I02 Algkbra. 



therefore come under Case I. When u and w are thus found, x 

 and )' can be detennined from the equations. 



J'==// — 2t'. 



The above work shows that Case II can ahvays be solved, but 

 we do not pretend that the method used is always the most eco- 

 nomical one to employ. The insight and ingenuity of the student 

 will often suggest special expedients for particular examples 

 which are preferal^le to a general method. 



17. Examples. Solve the following systems: 

 ( A--fxi'-fj'=65 

 i -rr =50. 



"• i.v" + :i-+r+j-=26. 

 -'■ I ^^' y 45 



^- i -rr=6. 



A common expedient for readily sioh^ing such a system is to 

 first transform it into the system 



I .^"^-2.^:r-fJ'-'=3ol- 

 from which the values of x—y and x-\-y can be found and conse- 

 quently the values of .r and y. 



x+y 9 

 .r+j'_i8 



18. Miscellaneous Systems. We have enumerated all the 

 classes of systems involving equations of a degree higher than 

 the first which can invariably be solved without a knowledge of 

 the solution of cubic and higher equations. The solvable cases 

 embrace but a small fraction of the S3'Stems which ma}^ arise. Of 

 the numbers remaining a still smaller proportion can be solved by 

 special expedients. The great mass of sy.stems involving quad- 

 ratic or higher equations are thus irreducible by straightforward 

 methods of solution, /. e., as a rule, such systems are insolvable, not- 



