Simplify Algebraic Expressions

Algebraic expressions are simplified when all like terms are combined. Like terms are values that have the same set of variables, each raised to the same degree (or power). Constants, or numbers without a variable expression, are also like terms with one another.

Examples

Like terms: *y*^{2} and 6*y*^{2} *Both terms have the same variable, raised to the power of 2 *

Like terms: 4 and -23 *Both terms are constants: they have a definite value.*

Not like terms: *y*^{3} and 3*y* *These variables have different degrees *

Not like terms: 6*xy* and 6*x* *These terms do not have the same set of variables.*

Properties of Real Numbers

When combining like terms to simplify algebraic expressions, some basic properties of real numbers are used.

Distributive Property

The most common application of the distributive property is the distribution of multiplication over addition.

Example

Distribute 5(2*x* + 4):

*x*) + (5 × 4)

*x*+ 20

Definition of Subtraction

Subtraction can be defined as "adding the opposite." A minus sign will also be distributed across multiplication.

Example

Distribute: 2(

*x*+ 5

*y*) − 2(3

*x*+ 7

*y*)

*x*+ 10

*y*+ (-2)(3

*x*+ 7

*y*)

2

*x*+ 10

*y*+ (-6

*x*) + (-14

*y*)

Commutative and Associative Property of Addition

Note that once the distributive property has been applied, terms that are added together can be arranged in any order.

Example

Combine Like Terms: 8

*x*+ 10

*y*− 6

*x*− 14

*y*

*x*+ 10

*y*+ (-6

*x*) + (-14

*y*)

8

*x*+ (-6

*x*) + 10

*y*+ (-14

*y*)

2

*x*− 4

*y*