Equation Properties for Expressions (EPE)
Early on in one's high school algebra courses one learns
several properties of equality  namely
 Addition/Subtraction Property of Equality:
One may add/subtract the same quantity to/from both
sides of a given equation, and the solutions of the
resulting equation will be the same as those of the
original (given) one.
 Multiplication/Division Property of Equality:
One may multiply/divide both sides of a given equation
by the same nonzero quantity, and the solutions
of the resulting equation will be the same as those of
the original (given) one.
Notice that both of these properties presuppose that we
start with an equation, usually one we are supposed
to solve (say, for ). These properties are helpful
in achieving that goal, as in:
Solve :
Add 1 to both sides: 

Divide both sides by 3: 

or,
solve
:
Multiply both sides by the neverzero quantity : 

Subtract 5 from both sides: 

Divide both sides by 4: 
. 
In contrast, these are not, generally speaking, properties
one uses when trying to simplify an expression. (There
are exceptions to this, such as in the simplifying of
and
using integration by parts, but these are relatively rare.)
Students asked to find the derivative of
may find it easier to work with the function
but they shouldn't be under any illusions that and
are the same functions, nor that they have
derivatives that are equal. If one is simplifying
an expression like
it may be tempting to multiply by , which gives
but, of course, multiplying by changed the expression.
One must both multiply and divide by (equivalent
to saying that we're multiplying by ) if the expression
is going to remain the same (but hopefully simplified):
Another example is in simplifying the difference quotient
.
which cannot be further simplified.
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Thomas L. Scofield
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