Algebra (Equation)

Date: 12/12/97 at 00:43:46
From: Jared
Subject: Algebra (equations)

What is the answer to the problem: 15-(5m-4) = 34?

I need to know how to do the 15- part - that is where I am having 
trouble. Please just give me the first couple of steps.

Thank you.

Date: 12/29/97 at 10:31:29
From: Doctor Mark
Subject: Re: Algebra (equations)

Hi, Jared -

Problems like this are incredibly easy to do once you know the secret.  
And here is the secret:

If you have a minus sign in front of (to the left of) a pair of
parentheses, remove the parentheses by *changing all the signs* of 
whatever is *inside* the parentheses.

So in your problem, you would find out what - (5m-4) is by changing 
all the signs in the quantity 5m - 4 (which is what is inside the 
parentheses). In doing this, you have to remember that the 5m has a 
secret "+" sign in front of it, which is not usually written.  So,

   - (5m - 4) = - ([+]5m - 4) = - 5m + 4.

Do you see how the + 5m got changed to - 5m, and the - 4 got changed 
to + 4?  Do you agree that all the signs of the thing inside the 
parentheses got changed when we removed the parentheses?

Once you have removed the parentheses, then you just collect terms. So

   15 - (5m - 4) = 15 - 5m + 4 = - 5m + 15 + 4 = - 5m + 19.

(Did you remember that the 15 was really just + 15?  Another secret 
"+" sign!)

I'll have to admit that remembering to change that last (-) sign [the 
one in front of the 4] when removing parentheses having a (-) sign on 
the left is hard. I forget it myself sometimes, and boy, do I feel 
like an idiot when I do!

Now back to the equation (this was just the left side)...

   15 - (5m - 4) = 34.

This becomes, from above, 

       - 5m + 19 = 34.

Now add 5m to both sides to get

              19 = 34 + 5m

And subtract 34 from both sides to get

         19 - 34 = 5m

which becomes

            - 15 = 5m.

Now turn it around so it looks normal:

              5m = - 15,

and divide both sides to get (finally!)

               m = - 3

You may have noticed that I got the m-term on the right.  This was so 
I did not have to divide by a negative number, which I find 
distasteful.

The only other thing you have to remember to do problems like this
correctly is that you should always write subtractions of "like" terms
last. Here is an example:

   - 4x - 3 - (8 - 7x) = - 4x - 3 - 8 + 7x

                       = 7x - 4x - 3 - 8 

                       = 3x - 11

Notice two things here:

1) when we removed the parentheses, we changed all the signs of the 
   thing inside the parentheses:

   8 - 7x --> - 8 + 7x,  i.e., - (8 - 7x) = - 8 + 7x.

2) we wrote

   - 4x + 7x = 7x - 4x 

   i.e., we put the subtraction "last" (on the right).

This idea of "put subtractions last" is just so that we can avoid 
mistakes.

I hope this was of help, Jared, and make sure you write back if you 
have any other questions.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/