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### 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,

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/
```
Associated Topics:
High School Basic Algebra
Middle School Algebra

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