Linear Equations in One Variable

Date: 10/11/2001 at 10:00:16
From: Rebecca England
Subject: Solving Linear Equations In One Variable

35 = 17 + 3(x-2)

Date: 10/11/2001 at 13:28:24
From: Doctor Ian
Subject: Re: Solving Linear Equations In One Variable

Hi Rebecca,

Let's imagine that the equals sign is a balance beam:

      35         17 + 3(x - 2)
   _________     _____________
              ^
              =

(That's not a bad way to think of it, actually.)  We can subtract 1 
from each side without changing the balance, right? 

    34 = 16 + 3(x - 2)

And we can keep doing that, 

    33 = 15 + 3(x - 2)

    32 = 14 + 3(x - 2)
     .
     .
    19 = 1 + 3(x - 2)

    18 = 3(x - 2)

Of course, we could have accomplished the same thing in one step by 
subtracting 17 from each side, but this lets you see _why_ that works. 

Anyway, so now we have this:

           18 = 3(x - 2)

which is really the same as this:

    6 + 6 + 6 = (x - 2) + (x - 2) + (x - 2)

So it's pretty clear that each (x-2) corresponds to a 6, right?  Which 
means that
  
            6 = (x - 2)

Again, we could have divided both sides by 3, but this shows you _why_ 
that little trick works. 

So now, can you think of a number such that, when you subtract 2 from 
it, you get 6?  If not, we can try our first trick again, except 
adding 1 instead of subtracting 1:

            6 = x - 2

            7 = x - 1

            8 = x

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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