Basic Rules of Algebra

Vocabulary of Algebraic Expressions

Understanding the vocabulary of algebraic expressions is foundational for manipulating and simplifying expressions in algebra. Key terms include terms, coefficients, and like terms.

• Term: A part of an algebraic expression separated by addition or subtraction. For example, in the expression 2x + 8, both 2x and 8 are terms.

• Coefficient: The numerical factor of a term. In 2x, the coefficient is 2.

• Like Terms: Terms that have exactly the same variable part. For example, 5x and 8x are like terms, but 2x and 3y are not.

Example: In the expression 6x + 2x + 11:

• There are 3 terms.

• The coefficient of the first term is 6.

• The constant term is 11.

• 6x and 2x are like terms.

Commutative Properties

The commutative properties state that the order in which you add or multiply numbers does not affect the result.

• Commutative Property of Addition:

• Commutative Property of Multiplication:

Example:

• Using the commutative property of addition: x + 14 = 14 + x

• Using the commutative property of multiplication: 7y = y7

Associative Properties

The associative properties state that the way in which numbers are grouped in addition or multiplication does not affect the sum or product.

• Associative Property of Addition:

• Associative Property of Multiplication:

Example:

• 8 + (x + 4) = (8 + 4) + x = 12 + x

• 6(5x) = (6 \cdot 5)x = 30x

The Distributive Property

The distributive property allows you to multiply a sum by multiplying each addend separately and then adding the products.

• Distributive Property:

Examples:

• 5(x + 3) = 5x + 15

• 6(4y + 7) = 24y + 42

Combining Like Terms

To combine like terms, add or subtract the coefficients of terms that have the same variable part.

• 7x + 3x = (7 + 3)x = 10x

• 9a – 4a = (9 – 4)a = 5a

Grouping and Combining Like Terms

When simplifying expressions, rearrange and group like terms using the commutative and associative properties, then combine them.

• 2x + 8 + 5x + 7 = (2x + 5x) + (8 + 7) = 7x + 15

• 3y + 2x + 4x + 7y = (2x + 4x) + (3y + 7y) = 6x + 10y

Simplifying Algebraic Expressions

To simplify an algebraic expression:

1. Use the distributive property to remove parentheses.

2. Rearrange and group like terms using the commutative and associative properties.

3. Combine like terms by adding or subtracting their coefficients.

Example: Simplify 7(2x + 3) + 11x

Example: Simplify 7(4x + 3y) + 2(5x + y)