Solving for One Variable of Many

Date: 05/30/2002 at 20:31:37
From: Makeia Gay
Subject: I need help

I don't understand how to solve formulas for specific variables.  For 
example, solving Ax + C = D for x. 

Date: 05/31/2002 at 14:25:22
From: Doctor Ian
Subject: Re: I need help

Hi Makeia,

Solving for one equation is a little like untying a knot.  You 
have an x in the middle of all this stuff, and you have to get it 
out.  The basic rules are:

  1. You can add anything to both sides of an equation.
  2. You can multiply both sides of an equation by anything.
  3. You can't divide by zero!

Note that sometimes 'add' means 'add a negative number' (which is 
like subtracting), and 'multiply' means 'multiply by a fraction' 
(which is like dividing).

You do additions and multiplications in order to move things from 
one side of the equation (where you don't want them) to the 
other.  So let's look at

  ax + c = d

We don't want the x on the same side as the x, so we can add -c 
to both sides of the equation. 

  ax + c + -c = d + -c

On the left side, the c and -c cancel each other out, so we get

  ax = d - c

Now we'd like to get rid of the a.  We can do that by multiplying 
by (1/a):

  (1/a) * ax = (1/a)(d - c)

           x = (d - c)/a

And now we have x by itself on one side, which is what we wanted. 

Note that the order of the operations is important!  We had to 
add -c, and then multiply by (1/a).  If we had done those things 
in the opposite order, we'd have ended up with a mess. 

Note also that when you multiply one side of an equation by 
something, you multiply _everything_ on that side.  Usually you 
do this by putting parentheses around it, to remind yourself.  If 
we'd done something like

  (1/a)*x = (1/a)d - c

we'd have ended up with the wrong answer.  But this is a common 
error, and you have to watch out for it. 

I hope this helps. 

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