Two-Step Equations and Equations with Variables

Date: 5/17/96 at 21:59:21
From: Shelly J Tyre
Subject: Equation

Hi. I am a 7th grade student. I usually get these things pretty well, 
but I suffer at 2-step equations and the ones with variables on both 
sides. I'm in pre-algebra. The question is, can you help me solve 
problems like:  5 - 2n = 8n -(-4n) ?  

I know that problem may not work out too easy, but I just made it up 
off the top of my head. I have asked many people for help, but I can't 
get it to work out right. Thanks!

Shelly Tyre :(

Date: 5/18/96 at 16:13:55
From: Doctor Ken
Subject: Re: Equation

Hello Shelly!

Don't worry too much about this stuff, because while it is pretty 
important, most people get the hang of it eventually.

In problems like the one you sent me, you basically just want to put 
all the terms with a variable on one side of the equation, and put all 
the terms without a variable on the other side.  That's the first part 
of doing these problems.  So if we do that to your problem, we get 
this:

5 - 2n = 8n - (-4n)       First, cancel the two negatives.
5 - 2n = 8n + 4n          Now add the two things on the right  
                            together.
5 - 2n = 12n              Now we want to get all the terms that have 
                            an "n" in them on one side.  So we'll add
                            2n to both sides.
5 - 2n + 2n = 12n + 2n    Simplify the left and right sides.
5 = 14n

So that's the first part of any problem like this.  On one side, we've 
just got a number, and on the other side we've got stuff involving n. 

The only thing left to do is to get rid of the 14.  To do that, we 
divide by 14:

 5    14n
--- = ---                 Cancel the 14's on the right side.
 14    14

5/14 = n                  That's the answer!!


Here's something that might help you solve equations like this.  Did 
you notice that in the first part of solving this problem, there was a 
-2n on the left side, and we turned it into a +2n on the right side?  
Well, that's the way it always works.  If you've got something on one 
side of the equation and you want to bring it over to the other side, 
it ends up being the negative of what it was on the first side.  So if 
you had this:
                           y + 5 = x
you could turn it into this:
                           y = x - 5.

The same kind of thing works for multiplication and division.  If 
you've got something like this:
                          5/x  =  3
you can turn it into this:
                            5  =  3x.
When it was on the left side, it was a "divided by x" and when we 
moved it to the right it became a "times x."  But be careful to make 
sure that when you use this, the WHOLE SIDE needs to be multiplied or 
divided by the thing you're moving.  So this doesn't work:
                      5/x + 4  =  7
                        5 + 4  =  7x  No No No No No No No No!!!!!
                But this does work:
                    (4 + y)/n  =  c
                        4 + y  =  nc

Hope this helps!

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/