96 Algebra. 



which is satisfied by -v=5, r=i. From this we may obtain the 

 system 



X—2J'=3 } 

 ^--4^=21 j 



From the first equation of the system 



-^=3 + 27. 

 Substituting this value for x in the second equation, it becomes 



9 + 121/ + 4y—4.y= 2 1 ; 

 whence y=i. 



Therefore, from the first equation of the system, 



In this case we see that no solution has been introduced. In 

 fact, the introduced systems become 



x—2y=T, ) 



x-2y=-o) 



jt-— 2J/=3\ 



7 = oj 



which are incompatible. 



9, Theorem. If fro in the system 



i..l;=r;rI ^"^ 



zve dei'ive the system 



the derivatio7i is questionaale if both L^ a^id R^ i7ivolve 7ink?iown 

 quantities, but legiti?nate if either is a known qiiantity 7iot zero. 



First, suppose that L, and R^ both involve unknown quantities. 



Then, by Art. 7, if we pass from (b) to (a) we gain solutions. 

 Hence to pass from (a) to (b) is to lose those solutions. 



Seco7id, suppose that either L^ or R^ is a known quantity. 



Then, by Art. 7, if we pass from (b) to (a) no solutions are 

 gained. Hence none are lost if we pass from (a) to (b). 



10. Examples. According to the above theorem it is legiti- 

 mate to divide one equation by another, member by member, if 

 one member is a known quantity not zero. 



