Order of Operations and Solving Linear Equations

Date: 03/01/2006 at 01:09:23
From: Kevin
Subject: How come PEMDAS does not work for solving two step equations

How do I explain to 6th graders that when solving x/-2 + 16 = 32 they
need to subtract 16 first, versus multiplying by -2, in order to solve
for x?  Students are taught to work through the proper order of
operations following PEMDAS, but the first step in solving this
equation should be subtraction, not multiplication.  Thanks!

Date: 03/01/2006 at 09:47:42
From: Doctor Peterson
Subject: Re: How come PEMDAS does not work for solving two step equations

Hi, Kevin.

Actually, the order of operations is precisely why you want to 
subtract the 16 first.

When you are EVALUATING an expression, you do the multiplication 
first, then the addition.  Here, however, you are not evaluating, but 
SOLVING, which really means undoing the evaluation.  If I knew what x 
is, I would be first dividing it by -2, and then adding 16.  The 
result is 32.  But we want to discover what that secret number is, so 
we have to work backwards.  First we undo the LAST step, the 
addition, to see what number I added 16 to in order to end up with 
32.  That number is 32 - 16 = 16.  Then, we undo the FIRST step, the 
division, to see what number I divided by -2 to get 16; we multiply 
by -2 and find that x is -32.

So solving requires following the order of operations backward!

I often illustrate this by describing how I put on my shoes and socks.
In the morning, I put on my socks first and then my shoes, following
the order of operations for dressing.  At night, I don't take off my
socks first!  I first undo the last thing I did, taking off the shoes,
and then the socks.  Evaluating is dressing a variable, solving is
undressing it!

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

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