Systems of Equations. 103 



withstanding a chance exception. In those S3\stems which may 

 be solved, special expedients are more to be sought for than gen- 

 eral methods. In fact, sharp inspection of the equations and a 

 knowledge of algebraic forms will often be the means of discover- 

 ing an ingenious solution for an apparently insoh^able system. 



The theorems of this chapter will be found useful either in just- 

 ifj'ing or in throwing doubt upon many of the common transfor- 

 mations during the ordinary solution of a system. As far as 

 possible the student should endeavor to take account of all ques- 

 tionable derivations at the time they are made and make allow- 

 ance for them in the result. In this connection the remarks at 

 the beginning of the last chapter should not be forgotten. No 

 matter how skillfull}' or ingeniously a set of values may have 

 been obtained, they must satisfy the or igiiial system, ot it is no sohc- 

 tiou. Whenever an}- derivation not allowed by the theorems is 

 used, however plausible it may seem, this ultimate test must be 

 applied. 



The following systems include examples of the cases already 

 considered, besides others requiring special treatment. The 

 method of Case II will be found to solve many symmetrical sys- 

 tems of high degree. 



• |.r5-fj'5=275. (2) 



Pvit .r =//-}-et' and y—u — iv, whence the system becomes 



( 2U^-\-20U\i''-\- IO«7i^=275. 



From the first of these equations /^=:]. Whence, substituting 



this in the second, we obtain 



\\2^ , 62=) ^ , 12s ^ 

 ^-—^4- ^i-'-h ^\i'-'=275 

 16 2 2 



which is a quadratic in terms of 7i'\ When 2t' is found x and r 



can be found from the equations .v=//-f-7t/ and y=u—7i'. 



i .v^K ==io-i^)'-|-900o 



j A---h,r=2oo. 



I x—y=2 



|-v'-y=8. 



].v3+j'^=65. 



2. 



i- 



