Beginning Algebra

Date: 02/13/2003 at 19:29:47
From: Jennifer
Subject: Algebra

How come x+3 = -2? 

The answer is x = -5

I don't understand how to get to the answer -5.

Date: 02/13/2003 at 22:29:55
From: Doctor Ian
Subject: Re: Algebra

Hi Jennifer, 

As with most problems in math, there are a lot of different ways to
get the answer. Let's start with the most obvious, and work our way
to the more subtle ones, okay? 

Suppose we know that 

  x + 3 = -2

We can start trying values for x, until we find one that makes the
equation true:

   x        x + 3     = -2?
  ---    ----------   -----
   0      0 + 3 = 3    Nope
   1      1 + 3 = 4    Nope
   2      2 + 3 = 5    Nope

Not only aren't we getting the right answer, we're getting farther
away from what we want. So let's try smaller values instead of larger 
ones:

   x        x + 3     = -2?
  ---    ----------   -----
   2      2 + 3 = 5    Nope
   1      1 + 3 = 4    Nope
   0      0 + 3 = 3    Nope
  -1     -1 + 3 = 2    Nope   (But getting closer)
  -2     -2 + 3 = 1    Nope   (But getting closer)
  -3     -3 + 3 = 0    Nope   (But getting closer)
  -4     -4 + 3 = -1   Nope   (But really close!)
  -5     -5 + 3 = -2   Yes    

So even if you don't know _any_ tricks from algebra, you can solve any
equation this way, just by trying out different values.  

On the other hand, depending on the equation, this might take a
l-o-o-o-o-o-o-o-n-g time. So what we want to learn in algebra are
quicker ways to accomplish the same thing. 

The number one rule in algebra is this:  If you start with two things
that are equal,

   this = that

then if you do the same thing to both of them, 

             this + 2 = that + 2

       3 * (this + 2) = 3 * (that + 2)

  15 - 3 * (this + 2) = 15 - 3 * (that + 2)

you still end up with two equal things. (Here is one exception: You
can't divide by zero.)

Does that make sense?  (Write back if it doesn't, because if this
doesn't make sense, then nothing else in algebra is going to, either.)

Okay, so now it becomes a matter of deciding _what_ we can do to both
sides of an equation to get us closer to the form we really want,
which usually looks like 

  variable = some number

So let's look at our equation:

   x + 3 = -2

Now, one thing we can do is add -1 to each side:

  x + 3 + -1 = -2 + -1

when we simplify this, we get 

       x + 2 = -3

This is closer to what we want, so let's do it again:

  x + 2 + -1 = -3 + -1

       x + 1 = -4

And let's do it one more time:

  x + 1 + -1 = -3 + -1

           x = -5

And this tells us what the value of x has to be!  

Now, look carefully at that last step.  We had 

  x + 1

on the left side, and we added -1 to it... and we were left with only
the x. So let's look at our original equation again:

  x + 3 = -2

If we can add -1 to get rid of a 1, we should be able to add -3 to get
rid of a 3, right?  Let's try it:

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

           x = -5

So we get to the same place, only more quickly. 

Does this help?

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