Algebra - Solving Equations

Date: 11/03/97 at 23:36:50
From: AVI MANTHA
Subject: ALGEBRA

Hi, Dr. Math,

I am in an enriched Algebra class, and I'm having a hard time solving 
some of my math problems. Please help me:

4 - 3X = 5 - 6X - 7

7X = 5(X - 12)

Please explain the solution step by step.

Thank you very much for your help.
Avi

Date: 11/10/97 at 09:28:19
From: Doctor Allan
Subject: Re: ALGEBRA

Hello Avi, and thanks for writing to Dr. Math!

You do not tell me exactly where you have problems, so I am not sure 
whether you have problems with these types of equations generally, or 
if it is just the two problems you mention here.

What I will do is give you techniques to use when solving these 
equations, and then help you with the first of your problems - then 
you can check out whether that helps you solving the second problem 
and getting the general picture.

When you want to solve an equation you want to find values such that 
if you substitute these values for x in your equation, you will get 
something that is true. Let me give you an example:

   x + 5 = 8

What you need to do is find a value to substitute for x such that you 
get 8. If you have five apples and you want to have eight apples how 
many more apples should you buy?

That's basically what an equation is all about.

So the smart thing to do is to isolate x on one side of the equation 
and that will tell you the exact value to be substituted.

In order to do this you have to remember some rules. The most 
important one is this: 

  Whatever you do to one side of the equation you should do to
  the other side. 

That means if you for instance add 3x to one side you should add 3x to 
the other side.

Furthermore, there is the distributive rule:

  x(a+b) = (a+b)x = xa+xb

This means that when you multiply terms within a parenthesis by x, you 
multiply x by each term of the parenthesis.

That should cover the basics. Now let's go to the first of your 
problems:

  4 - 3x = 5 -6x -7

First notice that 5 - 7 = -2 so you get

  4 - 3x = -2 -6x

Now you want to isolate x on one side of the equation, so let's 
subtract 4 from each side of the equation. We get

  4 - 3x - 4 = -2 - 6x -4

That is

  -3x = -6 -6x

Since you want the x on one side the thing to do now is to add 6x:

  -3x + 6x = -6 - 6x + 6x

That is

  3x = -6

So all you need to do now to isolate x is to divide by 3 on each side:

  (3x)/3 = (-6)/3

So you get

  x = -2


I hope this helped you. If you have further questions, please write 
again and I will try to help you out.

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