SiNGivE Equations. 77 



spoken of as a legitimate transformation or derivation ; when the 

 operation does have an effect upon the final result, it may be 

 called a questionable derivation, meaning thereby that the 

 operation requires examination. 



If there are two equations such that any solution of the first is 

 a .solution of xhe second, and also that any solution of the second 

 is a solution of the first, the two equations are said to l)e 

 equivalent. 



3. Theorem. The trayisformatioii of an equation by the addition 

 or subtraction from both members of either a kyiown quantity or a 

 functio7i of the unknown is a legitimate derivation. 



An equation containing one unknown quantity, as it commonly 

 appears with quantities on each side of the equation, may be 

 ^generalized in thought by the expression 



A function of x= Another function of x. 

 Or, using L to represent the left-hand side of the equation, what- 

 ever it may be, and R to represent the expression on the right- 

 hand side, we can represent any equation very conveniently by 



L = R. (I) 



Now suppose that T, which may be either a known quantity or a 

 function or the unknown, be added to both members of the equa- 

 tion, making 



L-^7 = R-\-T. (2) 



Now it is plain that (2) cannot be satisfied unless L=R and that 

 it is satisfied if L — R. Hence (2) means no more nor less than 

 (i). Therefore the derivation is legitimate. 



4. Corollary. Transposition of terms from one member to 

 the other, changing the si^ns at the same time, is legitimate. Thus 

 U L=R, to pass to /.—R=o is merely subtracting R from both 

 members. 



5. Theorem. Multiplying both members of an equation by the 

 same expression is legitimate if the expression is a knoicn quantity, 

 but questionable if the expression is a function of the unknown. 



