Single Equations. Si 



Division by T would give us but one of these, and consequently 

 solutions would be lost. Hence the division -by a function of the 

 unknown is a questionable derivation. 



12. Theorem. The extraction of the same root of both members 

 of an equation is equivalent to dividing by a function of the unknown 

 and hence is a questionable derivation. 



For we can pass from 



to L^R (2) 



by dividing both members of (ij by 



Hence, by Art. 11, root extraction is a questionable derivation. 



13. EXAMPI.ES OF THE InTEGRAUZING OF EQUATIONS. In the 



following equations the student should note the precise effect of all 

 questionable operations at the time they are performed. 



1. Solve i(^-5)(^--3,)('^'~2j=(x~s)(x-^:^)(x-\-2). 



^ , X-h 2 X~ 2 5 



2. Solve : =-. 



X— 2 X -{- 2 6 



3. Solve -— i -4- l-=i. 



^ 2(^.^—1; x'—i 4 



12 Jtr+i 



4. Solve ~-\ — = . 



X XX 



X"" X I 



5. Solve — ,= i 



X x-^i 13 



6. Solve f- = ^. 



X — I X 6 



,, , x^-^2X—i'^ -^*^- S 



7. Solve — ^ ^^= \ ^. 



x'—x—6 x-\- 2 



14. EXAMPI^ES OF THE RATIONAIvIZATION OF EQUATIONS. The 



most expeditious method for rationalizing any given equation de- 

 pends upon the peculiar make up of the equation, and can only 

 be determined by the student after a little experience with this 

 class of equations. 

 A— 10 



