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The Divide between Numbers and Variables

Date: 04/25/2011 at 19:56:00
From: Brandon
Subject: Diving 4x by 4 in 4x+7=5

In the equation 4x + 7 = 5, why do you divide 4x by 4? Why can't you just
subtract 4 from 4x, like this?

    4x - 7 =  5
    4x     = 12
   -4        -4
     x     =  8

I guess it's supposed to be this:

    4x - 7 =  5
    4x     = 12
    4x/4   = 12/4
     x     =  3

But why?

I have no idea about math. I slacked in school, but now I want to become
more educated, so I'm going to start college.

Date: 04/25/2011 at 23:08:22
From: Doctor Peterson
Subject: Re: Diving 4x by 4 in 4x+7=5

Hi, Brandon.

This is an excellent question. I suspect a lot of students think it but
don't ask it, and so they fail to learn this!

When you subtract 4 from both sides of the equation 4x = 12, you get the
new equation 4x - 4 = 12 - 4. This equation is equivalent to the original;
that's not the problem. It's what you do next.

The left side of this new equation, 4x - 4, is NOT equal to x. If that
isn't obvious (because they are not like terms and can't be combined),
just pick a value of x and see if the two expressions are equal. When 
x = 1, 4x - 4 is 0, which is not equal to x. So the two expressions are 
not equivalent, and you can't replace one with the other.

The reason we divide by 4 here rather than subtract is to accomplish our
goal of isolating x. The expression 4x/4 IS equal to x, because the
division undoes the multiplication: when you multiply by 4, and then
divide by 4, you get back the number you started with.

So the general rule in solving an equation is that you must identify what
operation you want to undo at each step, and then carry out that "undoing"
operation. In your first step, you wanted to undo a subtraction of 7, and
you did that by adding 7 (to each side). Now you want to undo a
multiplication, so you divide.

Another useful rule is to simplify the expressions on each side of the
equals sign methodically and honestly. Don't just write "x" because that's
what you WANT to see; actually simplify the expression so you'll realize
that 4x - 4 is NOT x, and that 4x/4 IS. That would prevent a lot of
students' mistakes!

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

- Doctor Peterson, The Math Forum 

Date: 04/26/2011 at 00:06:32
From: Brandon
Subject: Thank you (Diving 4x by 4 in 4x+7=5)

Excellent explanation! Thank you for that. You answered my question -- and more.
Associated Topics:
Middle School Algebra

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