# 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 Alice and Bill’s ages but we do know that Bill is 6 years older than Alice. We can let

*a*represent Alice’s age in years and

*b*represent Bill’s age in years and then the phrase “

*6 years older than Alice*” can be written concisely as the expression

6 +and the fact that “a

*Bill is 6 years older than Alice*” can be written concisely as the equation

Values ofb= 6 +a

*a*and

*b*that make this equation true are said to

**satisfy**the equation. For example the ages

*a*= 10 and

*b*= 16 cause the equation to read 16 = 16 and thus satisfy the equation. So do the ages

*a*= 50 and

*b*= 56, which cause the equation to read 56 = 56.

If we later learn that Bill is 3 times as old as Alice then we can express this fact using another equation

We now have two equations that must both be satisfied.b= 3 ·a.

This is called ab= 6 +a

b= 3 ·a

**system of equations**. You can verify that only the ages

*a*= 3 and

*b*= 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.