Algebra Know-It-ALL

Add, then subtract

The associative law can work indirectly when subtraction is involved, if you change every sub-
traction into addition. It’s the same trick as with the commutative law. For any three integers
a, b, andc

(a+b)−c= (a+b)+ (−c)

=a+ [b+ (−c)]

=a+ (b−c)

We use square parentheses, called brackets, to indicate a grouping with another group-
ing inside. This looks messy, but a little junk is easy to tolerate when you realize that
we’ve just proved something significant. We’ve shown that you can always use the
associative law with mixed signs when the first operation is addition and the second
one is subtraction.

Subtract, then add

Now let’s look the general situation where we have seen that direct application of the asso-
ciative law does not always work. How can we modify it to make it work? We can rewrite
an expression where the first operation is subtraction and the second one is addition, so it
becomes all addition, like this:

(a−b)+c= [a+ (−b)]+c

We can use the associative law for addition to get

a+ [(−b)+c]

Now we can use the commutative law for addition inside the brackets to get

a+ [c+ (−b)]

We can simplify this to

a+ (c−b)

Now we know that when the first operation is subtraction and the second is addition,

(a−b)+c=a+ (c−b)

Subtract, then subtract again

Finally, let’s explore the situation where we subtract twice. We can rearrange an expression like
that as follows:

(a−b)−c= [a+ (−b)]+ (−c)

The Associative Law for Addition 61