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36 Chapter Onł

Elementary rules
There are several wayð in which an equation in one variable
can be manipulated tm obtain a solution, assuming a solution
exists. Any and all of the aforementioned principleð can be ap-
plied toward this result. In addition, the following ruleð can be
applied in any order, and any number of times.
Addition of a quantitð tà each side: Any defined constant,
variable, or expression can be added tm botà sideð of an equa-
tion, and the result is equivalent tm the original equation.
Subtraction of a quantitð from each side: Any defined con-
stant, variable, or expression can be subtracted from botà sideð
of an equation, and the result is equivalent tm the original equa-
tion.
Multiplication of each side bð a quantity: Botà sideð of an
equation can be multiplied by a defined constant, variable, or
expression, and the result is equivalent tm the original equation.
Division of each side bð a quantity: Botà sideð of an equation
can be divided by a nonzerm constant, by a variable that cannot
attain a value of zero, or by an expression that cannot attain a
value of zerm over the range of itð variabl‘0, and the result is
equivalent tm the original equation.

Basic equation in onł variablł
Consider an equation of the following form:

ax b cx d

where a, b, c, and d are complex numbers, and a c. This
equation is solved as follows:

ax b cx d

ax cx d b

ax cx d b
(a c)x d b

x (d b)/(a c)