## What is Algebra

You know what Numbers are, and how to
combine them using the basic operations of addition, subtraction,
multiplication and division. That field of mathematics is called
**Arithmetic**.
The more advanced field of **Algebra** differs from arithmetic in that
in addition to specific numbers it involves entities called **variables** that have no particular value,
or an unknown value. These are usually denoted by upper or lower case
letters.
An **algebraic expression** is a
collection of letters and numbers combined by the four basic arithmetic operations. Here are some examples of algebraic expressions:

7x, 3x+y, 3x-4y, x/(x+y), x^{2}, (x+y)^{2}

The numbers in algebraic expression are called ** constants**.

If there are no variables in the algebraic expression then it
is called an **arithmetic expression**
(such as *3+4/7*).

Why bother with variables? They can be used to describe general
situations, and they can be used to solve problems that otherwise
would be much more difficult or even impossible. You'll see these
applications in action during this course, particularly when we
discuss and solve **word problems.**

An **equation** is an assertion that two
algebraic expressions are equal. This can have two different
meanings:

- The equation is true for all values of the variables. In that
case the equation is called an
**identity**. An example of an identity is
a + b = b + a

for all numbers *a* and *b*.
This is called **the commutative law of addition**. A less well known
identity is **the first binomial formula**:
(a+b)^{2} = a^{2} + 2ab + b^{2}

- The equation is true for
**some** values of the variables. In
that case the task is often to figure out the values of the variables
for which the equation is true. That is called **solving the equation**. We will spend a lot
of time studying ways of solving equations. An example of an equation
is
3x + 1 = 4

in which case obviously x = 1

is the solution. This is an example of a **linear equation.**
An example of a **quadratic equation** is
x^{2} - x - 2 = 0.
Less obviously this equation has the two solutions
x = -1 or x = 2.

In this class we will study how to manipulate algebraic
expressions, often with the purpose of solving certain equations.

To evaluate an algebraic expression means to substitute specific values
for its variables. Consider, for example, the (very simple) algebraic
expression E=2x+1

We give it the name *E* to be able
to refer to it easily. Evaluating *E*, for example at
*x=3*, gives *E= 2*3+1 = 7*. We say that the **value** of
*E* at *x=3* is *7*. We could actually evaluate
*E* at another expression. For example, evaluating *E*
at *x=2y+1*, where *y* is another variable, gives
E = 2*(2y+1)+1 = 4y+2+1 = 4y+3

## Equivalent Expressions

Two expressions are equivalent if their values are equal for all
possible evaluations of the two expressions. In other words, listing
them with an equality sign between them gives an identity.