Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

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), x2, (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:

  1. 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 = a2 + 2ab + b2
  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 x2 - 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.

Evaluating an algebraic expression

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.