3.2 - Addition and Subtraction of Expressions
Before studying this topic you may wish to review the
section on addition and subtraction of numbers.
Suppose that A and B are any two expressions.
Adding B to A means setting up the sum as
(A) + (B),
and then simplifying as much as possible.
Subtracting B from A means setting up the difference as
(A) − (B),
and then simplifying as much as possible.
The reason that we put brackets around A and B is
that they are expressions, not just numbers, and
the addition or subtraction is supposed to apply to whatever A and B may contain.
(We want the addition or subtraction to be at the end in the
order of operations.)
The simplifications that must be done are:
Example: If A = 4 x y + 7 and B = 8 y + 9 x y
then setting up the sum A + B gives:
(4 x y + 7) + (8 y + 9 x y)
and setting up the difference A − B gives:
(4 x y + 7) − (8 y + 9 x y)
Removing the brackets
Removing the brackets from (A) + (B) is easy:
(A) + (B) always simplifies directly to
A + B. The brackets can just be dropped.
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Here’s why: The acronym BEDMAS (Brackets, Exponents,
Division and Multiplication, Addition and
Subtraction) helps us remember the
order of operations within an expression.
Brackets are at the top of the priority list and addition and subtraction are
at the bottom.
In the expression A + B the addition is already
at the bottom of the list so we don’t need brackets around A
and B to move them any higher in the priority list.
The same argument applies to A − B
except in two cases. This is because in these two cases the
subtraction
is actually a multiplication by a negative,
which is not at the bottom of the priority list! Here is the rule:
(A) − (B) simplifies to
A − B except in these two cases:
Exception # 1: If B is a negative
monomial
then the subtraction of a negative must be
replaced by an addition.
Exception # 2: If B is a
multinomial then the − sign
of the subtraction must be distributed
over each term of the multinomial.
The result, as explained below,
is that each term of B has its sign reversed when the
brackets are dropped.
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Examples:
- (x y) + (x z 2 ) simplifies to
x y + x z 2
- (x y) + (w + x z 2 ) simplifies to
x y + w + x z 2
- (x y) − (x z 2 ) simplifies to
x y − x z 2
- (x y) − (− x z 2 ) simplifies to
x y + x z 2
(this is exception # 1; subtracting a negative is adding)
- (x y) − (a + b) simplifies to
x y − a − b
(this is exception 2; you must distribute the − sign)
- (x y) − (− x z 2 + 5) simplifies to
x y + x z 2 − 5
(this is exception # 2; you must distribute the − sign)
Explanation of Exception 2: Why does distributing a − sign cause
each term in the brackets to have its sign reversed? For example,
why does a − (b − c) simplify to
a − b + c ?
Here is the sequence of steps with an explanation of each step:
Combining like terms
Recall that like terms are terms
in an expression that differ only in their coefficients;
they have identical variable factors.
Like terms can be added or subtracted by adding or subtracting
their coefficients (essentially by factoring with the
distributive law).
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Here are some examples. The first two examples are done in detail.
Assume that x and y are variables.
In the last example assume that a and b are constants.
The like terms are shown in red.
Example: Simplify the expression
Example: Simplify the expression
Example: Simplify the expression
Example: Simplify the expression
Note: In this last example the two terms in red were considered to be
like terms because we were told that a and b are constants.
If instead a and b were variables then the two terms in red would
not be like terms and would NOT be combined.
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