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Algebraic Expressions

by Ron Kurtus (29 March 2008)

An expression in Algebra is a combination of numbers, variables, constants, operators and parentheses or grouping brackets that are stated as an entity. An equation consists of an expression on each side of the equals sign. Some expressions are made up of sub-expressions. Much of Algebra concerns simplifying expressions to facilitate the solution of equations.

Questions you may have are:

• What is the definition of expression?
• What are sub-expressions?
• How do you simplify expressions?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Definition of expressions

An expression consists of a combination of numbers, variables, constants, operators and/or parentheses. Examples include:

3

x

2y − 3

x² + (4x + 2)/2

25 + (x + 3)³/2z − [5 + z(x+ 1) − 7]

Operator alone is not an expression

Although an individual number, variable or constant can be an expression, an operator by itself is not an expression. In other words, 3, x and c can be expressions, but + and ÷ are not expressions.

Expressions used in equations

An equation consists of an expression on each side of the equals sign.

The equation x² + (3y + 2)/2 = 7z consists of the expressions x² + (3y + 2)/2 and 7z.

Sub-expressions

An expression can consist of several expressions or sub-expressions.

For example, the expression 2y − 3 consists of sub-expressions 2y and 3. Also 2y consists of sub-expressions 2 and y.

Likewise, x² + 2x + 1 consists of sub-expressions x², 2x, and 1. And x² consists of sub-expression x (since x² is x times x). And 2x consists of sub-expressions 2 and x.

Exercise example

What are the expressions in the equation y + 1 = x² + x(y + 1)?

Expressions on each side of the equation are y + 1 and x² + x(y + 1)

Expressions in y + 1 are y and 1

Expressions in x² + x(y + 1) are x² and x(y + 1)

Expression in x² is x

Expressions in x(y + 1) are x and (y + 1)

Expressions in (y + 1) are y and 1

Simplifying expressions

Expressions are meant to be used in equations. Thus, the expressions should be meaningful and in their simplest form to facilitate solving the equation. Much of Algebra concerns putting expressions in meaningful forms.

For example, the expression, x² + (4x + 2)/2 can be reduced to x² + 2x + 1 by completing the division by 2 in the expression (4x + 2)/2.

In some cases, it is useful to put an expression in a less simplified form, to facilitate the solution of an equation. For example, x² + 2x could be factored into x(x + 2) to make it easier to solve the equation x² + 2x = 0.

Summary

An expression is a combination of numbers, variables, constants, operators and parentheses or grouping brackets that are stated as an entity. An operator or grouping bracket alone is not an expression. An equation consists of an expression on each side of the equals sign. Some expressions are made up of sub-expressions. Much of Algebra concerns simplifying expressions to facilitate the solution of equations.

Answers to Readers' Questions

Be expressive when explaining things

Resources

The following resources provide information on this subject:

Websites

Algebra Resources

Books

Top-rated Algebra books

Miscellaneous

Mini-quiz to check your understanding

1. What are two expressions of the equation 3x + 1 = y?

2. What is more usable form of the expression (x − x)/2 + y?

3. In the expression 6x/3 + (5 − z), which sub-expression would you simplify?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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