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Ordinary school algebra is a game of numbers that are dealt with in the abstract. For example, if I say:

x + y = 4,

I'm using algebra to say that some number, x, and some number, y, add up to 4.

The statement deals in variables. If it had specific numbers (called constants) you'd be doing arithmetic. But here I don't say what the numbers are. They could be any pair. That is, x and y are holding places for numbers. In fact, x and y could be the same number; you could have 3 + 3, for instance. And the numbers could even be fractions. Remember fractions?

When I say x + y = 4, I'm making a whole bunch of statement at the same time. The expression is a statement form, a cookie cutter.. Variables are storage places for data, and equations are repositories of informatilon. For that reason we call it a sentence form, or a class of sentences. For instance, one of the sentences in the class is:

2 + 2 = 4,

which I'm sure you recognize. And you recognize it to be true.

Another one is:

-8 + 12 = 4,

which is also true.

Still another is:

3 + 6 = 4.

Oops! Is that right? Could this be one of the sentences in the class?  Doesn't 3 plus 6 equal 9?

Well, ... sure. What I mean is that '3 + 6 = 4' is a statement in our class. But I didn't say all of the sentences had to be true. In fact, in that whole package of information, some of it, like '3 + 6 = 4,' can be wrong. We say much that passes for good information, but isn't. False statements could even be deliberate lies, and often are. But they surely are noise.

So, now we've come to the crux of it.

The point is, in algebra you set up a general form and assume that the information in the form is correct -- that the statements in the form are true, like in logic. Then, using methods of algebra, following the rules of algebra, you proceed to find the true ones.

In other words, you draw conclusions from the statement forms. You calculate, or change the forms of true statements to other forms. When you apply the algebra to problems, you hope to draw out information that solves the problems. In other words, you try to narrow the package of information down to specific sentences. Sometimes you can in fact do it. But sometimes, not. And that's the breaks!

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Suppose you have a situation you describe with:

x + y = 4,

and you need to find the value for x.

Well, if your only information is that x and y together add up to 4, good luck! There isn't anything you can do. 'x' could be any number between plus infinity and minus infinity. Since x can have any value, then correspondingly, y can have any value. There's no way to choose.

But suppose you also happen to know that y = 5. That's a big step forward! It's big enough, in fact, that now you can deduce the value for x. Since you know x, you can determine y. You only have to use the rules of algebra to do it.

First you can say that:

x + y - y = 4 - y.

(That is, you can subtract y from both sides of the equation -- or add -y to both sides.)

Then you can cancel y from the left side to get:

x = 4 - y

and substitute 5 for y on the right side to get:

x = 4 - 5.

And of course this is the same as:

x = -1.

So you've solved the problem.

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