Systems of Equations. 91 



It will also be found that the system 



x+2>(^^2)==2(i-y)) 

 is a dependent one, the dependence being between the first and 

 third equations. 



We may then enumerate three conditions which must be ful- 

 filled by a system of equations in order that a solution may exist : 



There must be just as many equations as there are unknown 

 quantities. 



The equations must be compatible. 



The equations must be independent. 



3. Of course if any equation of a system be operated upon in 

 any manner during the solution, care rnust be taken that the 

 transfonuation be with a due regard to the theorems in VI, Arts. 

 3 — 12. Obviously, no operation which it is questionable to per- 

 form on an equation standing alone can be legitimately performed 

 upon one belonging to a system. But in addition to the reduc- 

 tions which single equations may undergo, equations of a system 

 permit of certain transformations peculiar to themselves, and it 

 remains to investigate the possible eifect of these on the solution 

 of the system. The following theorems are designed to point out 

 the effect on the result of the ordinary steps in the process of 

 elimination. 



4. Theorem. If from the system of equations 



'lJ„=R„ J 



7i'e derive the system, 



L.S-fLT=R,S-hR,T 1 ^^^ 



u=r1 .J 



where all but the second equation re?nai?i utichanged, the derivation 

 is legitimate if T is a kno2vn quantity, 7iot zero, but questionable if 

 T is a function of the unknow?i quantities, it being indifferent 

 2vhether S is a knoivn qua?itity or a function of the unknown ones. 



