Simplifying Algebraic Expressions
It will help you to learn algebra if you approach it as if you are learning a
new language. Actually, algebra is a language used to create mathematical
models and to solve problems that are beyond the realm of arithmetic. In the
language of algebra we use symbols, rather than words, to make a statement
or describe a situation.
Example: You have a piggy bank that contains $150. You are faithful and you deposit
$0.75 every day into the bank. You also have an addiction to soda, so you have to take
out $1 every day and give it to the vending machine in your dorm. How much money is
in your bank on a given day?
An algebraic expression is a collection of one or more algebraic terms that
are formed and combined using variables, constants, and binomial
operations (addition, subtraction, multiplication, division, etc.)
Examples:
* ‚ Ð  #Ñ  & ‚ Ð  #Ñ  *  "!
BC  &C  B  "!
A term is any part of an algebraic expression that is separated from all other
parts of the expression by addition or subtraction.
In an algebraic expression, a variable is a number that can take on more
than one value. Variables are often represented using letters.
A coefficient is the non-variable part of a term in a variable expression. If
only variables appear when a term is written then the coefficient for that
term is 1 or  ".
Constants are the terms in an algebraic expression that contain only
numbers. That is, they're the terms without variables. We call them constants
because their value never changes.
Review: Working with Expressions , Order of Operations
T IQ HEW
1) Parentheses Grouping Symbols ÐÑ Ö× ÒÓ
2) Exponents
3) Multiplication/Division
4) Addition/Subtraction
Simplify the following:
 *  * ƒ Ð  $Ñ † $
"&  %# ƒ Ð&  $Ñ † $
 %#
"
#
ƒ
"
&
(  % )#

"
'
ƒ
"
)
$ † Ò$  Ð  #Ñ  # ƒ #Ó#
Simplifying expressions with variable terms:
We work with variable expressions in the same way we work with nonvariable expressions, using the same rules, but we cannot simplify variable
expressions as completely as we do non-variable expressions. In algebra we
learn properties that allow us to change the way we write an expression
without changing the quantities associated with those expressions. When we
do this, we say that we have created an equivalent expression.
Some important properties of real numbers that we use to
determine equivalent forms of expressions:
For all real numbers +ß ,ß and +, œ,+
ÐCommutative Property of AdditionÑ
+ † , œ , † + ÐCommutative Property of MultiplicationÑ
Ð+  ,Ñ  - œ +  Ð,  -Ñ
Ð+ † ,Ñ † - œ + † Ð, † -Ñ
+ † Ð,  -Ñ œ + † ,  + † -
ÐAssociative Property of AdditionÑ
ÐAssociative Property of MultiplicationÑ
‡‡
(The Distributive Property)‡‡
Like terms (or similar terms) are terms that contain the same
variables raised to the same powers.
Example: Identify the like terms in the expression
$  B# C  #BC  %B  "!B# C  "#  #B  $BC#  "!
To make expressions easier to work with, we will want to simplify
an expression. One way we can simplify an expression is to use
the commutative property of addition along with the distributive
property to combine like terms.
Example: Simplify the expression below by combining like terms.
$  B# C  #BC  %B  "!B# C  "#  #B  $BC#  "!
Example: Simplify the expression below by combining like terms.
$B#  #B  &  $B  C  "
To evaluate an expression at specified values of the variables, you
substitute the specified values for each variable and carefully
follow the order of operations. It is usually easier to evaluate an
expression if you first simplify the expression.
Evaluate the expression below using B œ  # and C œ  $
%ÐB#  #CÑ  Ð&B#  $Ñ  B  C  #
Big Idea!!
In arithmetic we learn that the same number can be obtained through many
different binomial operations and combinations of those operations.
Likewise, in algebra we learn that a quantity that is represented by an
expression can be represented in many forms. The main goal for all of unit 1
of this course is to learn how to make changes to an expression to create an
expression that is simpler or more useful without changing the quantity
represented by that expression.
Examples: