r=-2 J 



90 Algebra. 



and by no other method of elimination can we get anything but 

 an absurdity from the given system. Equations of this kind are 

 said to be incompatible because one equation affirms what another 

 denies. We will .see this to be so in the above system if, by 

 proper transformations in equation (\), the system be written 



x—y=i (4) \ 



These equations are necessarily contradictory and can have no 

 solution. 



Another example of an incompatible system is 



A— J/ =4 ,- 



x-^y—2: =2 ) 

 From the second of these equations it is seen that 



.r=4+j'. 

 Substituting this value of x in the first and third of the equa- 

 tions in order to eliminate x, we obtain the system 



2y — z- 

 and, since these are incompatible, we can go no further. 



There is still another case in which a system may have no 

 solution. Consider the equations 



From the first equation we find 



^- 8 9 ^, 



-^— 2 TJ- 



Substituting this value of x in the second equation we obtain 



which reduces to 0=0, 



and we get no solution. Equations of this kind are said to be 



dependent because the equations really make the same statement 



about the unknown quantities. This will be seen when, by 



proper transformations in the equations, the above system is 



written. 



It is now seen that the equtions of the system do not state 

 independent truths, and consequently the system has no more 

 meaning than a single equation containing two unknown 

 quantities. 



