8o AlvGEBRA. 



be transposed to one .side of the equation, their common de- 

 nominator found and then added together. This will now give 

 but one fraction in the equation, and, when this is reduced to its 

 lowest terms, we will have an equation of the form 



N 

 which, since — is in its loivest te?^rns by supposition, will take on 



no additional solutions when multiplied through by D, according 

 to Art. 7. 



lO- Theorem. llie raisijig of both itwmbers of an equation to 

 the sa?ne power is equivalent to multiplyi7ig through by a funetion 

 of the unkfioum ayid henee is a questionable derivatio7i. 

 Take the equation 



L = R (i) 



and raise both members to the nth power, obtaining 



L"=-R". (2 



Now r I j is equivalent to 



L-R^o 

 and (2) is equivalent to 



L"-R"^o, a; 



But (\) can be derived from (^3^ by multiplying both members by 

 L"-'-VL"-^~R-\-L"-'R'-^. . .-VL'R"-^^-LR"-'-^R'-' 



whence (2) \s equivalent to the tzvo equations 



( L = R ) 



i L"-'-\-L"-'R+L"-'R-'+. . .■^L"R"-' + LR"-' + R"-' = o. \ 



II. Theorem. Dividing both members of a7i equation by the 

 same expression is legiti?nate if the expressio?i is a known quantity, 

 but questionable if it is afunctio?i of the unknown. \ 



Suppose both members of the equation to be divisible by T and 

 write the equation 



LT^RT. (1) 



Now if T'is a known quantity, then by Art. 5 this equation is 

 equivalent to 



L=R (2) 



whence division by 7' would be legitimate. But if 7^ is a func- 

 tion of the unknown quantity, then (^i j is equivalent to the two 



equations ^ L=R 



{T=o. 



