Single Equations. 79 



8. As an example under the above theorem take the equation 



II — 2A- 3^1"— I 



These fractions are in their lowest tenns and their denominators 

 are prime to each other. The least common multiple of the de- 

 nominators is (ii—2.r)(T,.v~i ). Multiplying through by this 

 we obtain 



r.v— or?— -tj+rii— 2A-;r4-i— 5y>=2('ii— 2A-;(^3x— ij. (2j 



Now we can see that although ( i ) has been multiplied through 

 both by (ii — 2x) and (2>^—i), yet neither of the.se has been in- 

 troduced as a factor through the equation. Hence there is no ad- 

 ditional solution introduced. The roots oi (2) will in fact be 

 found to be 4 or —10, w^hich values also satisfy (\). 



But an extraneous solution may be introduced if the denomin- 

 ators are not prime to each other, or if .some of the fractions are 

 not in their lowest terms. Thus 



•^"-3 ^^"+3 -^"-3 

 has two denominators alike, and consequently not prime to each 

 other. Multipl3dng through by the common denominator -t^— 9, 

 we obtain 



3.ir.v+3;=6rx-3;+9r-r-h3; a; 



or, reducing, .v"— 2.1=4 (s) 



whose roots are 3 and — i. Now if we put the original equation 

 (^3J in the form 



-r— 3 .rH-3 



that is 3= ^ (6) 



•V -|- 3 



it is seen that it is .satisfied only for .1 = — i. Hence a .solution 



was introduced in clearing (t^) of fractions. It is easy to see that 



(t,) is really equivalent to (6) and hence that in clearing (2,) of 



fractions by multiplying by .r-— 9 we multiplied by .v— 3 when it 



was not necessary ; this is where the solution .1=3 was introduced. 



9. Theorem. Jiverv equation ran be inte^ralized U\e:itimately. 

 For if the several fractions in the equation are not in their low- 

 est terms thev can be .so reduced. Then these fractions can all 



