Beginning Algebra
Tutorial 4: Introduction to Variable Expressions and Equations
Learning Objectives
After completing this tutorial, you should be able to:
 Evaluate an exponential expression.
 Simplify an expression using the order of operations.
 Evaluate an expression.
 Know when a number is solution to an equation or not.
 Translate an english expression into a math expression.
 Translate an english statement in to a math equation.

Introduction
This tutorial will go over some key definitions and
phrases used when
specifically working with algebraic expressions as well as evaluating
them.
We will also touch on the order of operations. It is very
IMPORTANT
that you understand some of the math lingo that is used in an algebra
class,
otherwise it may all seem Greek to you. Knowing
the terms and concepts on this page will definitely help you build an
understanding
of what a variable is and get you more comfortable working with
them. Variables are a HUGE part of algebra, so it is very important for you
to
feel at ease around them in order to be successful in algebra. So
let's get going and help you get on the road to being variable savvy.

Tutorial
An exponent tells you how many times that you write a
base in a PRODUCT.
In other words, exponents are another way to write MULTIPLICATION.
Let’s illustrate this concept by rewriting the
product (4)(4)(4)
using exponential notation:
In this example, 4 represents the base and 3 is the
exponent.
Since 4 was written three times in a product, then our exponent is
3.
We always write our exponent as a smaller script found at the top right
corner of the base.
You can apply this idea in the other direction.
Let’s say you
have it written in exponential notation and you need to evaluate it. The
exponent will tell you how many times you write the base out in a
product. For example if you had 7 as your base and 2 as your exponent and you
wanted
to evaluate out you could write it out like this:

Example
1: Evaluate 
In this problem, what is the base?
If you said 5, you are correct!
What is the exponent?
If you said 4, you are right!
Let’s rewrite it as multiplication and see what we
get for an answer: 

*Rewrite the base 5, four times in
a product
*Multiply

Example
2: Evaluate 
In this problem, what is the base?
If you said 7, you are correct!
What is the exponent?
If you said 1, you are right!
Let’s rewrite it as multiplication and see what we
get for an answer: 

*Rewrite the base 7, one time
in a product 
Example
3: Evaluate 
In this problem, what is the base?
If you said 1/3, you are correct!
What is the exponent?
If you said 2, you are right!
Let’s rewrite it as multiplication and see what we
get for an answer: 

*Rewrite the base 1/3, two
times in a product
*Multiply

Note that when you have a 2 as an exponent, which
is also known
as squaring the base. In this problem we could say that we are
looking
for 1/3 squared. 
Order of Operations
Please Parenthesis or grouping symbols
Excuse Exponents (and radicals)
My Dear Multiplication/Division
left to right
Aunt Sally Addition/Subtraction
left to right 
When you do have more than one mathematical operation,
you need to
use the order of operations as listed above. You may have already heard
of the saying "Please Excuse My Dear Aunt Sally". It is just a
way
to help you remember the order you need to go in when applying the
order
of operations. 
Example
4: Simplify . 
Example
5: Simplify 

*Inside ( )
*Exponent
*Multiply
*Add 
Example
6: Simplify . 
Note that the absolute value symbol   is
a fancy grouping
symbol. In terms of the order of operations, it would be
including
on the first line with parenthesis. So in this problem, the first thing we need to do is
work the inside
of the absolute value. And then go from there. 

*Inside  
*Exponent
*Add in num. and subtract in
den.

A variable is a letter that represents a
number.
Don't let the fact that it is a letter throw you. Since
it represents
a number, you treat it just like you do a number when you do various
mathematical
operations involving variables.
x is a very common
variable that is used
in algebra, but you can use any letter (a, b, c, d,
....) to be a variable. 
An algebraic expression is a number, variable
or combination
of the two connected by some mathematical operation like addition,
subtraction,
multiplication, division, exponents, and/or roots.
2x + y, a/5,
and 10  r are all examples of algebraic
expressions. 
You evaluate an expression by replacing the
variable with the
given number and performing the indicated operation. 
When you are asked to find the value of an
expression, that
means you are looking for the result that you get when you evaluate the
expression. 
So keep in mind that vary means to change  a
variable allows an
expression to take on different values, depending on the situation.
For example, the area of a rectangle is length times
width. Well,
not every rectangle is going to have the same length and width, so we
can
use an algebraic expression with variables to represent the area and
then
plug in the appropriate numbers to evaluate it. So if we let the
length be the variable l and width be w,
we can use the expression lw. If a
given
rectangle has a length of 4 and width of 3, we would evaluate the
expression
by replacing l with 4 and w with 3 and multiplying to get a value of 4 times 3 or 12.
Let’s step through some examples that help illustrate
these ideas. 
Example
7: Evaluate the expression
when x = 4, y =
6, z = 8. 
Plugging in the corresponding value for each
variable and then evaluating
the expression we get: 

*Plug
in 4 for x, 6 for y, and 8 for z
*Exponent
*Multiply
*Add
*Subtract 
Example
8: Evaluate the expression
when x = 3, y =
5, and z = 7. 
Plugging in the corresponding value for each
variable and then evaluating
the expression we get: 

*Plug in 3 for x,
5 for y, and 7 for z
*Exponent
*Multiply
*Add

Equation
Two expressions set equal to each other.

Solution
A value, such that, when you replace the variable with
it,
it makes
the equation true.
(the left side comes out equal to the right side)

Solution Set
Set of all solutions.

Example
9: Is 2 a solution of ? 
Replacing x with
2 we get: 

*Plug in 2 for x
*Evaluate both sides

Is 2 a solution?
Since we got a TRUE statement (7 does in fact
equal 7), then 2
is a solution to this equation. 
Example
10: Is 5 a solution of ? 
Replacing x with
5 we get: 

*Plug in 5 for x
*Evaluate both sides 
Is 5 a solution?
Since we got a FALSE statement (16 does not
equal 14), then 5
is not a solution. 
Translating an
English Phrase Into an
Algebraic Expression

Sometimes, you find yourself having to write out your
own algebraic
expression based on the wording of a problem.
In that situation, you want to
 read the problem carefully,
 pick out key words and phrases and determine their equivalent
mathematical
meaning,
 replace any unknowns with a variable, and
 put it all together in an algebraic expression.
The following are some key words
and phrases and
their translations: 
Addition: sum, plus, add
to, more than,
increased by, total 
Subtraction: difference
of, minus,
subtracted from, less than, decreased by, less 
Multiplication: product,
times, multiply,
twice, of 
Division: quotient
divide, into,
ratio 
Example
11: Write the phrase as an algebraic
expression.
The sum of a number and 10. 
In this example, we are not evaluating an expression,
so we will not
be coming up with a value. However, we are wanting to rewrite it
as an algebraic expression.
It looks like the only reference to a mathematical
operation is the
word sum. So, what operation will we have in this
expression?
If you said addition, you are correct!!!
The phrase 'a number' indicates that it is an unknown
number.
There was no specific value given to it. So we will replace
the phrase 'a number' with the variable x.
We want to let our variable represent any number that is unknown
Putting everything together, we can translate the
given english phrase
with the following algebraic expression: 
The sum of a number and 10

*'sum' = +
*'a number' = variable x

Example
12: Write the phrase as an algebraic
expression.
The product of 5 and a number. 
Again, we are wanting to rewrite this as an algebraic
expression, not
evaluate it.
This time, the phrase that correlates with our
operation is 'product'  so what operation will we be doing this time? If you said multiplication,
you are right on.
Again, we have the phrase 'a number', which again is
going to be replaced
with a variable, since we do not know what the number is.
Let’s see what we get for this answer: 
The product of 5 and a number

*'product' = multiplication
*'a number' = variable x

Translating a Sentence into an
Equation

Since an equation is two expressions set equal to each
other, we will
be using the same mathematical translations we did above. The
difference
is we will have an equal sign between the two expressions. The following are some key
words and phrases
that translate into an equal sign (=): 
Equal Sign (=) : equals,
gives, is,
yields, amounts to, is the same as 
Example
13: Write the sentence as an
equation.
Let x represent the unknown
number.
The quotient of 3 and a number is ½. 
Do you remember what quotient translates
into? If
you said division, you are doing great.
'Is' will be replaced by the symbol =.
Let’s put together everything going left to right: 
The quotient of 3 and a number is ½ 
Example
14: Write the sentence as an
equation.
Let x represent the unknown number.
7 less than 3 times a number is the same as 0. 
Do you remember what less than translates
into?
If you said subtraction, you are doing great.
Do you remember what times translates
into? If you
said multiplication, you are correct.
'Is the same as' will be replaced by the symbol =.
Let’s put together everything going left to right: 
7 less than 3 times a number is the same as 0.

Practice Problems
These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1b: Evaluate.
Practice
Problems 2a  2b: Simplify each expression.
Practice
Problem 3a: Evaluate the expression if x = 1, y =
2, and z = 3.
Practice
Problems 4a  4b: Decide whether the given number is a
solution of the
given equation.
Practice
Problems 5a  5b: Write each phrase as an algebraic
expression.
Let x represent the unknown number.
Practice
Problems 6a  6b: Write each sentence as an
equation. Let x represent
the unknown number.
Need Extra Help on these Topics?
Last revised on July 42, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.




