Solving Literal Equations

Date: 09/08/2001 at 14:15:31
From: Jenny
Subject: Literal equations

I just don't get how to do literal equations. I can do equations fine 
with just numbers, but doing them literally with letters is confusing 
me. Do you think that you can explain literal equations to me, please?

Thanks!

Date: 09/08/2001 at 22:43:18
From: Doctor Peterson
Subject: Re: Literal equations

Hi, Jenny.

I'd like to see just how you are confused, so I can help better. The 
explanation may not be satisfying to you: You do exactly the same 
thing to solve a "literal equation" as for one with only one variable. 
In fact, you do a little less, since you can't actually carry out 
multiplications and additions, but just simplify instead. (Once you 
know algebra well enough, this is so commonplace that the term 
"literal equation" seems unnecessary; I'd never heard it until fairly 
recently.)

But your confusion is very common; it does seem like a weird thing to 
do when you first see it. So how can you get unconfused?

It may help if you try replacing the variables (except the one you are 
solving for) with actual numbers, solve it that way (without doing any 
actual arithmetic) and then do the same thing using the variables.

Here's an example: Let's solve

    P = 2(L+W)

for W. I'll practice by solving this equation using numbers:

    12 = 2(5 + W)

To solve this, I have to undo everything that was done to W, one thing 
at a time. It was added to 5, then multiplied by 2; to undo those, in 
reverse order, we have to first divide by 2 and then subtract 5:

    12/2 = 5 + W    divided by 2

    12/2 - 5 = W    subtracted 5

Notice that I didn't bother to divide 12 by 2 to get 6, or subtract 5 
to get 1, as I would normally do; I turned off the arithmetic part of 
my mind, and left only algebra running. But if I did the arithmetic, I 
would get the actual answer, 1.

Now let's do exactly the same thing with the variables;

    P = 2(L + W)

    P/2 = L + W     divided by 2

    P/2 - L = W     subtracted L

That's the answer:

    W = P/2 - L

Now, of course, if I plugged in the numbers P = 12 and L = 5, I would 
get W = 1. But having done all the algebra already, I could use 
different numbers and just use arithmetic (evaluate this expression) 
to find W again. That's what "literal equations" do for you; they 
predigest the algebra, leaving you needing only arithmetic to solve 
many similar problems.

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