Linear Expressions

Date: 08/17/99 at 14:29:45
From: Evan Eubanks
Subject: Linear expressions

What are linear expressions, and how will I use them in my life?

Date: 09/14/1999 at 18:59:20
From: Doctor Maureen
Subject: Re: Linear expressions

Dear Evan,

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 equal sign or inequality sign (< or >) and at 
least one term on either side. For instance: 5 + 3 = 8 is a 
mathematical sentence, called an equation, while 9 < 100 is an 
inequality.  

When we study English, we may talk 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 equal sign or inequality sign. For example, 8 is 
an expression, and so are 5+3 and 4/3.  

Some expressions include letters that stand for something else. These 
kinds of expressions are used every day in all sorts 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. You could represent 
the total number of miles you can drive based on how much gas you put 
into the tank using the expression 15g, where g stands for the number 
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 involves cars and mileage.  Imagine that you are 
driving down a highway at a steady rate of 50 miles per hour. 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 miles?   

Now for the linear part. If you have an expression with a letter in 
it, we call the letter a variable. Here are some examples of 
expressions with variables in them: 5a; x + 4; m/50. You may have 
figured out already from the two car examples above that "variable" is 
a good name for these letters because the letter can represent 
different amounts depending on the circumstances - so the letter 
represents a varying amount. A linear expression is an expression with 
a variable in it, but there is a special condition involving 
exponents.  

You may not have learned about exponents yet, so I'll give a brief 
explanation. We use exponents to symbolize many multiplication 
operations using the same number. For instance, if you had square 
chart with 6 rows and 6 columns and you wanted to place a sticker on 
each block of the chart, you could determine the number of stickers 
needed by multiplying 6 x 6. Exponents allow us to shorten this 
expression by writing 6^2. Actually, in your math book you will not 
see the ^. Instead you will see a smaller 2 above and to the right of 
the 6, but I can't write that here in email.    

Here is another example: perhaps you must multiply 3 by 3 by 3 by 3 by 
3. We would write this as 3^5, which means that 3 is multiplied by 
itself five times.  We read the expression 3^5 as 3 raised to the 
fifth power. Any number raised to one is simply that number: 4^1 = 4; 
(5a)^1 = 5a. We usually omit writing the ^1 when a number is raised to 
the first power.  

A linear expression is an expression with a variable in it; however, 
the variable is only raised to the first power. For example, 5a is a 
linear expresion, because it is understood that a is raised to the 
first power.  9t^2 +8 is not a linear expression because t is raised 
to the second power. 10y + 1 is also a linear expression. In the last 
expression, where 10 is multiplied by the variable y, the 
multiplication symbol is omitted. This is a common way of writing 
multiplication of a variable and number.  

There are many other examples of linear expressions in addition to the 
two I gave you. Another common one is computing how much money you 
might earn working at a resturant for $6.00 per hour. Your total pay 
for a week could be expressed as 6h, where h represents the number of 
hours you worked this week. Can you think of more examples now? Try 
writing an expression that could be used to compute the total cost of 
apples if each one costs .50 (50 cents).

I hope this explanation helps.

- Doctor Maureen, The Math Forum
  http://mathforum.org/dr.math/