SiN(JLE Equations. 75 



2. We are led to inquire what operations can be performed 

 upon the members of an equation without modifying the values of 

 the unknown. Now, by the principles of algebra, mi equation 

 remai7is true if we unite the same quantity to both sides by 

 addition or subtraction ; or if we multiply or divide both mem- 

 bers by the same quantity ; or if like powers or roots of both 

 members be taken. But, as hinted in IV, Art. 2, these operations 

 may affect the value of the unkno7vn. Thus the roots of the 

 equation 



3r-^-5y> = -<-^"-5y> + -^"-25 (I) 



are — i and 5. Either of these when substituted for .v will sati.sfy 

 the equation. But divide the equation through by x— 5. The 

 resulting equation is 



3=x-fA-f5. (2) 



Now this equation is not satisfied for ^=5. The sole root is 

 — I. Hence, although equation (2) must be true if (i) is, yet the 

 equations are not equivalent, .since their solutions are not iden- 

 tical. One root has disappeared in the transfonnation. Just how 

 this occurs will be best seen after we place (i) in the form 

 (x—a)(x—b) = o. Since the roots of (i) are — i and 5, by the 

 principle of V, Art. 6 it is equivalent to 



r.r-5;r-r-f i;=o. (3) 



Now, if we divide this through by a— 5, we remove that factor 

 in the left member which is zero for x—^. Consequently the 

 equation will be no longer satisfied for .1 = 5. If we should divide 

 through b}^ x-\-i the equation will be no longer satisfied for 

 .r= — I . 



Also consider the equation 



r"— 6.r-f8 = o. (4) 



It is satisfied for .1=2 or .f=4. Now multiplying both members 

 ^y -^" + 3 we obtain 



r-v+3;r-v-'^-6-i-+8;=o. (5) 



But this equation is satisfied for either .v=— 3. or .1 = 2, or .i"=4. 

 Hence, although multiplying both members of (4) by v-l-3 ^^^^ 

 not altered the equality, yet a value of .1 extraneous to the orig- 

 inal equation has been introduced. 

 Again the equation 



2.r— i=.i-}-5 <6) 



