Linear Equations and Standard Form

Date: 04/04/2002 at 23:30:42
From: Anthony Hsu
Subject: Standard form of a linear equation

My question is about the standard form of a linear equation. I already 
know it's ax+by=c, but I need to know what the variables mean and if 
there are any conditions to them. I tried looking in a textbook called 
_College Algebra_, but they gave this other fomula, ax+b=o. I already 
know a little about how it can be converted into slope-intercept form 
but I'm still a little fuzzy on its relation to ax+by=c. 

Also, what is it about something function f f(x)+ mx+b ? 

I would be very grateful if you could answer my questions.
-Anthony Hsu

Date: 04/05/2002 at 09:19:20
From: Doctor Rick
Subject: Re: Standard form of a linear equation

Hi, Anthony. I will try to help.

The numbers represented by a, b, and c don't have meanings like m, the 
slope, and b, the y-intercept, in the slope-intercept form y = mx + b. 
Notice that if you multiply all three numbers a, b, and c by 
the same amount, you get a different equation that has the SAME graph 
- that is, an equivalent equation. For instance,

  2x - 3y = 7

and 

  4x - 6y = 14

are both equivalent to the slope-intercept form

  y = (2/3)x - 7/3

In general, the equation ax + by = c is equivalent to the slope-
intercept form

  y = (-a/b)x + c/b

You see that the slope is -a/b and the y-intercept is c/b. It's the 
ratios of a and c to b that have meaning.

Why, then, do we bother with the "standard form"? Isn't the slope-
intercept form better? It's simpler, and the numbers have meaning. 
Well, the advantage that the standard form has over slope-intercept 
form is that EVERY linear equation can be written in standard form, 
but not every linear equation can be written in slope-intercept form. 
For instance, this is a 
linear equation:

  x = 5

This is the equation of a straight line - in particular, a vertical 
line. Its slope is "infinite"; that is, it is not a number. You can't 
write it in the form y = mx + b if m is not a number. Just try it!

Thus, the standard form is important when you are trying to describe a 
line that may be vertical. If you know the slope, and it is a number 
(not infinite), then the slope-intercept form is fine. But suppose you 
are describing a rotating line: at some time it will be vertical, and 
then the slope-intercept form won't work. You can either make that a 
special case, or you can use the standard form so the same description 
will work all the time.

I don't know exactly what you saw about functions. Perhaps you meant 
to type

  f(x) = mx + b

This is a linear function. You can think of a function as a machine 
that takes in a number, does some sort of work on it, and puts it out 
as another number at the other end of the machine. In this case, you 
put in some number x; the function multiplies it by m, then adds b to 
the result, and puts out this new number, which is called f(x). 

If you give that new number (the output of the function) the name y, 
then you have your familiar linear equation in slope-intercept form: 
y = mx + b.

A big difference between the function and the equation is that a 
function is only allowed to put out ONE number for each number that 
goes in. If you think about a vertical line such as x = 5, you can't 
make this into a function. You can only put in the number 5, and ANY 
number can come out. (The point (5,y) is on the line for ANY value of 
y.) A function isn't allowed to do this. Therefore, a linear FUNCTION 
is never graphed by a vertical line. The slope-intercept form can 
describe any linear function. 

A linear EQUATION, on the other hand, may be graphed as a vertical 
line; the slope isn't always a number, so the slope-intercept form 
doesn't always work. That's why you need the standard form for a 
linear equation.

Does this help? If I have raised even more questions in your mind, go 
ahead and ask them - we really like it when students wonder about math 
and discover questions that they don't know the answer to. That's the 
first step to discovering answers!

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

Date: 04/06/2002 at 13:38:07
From: Anthony Hsu
Subject: Re: Standard form of a linear equation

Dr. Math,

I just received your e-mail. I found it to be very helpful and 
informative. You don't know how much this helped me. Thank you very 
much!

-Anthony Hsu