470 



M VTMKM \TI' <. ALGEBRA. 



[SIMPLE EQUATIONS. 





At already remarked at page 467, we may. if wo please, 

 always ekonye Uttiiyni of numerator aim danoaunftto 

 of a fraction, so Uiat it would be equally correct to writo 



1 _ ( + !)< 

 the preceding exprettion for * thai : * - (a l)a 



= f A Vx 



Truopming, 

 Squaring. 



/r4-=r. Tnntpaing, </ + * = < 



.4-* = (< *) f 2 < e *> v ' JP 

 ing. 2(c i) </x = (e b)* a. 



4(c 4)'*={(e )* " 



This rxanifle is in appearance difficult, but in reality it 

 i is easy. If you look at the operations actually performed, 

 you will MO that they are very trilling ; the work is for 

 the most part merely indicated, not executed ; and such, 

 I in general, is the case when wo have to deal exclusively 

 with liifral, and not with numerical quantities. The 

 work of an algebraic problem is usually tho greater, as 

 the merely numerical operations are greater ; because 

 these most be actually performed : it is the arithmetic, 

 not the algebra, that occasions the labour. An example 

 purely algebraical, like that above, requires only a little 

 address in the deduction of one step from another ; a 

 little caution in the management of signs, and some care 

 in indicating processes, which arithmetic must per/arm 

 when the symbols are interpreted by numbers. It is 

 not easy, however, to embody in a rule all the expedients 

 and artifices which may with advantage be resorted to in 

 the solution of a simple equation ; these most bo acquired 

 by observation and experience, by tho exercise of your 

 own ingenuity, and the suggestions of common sense. 

 A little thought and reflection will often do more than 

 the most elaborate rule. The following example will 

 serve to illustrate these remarks : 



-*-. 

 * x 6 



In this example, the removal 



of the radicals is evidently the first thing wo should try 

 to effect ; and it is easy to see that if we were to attempt 

 this by implicitly following the precepts in the rule, we 

 should soon have to deal with complicated expressions. 

 Let us try to evade these by a little ingenuity. You see 

 that if wo were at liberty to subtract the denominator of 

 the first member from the numerator, the upper radif.il 

 would bo removed at once : this suggests the subtracting 

 1 from each side ; wo therefore make the following the 

 first step, namely 



1 



-- 1 ; that is, 



c 





be 



And as it is in general loss inconvenient to have a radical 

 in tli>' numerator than in the denominator of a fraction, 

 we shall take the reciprocals of these fractions ; that is, 

 ahall simply turn them upside down, writing tho equation 

 thus __ 



** _?_. _ ( 2e +l) 



o c 

 . squaring each do, a' 



_( 6 . C V 

 V J 



. 



- - 



4oo 



TV stop marked [A] may be reached a little differently 

 by help of a property of two emial fraction*, which you 

 will do well to remember, and apply in examples Uke 



that above. Let f' * bo two equal fractions (t 



stand f<>r tho first member of the given equation above); 

 .- .titntettn-j 1 from each, we have 



Also, adding 1 to each, wo have 



q e 



Xow divide these results by the former, and we get 



By applying this property to the given equation, we get 

 at once 



-l5, and thence 

 * 6 c 



EXAMPLES FOR EXERCISE. 



NOTE. Example 8, 0, 10, 11, anil 10, hive boon wlcctoil with a T!PW to 

 tho application of tho principle Jut explained. Remember, In Kx. li, 

 that sum. X diff. a dijf. o} iqu 



10 y-lT* 1 * " 2 



+ b J* + 

 12. ,/( 24)= N /x 2 



Simple Equations with Two Unl-noim Quantities. 



There are three methods of solving a pair of simple 

 equations containing two unknown quantities. Wo shall 

 llustrato them by an example : 

 L Find tho values of x and y in the equations, 

 < 2^+51^=23, 

 \3x 2y= 0. 



From the first equa., by trans., 2x=23 By /. 

 23-Sy 



2 

 From tho second equa., by trans, 3x = 6 + 2y .'. 



we 



3 



Equating these two different expressions for x, 



lave ' = . . (clearing) C9 lay n 12 + 4y 



'. trans. T9y= 57 . '. */=3. Substituting this value 

 or y in either of tho expressions for x, tho second, for 



nstance, we have x = -^T = 4 . . x 4, y = 3. This 



o 



method, you see, consists in equating tho expressions for 

 ho tame unknown quantity, as deduced from tho two 

 quations. 



2. Having found an expression for x from one of the 

 quations, as, for instance, = i ! ' substitute it for 



i tho other equation, and wo get + 2 ^-fSy=23; . . 



3 

 clearing and removing brackets, 



12 + 4i/+15y= 09.-. 19./=67 .'. y - 3, 



