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Introduction to Linear Equations and Slope/Intercept Form

Date: 03/19/2005 at 02:36:01
From: Vi
Subject: A curious question I have about equations

Dear Dr Math,

Hi! I am in Year 8 in Australia.  I was looking through your middle 
school archives (which is equivalent to junior high school here) and I
even tried the high school ones, even though I won't be in Year 9 
until next year :) 

In the algebra section, you talk of 'linear equations'.  Can you 
explain to me what they are?  I've never heard my teachers talk about
them, and the answers aren't really clear to me.  Can you help, step 
by step, in an easy way that doesn't baffle the brain?  Once I get it. 
I could probably solve them easily, as algebra isn't that hard for me! 
                       
Thanks for any replies!  



Date: 03/19/2005 at 11:43:54
From: Doctor Ian
Subject: Re: A curious question I have about equations

Hi Vi,

When you first get into algebra, you start with equations where you 
have one 'unknown', or 'variable'.  That might look like

  3x + 2 = 5              '3x' means '3 times the quantity x'

With an equation like this, your job is to find the value of x that 
makes the equation true, e.g., 

  3*1 + 2 = 5   

so x = 1 is the value of x that makes it true.  Sometimes you can have
more than one value, e.g., 

  x^2 - 1 = 3              'x^2' means 'x squared' or 'x*x'

where x = 2 and x = -2 both make the equation true. 

So far, so good?  Eventually, you add a second variable, to get 
equations that look like 

  y = 3x + 2

Now there is no longer a limited number of values that make the 
equation true.  Why?  Suppose I _decide_ that x is 1.  Then 

  y = 3*1 + 2

  y = 5

and if I make y equal to 5, the equation is true.  But if I decide 
that x is 2, then

  y = 3*2 + 2

  y = 8

and if I make y equal to 8, the equation is also true.  I think you 
can see with a little thought that I can give x any value I want, and
I'll always be able to come up with a value for y that 'saves' the
equation.  

So the solutions to an equation like this are pairs of related (x,y)
values.  The two solutions we just found are (x=1,y=5) and (x=2,y=8).
By convention, we always write them in (x,y) order, so we can just
abbreviate those as (1,5) and (2,8).

Now, here's the really interesting part.  If I set up a coordinate
plane with two axes, one for x and one for y, then I can place a mark
at the location (1,5) by starting at the origin (where the axes
cross), moving 1 unit to the right, and 5 units up:


              |
              -
              |
              -
              |
              -
              |
              -
              |
              -  * (1 over, 5 up)
              |
              -
              |
              -
              |
              -
              |
              -
              |  
   --|--|--|--+--|--|--|--|
              |
              -
              |
              -
              |

And I can plot the point (2,8) the same way:

              |
              -
              |
              -     * (2 over, 8 up)
              |
              -
              |
              -
              |
              -  * (1 over, 5 up)
              |
              -
              |
              -
              |
              -
              |
              -
              |  
   --|--|--|--+--|--|--|--|
              |
              -
              |
              -
              |

Now, let's stop and ask a question:  What if I plot _all_ the possible
solutions?  At first, that seems kind of silly.  I wouldn't have time
to figure them all out!  It would take forever.  

But here's what's very, very cool.  Simply by looking at the form of
an equation, I can say a lot about what the plot of all the solutions
would look like.  For example, looking at our equation

  y = 3x + 2

I can say that all the solutions will lie on a straight line.  That
is, if I place a ruler over the two points we've plotted so far, and
draw a line through the points that extends forever in each direction,
every solution to the equation will be a point on that line.  We say
that the line is the graph of the equation.

An equation like that, we call 'linear'.  :^D

There are other kinds of equations.  For example, an equation like

  y = (x - 3)^2 + 4

is always going to be the equation of a parabola.  The equation

  (x - 2)^2 + (y - 1)^2 = 4^2

is always going to be a circle.  And so on.  So a lot of what you'll 
be doing in algebra is learning how to look at an equation and know
something about what its graph will look like.  You also learn how to
look at a graph and know something about what the corresponding
equation will be. 

For example, with linear equations, you'll learn that you can describe
any line using a quantity called 'slope' (which tells you how steeply
the line ascends or descends as it goes to the right), and another
quantity called the 'y_intercept' (which tells you the value of x
where the line crosses the y-axis).  If you know those two values, you
can just write down the equation of the line:

  y = slope*x + y_intercept
              
In our case, 

  y = 3x + 2

the slope is 3 (that is, for each step to the right, we move three
units up), and the y_intercept is 2 (if you extend the line, you'll
see that it crosses the y-axis at y=2).  

Going in the other direction, if I know the slope and the y_intercept,
I can plot one point at 
  
  (0, y_intercept)

and a second point at 

  (0+1, y_intercept+slope)

and that gives me the two points I need to fill in the whole line.

I think you might be getting a sense for how easy it is to work with
lines, compared to other kinds of graphs we might see.  This is why we 
place a lot of emphasis on linear equations.  In fact, they're so nice 
to work with that in the real world, even when we know that something 
is too complicated to be described using lines, we'll pretend that 
it's simpler than it is, just so the math works out nicely.  

(This is the same sort of idea that you use in learning to estimate
39*48 as 40*50--you give up some accuracy, but you also avoid a lot of 
work.  It's a trade-off that occurs at all levels of mathematics!)

Take a look at what I've written above, and let me know if it's clear.
Or if any parts aren't clear, tell me what they are, and we'll keep
going until we get an explanation you can follow.  Then we'll put it 
in our archive, and other students who are in your situation can 
benefit from it too, okay? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Linear Equations
Middle School Equations

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