Doing the Same Thing to Both Sides of an Equation

Date: 10/20/2003 at 22:07:18
From: Mike
Subject: equation

When I subtract something from one side of an equation do I add it to 
the other side or do I subtract it from that side also?  I'm working
on the equation 6(c + 4) = 4c - 18.

Date: 10/21/2003 at 11:36:25
From: Doctor Douglas
Subject: Re: equation

Hi Mike -

Thanks for writing to the Math Forum.  Good question!

When manipulating equations, you should always do the same thing to
both sides.  For example, you should subtract the same thing from BOTH
sides:

     6(c + 4) = 4c - 18
 6c + 24      = 4c - 18            no problem yet.
 6c + 24 - 4c = 4c - 18 - 4c       subtract 4c from BOTH sides
 2c + 24      =    - 18               

But there is another way to look at it, too.  You can MOVE something 
from one side of the equation to the other side, provided you change 
its sign:

 6c + 24      = 4c - 18
 6c + 24 - 4c =    - 18           We move the +4c to the left,
 2c + 24      =    - 18           changing the sign in the process.
 
It may seem like this moving process is not performing the same
operation on both sides, but it really is.  Both sides have had 4c 
subtracted, just like in the first example.  Usually the moving method
is a shortcut because you don't have to actually do the arithmetical
work on one side.  In the second example, we don't have to do the 4c -
4c on the right because we know it must be zero.

I hope that explains what is going on.  Please write back if you have
more questions about this.

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