## 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 pre-suppose 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 never-zero 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 2003-09-04