Solving Linear Equations without PEMDAS

Date: 08/13/2002 at 11:00:45
From: Laura
Subject: Solving linear equations without PEMDAS

Is there a linear equation that can be solved using both the PEMDAS 
order of operations and NOT using the order of operations and still 
come out with the same answer?

Date: 08/13/2002 at 12:15:53
From: Doctor Peterson
Subject: Re: Solving linear equations without PEMDAS

Hi, Laura.

This is sort of like asking, Is there any English sentence that still 
means the same thing if I don't use English grammar? That's because 
the order of operations rules are an essential part of the grammar of 
algebra; without them, we don't know what a "sentence" (equation) 
means! You have to use _some_ grammar; the question is, what grammar 
do you use if not the accepted one?

Let's take my analogy further. Here's my English sentence:

    The dog bit Jack.

Using English grammar, this means that Jack is hurting. But someone 
else, who knows English words but follows the grammar of his native 
language, might think that Jack just has some interesting dietary 
habits, because he uses a reverse word order. Or, he might not be able 
to make sense of it at all, because he might always expect the verb to 
be at the end of the sentence, or might expect every noun to have an 
ending showing how it is used in the sentence. So this sentence will 
mean the same using SOME alternative grammars, but there are other 
grammars in which it would either mean the opposite, or have no clear 
meaning at all.

Now look at an algebra sentence:

    2 + 3 * 4 = 14

Using normal ("PEMDAS") grammar, this is a true sentence; we multiply 
first, and add last.

Using the "grammar" of a simple calculator, which takes each operation 
as it comes, the left side would equal 20, since we first add and then 
multiply.

Using Reverse Polish Notation, the "grammar" of some calculators, this 
sentence means nothing; in that form it should be written as

    2 3 4 * + 14 =

to say that we stack up the numbers 2, 3, and 4, then multiply the 
last two and replace them with 12, then add the top two numbers (now 
2 and 12) and replace them with 14, then stack a 14 on top of it and 
compare the two 14's to see if they are equal.

So the answer to your question is this:

It is impossible to read an equation without using _some_ order of 
operations rules. If you don't use PEMDAS, you have to choose an 
alternative; and then it may either be meaningless, or mean the same 
thing, or mean something different. There is no equation that will 
mean the same thing (and therefore have the same solution) using 
_all_ possible "grammars." But if you choose one particular set of 
rules, such as those used by a simple calculator, then there are many 
equations that mean the same thing either way. For example,

    2x + 3 = 5

means 2 times x, plus 3, equals 5, whether we read left to right or 
take multiplication first. But

    3 + 2x = 5

means different things each way, and will have a different solution.

If you have any further questions, feel free to write back.

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