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### Importance of Linear Functions

```
Date: 02/06/2002 at 16:38:51
From: Abreeka Moore
Subject: Importance of Linear Equations

Hello,

I was wondering what the importance of linear functions in the "real
world" is, and how they are used.

Thank you.
```

```
Date: 02/07/2002 at 09:46:12
From: Doctor Ian
Subject: Re: Importance of Linear Equations

Hi Abreeka,

Most 'real world' functions are approximations, as the world usually
contains too many complications to be modeled exactly by mathematical
functions.

The really nice thing about linear functions is that they are easy to
work with. They are easy to solve, easy to plot, and easy to
understand. So when you're looking for a function to approximate the
behavior of something in the real world, you usually try to use a
linear function first; and only if that proves to be too simple a
model do you look for other kinds of functions to use.

Often, in order to use a linear function to model something, you need
to find the 'best' line that models your data. For example, if your
data look like

|             *
|         * *
|   *  *  *
|    * * *
|  *
|
+-----------------

each pair of points that you could choose would give you a different
line; but there is _no_ line that contains all the points. What you
can do in a case like this is find the line that is 'closest' (in some
sense) to all the points:

|                  L
|             *
|         * *
|   *  * L*              The 'best' line is the one that passes
|    * * *               through the L's.
|  *
|L
+-----------------

This line is normally found using a process called 'least squares'
analysis, or 'linear regression'.  (The basic idea is this:  Choose a
line; find the distance of each data point from the line; add those
distances up to get a score for the line. Try other lines, until you
find the one that gives you the lowest score. That's the 'best line'.)

Once you've found a linear function that you think models your data,
you can use the function to make predictions. For example, if the data
above showed the amount of wheat harvested as a function of inches of
rain in past years, then knowing how much rain has fallen _this_ year
can let you predict the amount of wheat that will be harvested, even
before it's finished growing.

Note that when you've reached this point, one little function

wheat = m*rain + b

replaces a whole collection of data, so it's more efficient in terms
of space.

Of course, there is a danger in doing this: It might turn out that the
function doesn't consider all the relevant factors. (For example,
interest rates and subsidies have as much to do with the production of
wheat as rain does.) In which case, any predictions that it makes
might be wrong. But this is always the trade-off - ease of use versus
precision - that you make when you try to use math to describe the
real world.

As someone once said, you want to make your mathematical models as
simple as possible... but no simpler. A linear function is about the
simplest kind of model that it's possible to make.

anything else.

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

```
Date: 02/07/2002 at 23:21:53
From: Abreeka Moore
Subject: Importance of Linear Equations

Thank you! That was a huge help to me. I look forward to sending
questions in the future.
```
Associated Topics:
High School Exponents
High School Functions

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