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Basic Tips on Solving for X


Date: 11/16/2000 at 22:25:13
From: Megan
Subject: How to solve for the variable "X"

The following problems have tripped me up. I can't seem to figure out 
how to solve them. I've tried, several times, and had my algebra 
teacher explain them, but I still can't get the answers.

     3x - 11 = 6              
     2x - 7  = 3 + x
     3 + 3x  = 9 + 2x          
     3x - 3  = 6 + x

Thanks, 
Megan


Date: 11/17/2000 at 11:03:31
From: Doctor Ian
Subject: Re: How to solve for the variable "X"

Hi Megan,

Suppose I start with something like

     x = 5

That's no fun, is it? There's nothing to figure out. But I can 
multiply both sides of the equation by 3 without changing the truth of 
the equation, right?

     3x = 15

And there are a lot of different ways I can write 3:

     3 = 1 + 2

       = 7 - 4

       = -5 + 8

So I can change my equation to look like this:

           3x = 15

         x(3) = 15

     x(7 - 4) = 15

      7x - 4x = 15

And since I can add or subtract anything on both sides of an equation 
without changing the truth of the equation, I can add 4x to both 
sides:

          7x - 4x = 15

     7x - 4x + 4x = 15 + 4x

               7x = 15 + 4x

And of course, I can do the same thing with any number I want:

         7x = 15 + 4x

     7x - 5 = 15 + 4x - 5

     7x - 5 = 15 - 5 + 4x

     7x - 5 = 10 + 4x

Now I have something that looks just like your problems, don't I?

The trick, then, is to reverse the steps that I just did, to get back 
to the original, simple, boring equation:

         7x - 5 = 10 + 4x

     7x - 5 + 5 = 10 + 4x + 5           Add 5 to both sides

             7x = 15 + 4x

        7x - 4x = 15 + 4x - 4x          Subtract 4x from both sides

             3x = 15

         (3x)/3 = 15/3                  Divide both sides by 3

              x = 5

It's easy to forget what's really going on when you're solving an 
algebraic expression for a variable. You're really just trying to 
rewrite a complicated expression in a simpler way, without doing 
anything to change the meaning of the expression. It's a little like 
what you would do to change a complicated sentence like:

     "Under a mammal of feline persuasion is located, at this time, 
     in an unspecified location, a rectangle of woven fabric."

into a simpler one like:

     "The cat is on the mat."

Algebra is basically tying knots and then untying them again.

With that in mind, let's look at one of your example problems:

     3 + 3x = 9 + 2x

The first thing to notice is that we have numbers on both sides. We 
know that someone put them there by adding a number to both sides of 
the equation, so we can un-put them there by subtracting a number from 
both sides:

           3 + 3x = 9 + 2x
  
       3 + 3x - 3 = 9 + 2x - 3

     (3 - 3) + 3x = (9 - 3) + 2x
           
               3x = 6 + 2x

Now we have multiples of x on both sides. We can do the same kind of 
thing to fix that:

           3x = 6 + 2x

      3x - 2x = 6 + 2x - 2x

     (3 - 2)x = 6 + (2 - 2)x

            x = 6          

Can you follow the same kinds of steps to solve the rest of the 
problems?

I hope this helps. Write back if you'd like to talk about this some 
more, or if you have any other questions.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Middle School Algebra

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