### Introduction: Exactly what is algebra?

Algebra is the language of mathematics. In algebra we use letters to represent numbers.
Then we can do several things:

First, we can make statements that are generally true without having to be specific.
For example if *a* and *b* represent any two numbers then we can say that
*a + b = b + a* rather than just saying that for example
5 + 3 = 3 + 5.

Second, algebra is more brief than any human language. It is more brief to say *a + b = b + a*
than to say “*when adding two numbers together, it does not matter which number
is added to which; the result is the same*”.

Finally and most importantly, we can use algebra to solve problems.
We use **expressions**
to describe combinations of numbers and we use **equations**
to describe mathematical facts.

For example, suppose that we don’t know Dave and Bill’s ages but we do know that Dave is 6
years older than Bill. We can let *d* represent Dave’s age in years and *b*
represent Bill’s age in years and then the phrase “*6 years older than Bill *” can
be written concisely as the expression
6 + *b*,

and the fact that “*Dave is 6 years older than Bill *” can be written concisely as the equation
*d* = 6 + *b*.

Values of *b* and *d* that make this equation true are said to **satisfy** the equation. For example
the ages *b* = 3 and *d* = 9 cause the equation to read 9 = 9 and thus satisfy the equation.
So do the ages *b* = 33 and *d* = 39, which cause the equation to read 39 = 39.

If we later find out that Dave is also 3 times as old as Bill then we can express this fact
using another equation
*d* = 3 · *b*.

We now have two equations that must be satisfied **simultaneously**. This is called a
**system of equations**. You can verify that only the ages *b* = 3 and *d* = 9
satisfy both equations.

Algebra can be classified into three broad topics:
- simplifying expressions,
- manipulating expressions into other forms, and
- solving equations or systems of equation.

Simplifying expressions involves learning the properties of exponents, logarithms, various functions, etc.
Manipulating expressions involves factoring, adding fractions, completing the square, etc.
And solving equations involves learning about quadratic equations, exponential equations, etc.
Each of these categories will take about the same amount of time for you to learn (and each
takes about 10,000 lines of code to implement in the Algebra Coach.) Let’s get started!