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### Two-Step Equations and Equations with Variables

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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!

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!!

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
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/
```
Associated Topics:
Middle School Algebra

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