Variables, Expressions and Equations Study Guide

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Updated on Oct 3, 2011

Introduction to Variables, Expressions and Equations

Algebra is, properly speaking, the analysis of equations.

— Joseph Alfred Serret (1819–1885)

This lesson will break down the basic algebra group of variables, expressions, and equations.

Algebra is a type of math that uses variables to represent values. If numbers were secret agents, variables would be the disguises they wear. In algebra, you can use a letter to represent a number. And if you come across a variable, you can find its real identity by using any clues you find. In algebra, letters, called variables, are often used to stand in for numbers. Variables are usually written in italics. The z in z + 21 = 8 is a variable.

Any letter can be used to represent a number in an algebraic expression. The letters x, y, and z are commonly used.

Variables can represent any quantity—an integer, a decimal, or even a fraction. A variable can also represent a positive or a negative number.

Once you realize that these variables are really numbers incognito, you'll see that they follow all the rules of mathematics, just like numbers do. This can help you figure out what number the variable you're focusing on stands for.

A term is a number or a number and the variable(s) associated with it. For example:




When a number is placed next to a variable, the number is the coefficient of the variable. When you are multiplying a number and a variable, you just have to write them side by side. You don't need to use a multiplication symbol. Let's see a few examples.

8c      8 is the coefficient of the variable c.

6ab   6 is the coefficient of both variables, a and b.

If two or more terms have exactly the same variable, they are said to be like terms. Look at these examples:

7x + 3x

45d – 31d

When you combine 12x and 14x into 26x, you are combining like terms. What does this mean? Here, 12x and 14x are considered like terms because they both involve the variable x. The numbers 3 and the 4 are called coefficients of the x term. For the previous examples,

7x + 3x = 10x

45d – 31d = 14d


A term with no coefficient actually has a coefficient of one: y = 1y.

Combine like terms carefully—check to make sure that the variables are exactly the same!


An expression is a mathematical statement that can use numbers, variables, or a combination of the two. Sometimes, there are operations, but there is never an equal sign. Expressions include

    8x + 9
    8x + 9 ÷ 2y

Notice that the value of an expression changes as the value of the variable changes. Let's see how this works:

    If x = 1, then x + 6 = 1 + 6 = 7
    If x = 2, then x + 6 = 2 + 6 = 8
    If x = 3, then x + 6 = 3 + 6 = 9
    If x = 100, then x + 6 = 100 + 6 = 106
    If x = 213, then x + 6 = 213 + 6 = 219

If you are given a phrase, you can turn it into an expression by looking for operations, variables, and numbers. Suppose you were asked for the math expression equal to the quotient of three divided by a number plus the difference between three and two.

The word quotient means to divide, so you write "3 ÷." The word number represents your variable, so use the variable x: 3 ÷ x. The word plus means to add, so now you have "3 ÷ x +." Finally, the word difference means to subtract, so the final expression will be 3 ÷ x + (3 – 2).

Sometimes, you will be able to simplify expressions (especially if there are any like terms). In order to simplify expressions, there are several things you have to know. When simplifying expressions, you should start by working inside the parentheses. Let's take a closer look:

    (3x + 4y – 7) – (5x – 3y – 7)

Start inside the parentheses. If you'd like, you can change any subtract to "plus the opposite." For example, you could change (5x – 3y – 7) to (5x + –3y + –7). Some people think that addition is easier to work with when you are combining like terms:

    3x + 4y + –7 + –5x + 3y + 7

Now, combine all the like terms, xs, and ys:

    –2x + 7y


When you simplify expressions, don't forget the order of operations from Lesson 2:

P Do operations inside Parentheses.
E Evaluate terms with Exponents. (You will learn more about exponents in Lesson 13.)
M D Do Multiplication and Division in order from left to right.
A S Add and Subtract terms in order from left to right.

Expressions have known values only when you know the values of the variables. To find the value of an expression, plug in the known values of its variables. This is called evaluating an expression. Try to evaluate 25 ÷ (x + y), when x = 2 and y = 3.

Plug the known numbers into the expression:

    25 ÷ (2 + 3)

Now, just use the order operations to solve this expression:

    25 ÷ (5)
    25 ÷ 5 = 5
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