truth, than that x + y = a should be a 

 just equation. 



PROBLEMS WHICH PRODUCE SIMPLE 

 EQUATIONS. 



From certain quantities which are 

 known, to investigate others which have 

 a given relation to them, is the business 

 of Algebra. 



When a question is proposed to be re- 

 solved, we must first consider fully its- 

 meaning and conditions. Then substi- 

 tuting for such unknown quantities as ap- 

 pear most convenient, we must proceed 

 as if they were already determined, and 

 we wished to try whether they would an- 

 swer all the proposed conditions or not, 

 till as many independent equations arise 

 as we have assumed unknown quantities, 

 which will always be the case, if the ques- 

 tion be properly limited; and by the so- 

 lution of these equations, the quantities 

 sought will be determined. 



Prob. 1. To divide aline of 15 inches 

 into two such parts, that one may be three- 

 fourths of the other. 



Let 4 x = one part, 

 then 3 x = the other, 



7 x = 15, by the question, 



15 

 *= f 



60 4 

 4or= - = 8 -, one part, 



45 3 

 3 x=.-~- = 6 , the other, 



Prob. 2. If Jl can perform a piece of- 

 work in 8 clays, and n in 10 days, in what 

 time will they finish it together ? 



