76 Al^GKBRA. 



is satisfied only by the value .1 = 6. Now square both sides of the 

 equation, obtaining 



4-1-"— 4.rH- 1 =ji-=-|- io.r+ 25, (7) 



which is satisfied for either x=6 or x——\. Here, obviously, an 

 extraneous solution has been introduced by the operation of 

 vsquaring both members. 



In a like manner notice the effect of taking a root of both 

 members of an equation. Thus suppose 



^■^=(x-6)\ (8) 



This is satisfied for either x=2 or —6. Take the square root of 

 each member and we obtain 



2X=.V— 6, (9) 



which is satisfied only by .r=— 6. We have lost one of the solu- 

 tions of the equation during this transformation, Equation (?>) 

 is really not equivalent to (()), but to the two equations 



\2X= + (x-6)\ . 



{2X=-(x-6) \ ^^^^ 



We have given examples enough to show that certain opera- 

 tions upon an equation may modify the solution. Thus we see 

 that during a series of transformations which sometimes an equa- 

 tion must undergo before we can reach the values of the unknown 

 it is possible that the solutions that satisfy the original equation 

 may all be lost and that any number of new ones may be intro- 

 duced, so that the final results may have no relation at all to the 

 problem in hand. It is now proposed to formulate certain propo- 

 sitions which will enable us to tell the exact place in the process 

 of any solution where roots may be lost or new ones may enter. 

 We will then be able to perfonn the different operations on the 

 members of an equation if we will note at the time their effect on 

 the solution and finally make allowance for it in the result. This 

 fact must be emphasized : the test for a?ty solution of an equation 

 is that it satisfy the original equation. '' No matter how elaborate 

 or ingenious the process by which the solution has been obtained, 

 if it do not stand this test it is no solution ; and, on the other 

 hand, no matter how simply obtained, provided it do stand this 

 test, it is a solution." — Chrystal. 



When one equation is derived from another by an operation 

 which has no effect one way or another on the solution, it ma}^ be 



