Variables (Level 1) Algebra and mathematics in general, is a symbolic
language. Your ability to read these symbols and your commitment to learn a new mathematical
language will determine your success in algebra. In addition, remember that algebra builds
upon itself so if you don’t feel confident on a specific concept do not proceed until
you have mastered it. You need to learn how to walk before you can run in other words
don’t rush into doing things until you know how to do it.
Alright let's learn about variables. Let’s say that you recently got hired as a math
tutor at your local tutoring center. Your current hourly rate is $12.00 an hour the amount
of money you will receive depends on the number of hours that you work. Let's create a table
with one column representing the number of hours that you worked and another column with
your pay rate and the final column representing your total payment.
If you don’t work don’t expect to get paid anytime soon so if you work for 0 hours
multiply that by your payrate you get zero as expected. If you work for one hour you
will receive 12 dollars. For two hours of tutoring you will receive 24 dollars, 3 hours
you will get 36 dollars and with 4 hours of tutoring you will get 48 dollars not bad.
Tutoring math pays off. Alright notice that we can continue finding out the amount of
money we will receive by following the pattern, it turns out that your total payment has a
consistent pattern specifically, the total payment is equal to the pay rate times the hours worked.
In algebra we can represent this expression as follows, 12 times h.
The letter h stands for the hours shown in the 1st column of the table; in addition h
can stand for other hours not in the table like 5, 6 and even 80 hours. In algebra we
call the letter h a variable. A variable is nothing more than a symbol usually
a letter of the alphabet used to represent one or more numbers that’s all they are.
The numbers that are represented by the variable are called values of the variable.
An expression that contains variables such as 12 X h is called a variable expression.
If we decide to let h represent a number let’s say that we decide to let it equal to 4, our
variable expression now becomes a numerical expression notice that we have nothing but
numbers in our expression that is to say the expression does not have any variables it
only contains numbers hence it is a numerical expression.
In algebra the most common and over used variable is the letter x. Let's go back to our variable
expression 12 times h, if we decide to use x instead of h in our variable expression it will look like this.
This does not look right. Which one is the variable the first X or the second x?
Because of this reason the symbol for multiplication needs be changed now that you are in algebra
the appropriate way to indicate multiplication for numerical expressions is as follows,
We use a raised dot that is slightly bigger than a decimal point. In general products
that contain a variable are usually written without the multiplication sign, again because
the multiplication sign looks like the variable x. For variable expressions the way we denote
multiplication is as follows, 12 times h can be written as 12h likewise
a times b can be written as ab, ½ times x can be written as ½*x start
getting use to writing multiplication in this way for variable expressions and with a dot
for numerical expressions. Let's go back to our numerical expression 12 times 4.
This seems obvious but another way of writing 12 times 4 is by writing 48. What
I want you to notice is that in algebra there are different ways to represent the same number
48 can be expressed as 12 times 4 since both 12 times 4 and 48 have the same value we use
an equal (=) sign to symbolically show this relationship. So we write 12 times 4 = 48
and you read this expression as “12 times 4 equals 48” or "12 times four is equal to
48". Another way of reading this expression is "12 times four is 48".
Either forms are acceptable ways of reading this expression.
Ok now what if I were to ask you if 12 times 3 and 30 have the same value that is to say
are they equal? Obviously they don’t have the same value 12 times 3 is clearly 36 so
the correct way to express this statement is by using the does not equal symbol and
we write it as follows,
the way we read this is "12 times 3 is not equal to 30".
At times when we are trying to compare numerical expressions we need to simplify the expression
into its simplest value. For example if we are trying to compare the numerical expression
12 times 2 and the numerical expression 6 + 3 we need to simplify each numerical expression
to properly evaluate them. The left numerical expression simplifies to 24 and the right
simplifies to 9. Both numerical expressions are now written in there simplest value now
we can compare them, and it turns out without a doubt that these two values are not equal
to each other. At times a numerical expression can have more
than 2 mathematical operations in its numerical expression like the expression 36 divided
by 6 plus 8, notice that we have a parenthesis.
The parenthesis shows how the numerals in the expression are to be grouped.
The expression within the parenthesis is simplified first. In this case we first simplify 36 divided
by 6 because it’s grouped with the parenthesis the result is 6 then we add 8 the numerical
expression is simplified to its simplest value of 14.
Alright now let's go back to variable expressions. Say we have the variable expression 10 times a
or 10a for short. Remember that the variable represented by the letter “a” represents
one or more numbers if we replace the variable in the variable expression by a given value
and then simplify we say that we have evaluated the expression or found the value of the expression.
For example say that we are asked to evaluate the expression 10a if a = 5 what we want to
do is substitute or replace the variable “a” usually with a parenthesis with the numerical
value they want us to evaluate with in this case the number is 5 doing that gives us the
following and simplifying the numerical expression gives us that the final answer is 50.
In our next video we will go over a few examples.