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### 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/
```
Associated Topics:
High School Basic Algebra
High School Polynomials
Middle School Algebra

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