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Pre-algebra variable expressions

Date: 09/01/99 at 20:04:16
From: Vanessa
Subject: Pre-algebra variable expressions 

Can you please send me step-by-step instructions on how to do variable 
expressions problems?

Date: 09/10/99 at 22:28:40
From: Doctor Maureen
Subject: Re: Pre-algebra variable expressions 

Dear Vanessa,

Thanks for your question. 

Let me start with the "expressions" part of your question. I am going 
to explain mathematical expression by comparing it to something you 
probably understand from studying English or grammar in school. 

When we write, we use sentences to write a complete thought, and in a 
sentence there must be a noun and verb, and often there are extras 
like describing words. 

In mathematics, we also frequently write in sentences, but we use 
numbers and symbols to convey a thought. A complete mathematical 
sentence includes an equals sign or an inequality sign (< or >) and at 
least one number on either side. For instance: 5 + 3 = 8 is a 
mathematical sentence, called an equation; 9 < 100 is an inequality.

When you study English, you may have talked about phrases, which are 
groups of words, but are not always complete sentences, like "just do 
it" or "good sport." In mathematics, we have phrases called 
expressions, which can be just one number or several numbers and some 
symbols; however, there is no equals sign or inequality sign. For 
example, 8 is an expression, and so is 5+3 or 4/3.

Some expressions include letters, called variables, which are symbols 
that represent numbers. These kinds of expressions are used every day 
in all kinds of situations. Here is an example: let's say that you 
have a car and it can travel 15 miles for every gallon of gas in the 
tank. Then you could represent the total miles you can drive based on 
how much gas you put into the tank using the expression 15g, where g 
stands for the numbers of gallons you put in the tank. Once you put 
gas into your tank, you can figure out how far you will be able to go 
by replacing g with the number of gallons you purchased and pumped 
into your tank.

Another example involving cars is determining your travel time using a 
variable for distance. If you are driving down a highway at a steady 
rate of 50 miles per hour, then you can determine how long it will 
take you to get to your destination by representing the time as an 
expression m/50 where m represents the number of miles you have to 
drive to reach your destination. If your destination is 250 miles 
away, then by replacing m with 250 miles, you can calculate that the 
travel time will be 250/50, or 5 hours. If your trip is 525 miles, it 
will take 10 1/4 hours. How long will it take if the trip is 775 

One more example: Let's say you baby-sit and you charge $4 per hour. 
Then you could express your total earnings in a week as 4h where h is 
the number of hours you baby-sit. Maybe you work for a health club and 
you are paid $25 per week to be available when the club needs you and 
then $4 per hour to actually baby-sit children while club members work 
out. We could express your weekly pay as 25 + 4h, where again h = 
hours worked.

Now for solving problems using variable expressions. You can only 
solve problems involving variable expressions if you are given a value 
for the variable. For instance, in the travel time example above, you 
cannot actually determine a travel time until you are given a value 
for distance. If you are told that the distance is 250 miles, then you 
substitute 250 for the variable, m, and solve the expression: 
250/50 = 5 hours.

In the babysitting examples, you can only determine a weekly pay total 
if you know how many hours you worked. I suggest to my students that 
they first substitute, then compute the arithmetic, then check their 
work. For instance, if you worked 4 hours for the health club, then 
you would substitute 4 for h in the expression 25 + 4h. It would look 
like this:

     25 + (4)(4) = 41. 

Solving variable expressions is like using a formula. The expression 
is like a formula and the value you are given is to be substituted 
into the formula.

I hope this explanation helps. Write back if you need more 
explanation. Maybe you could give an example of a problem with which 
you are having difficulty.

- Doctor Maureen, The Math Forum
Associated Topics:
Middle School Algebra

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