Systems of Equations. 93 



being satisfied. Therefore (d) is not equivalent to (c) but to the 

 two systems 



L-R=o 



L-R.=o 



5. Examples. The derivation discussed in the above theorem 

 is the one so frequently used in elimination. Thus take the 

 system 



2.v+j'=i7 ro I 



5Jtr— ioy= 5 (2) S 



Multiply (\) through by 5 and (2.) through by 2 and obtain a 

 new equation b}^ subtracting the former from the latter and the 

 system becomes 



2-r+j=i7 (t,)} 



^sy^is (4) i 



We have eliminated x from the second equation and consequently 

 J' is readily found to equal 3. 



From (7,), X is then found to equal 7. 



The theorem shows it is also legitimate to transfonn 



-v+3=J 1 

 j(f-\-6^=^xy ) 



into ^ ; 



6— 3-r=o j 



by multiplying the first equation through by x and subtracting 



the resulting equation from the second. 



An example of the use of the following theorem will be found 



in V, Art. 12 (c). 



6. Theorem. It is legitimate to derive from the system 



the system 



SLr+Tlv,=SR,--fTR. \ < ^^ 



//" 7' is a know?i quantity, 7iot zero. 



