Simplifying Linear Expressions and Algebraic Laws

Algebraic Laws for Addition and Multiplication

• Commutative Law for Addition: $a + b = b + a$
• Commutative Law for Multiplication: $ab = ba$
• Associative Law for Addition: $a + (b + c) = (a + b) + c$
• Associative Law for Multiplication: $a(bc) = (ab)c$

Terminology and Definitions

• Term: A constant, a variable, or a product of a constant and one or more variables raised to powers.
• Variable Terms: Terms containing variables.
• Constant Terms: Terms that do not contain variables.
• Equivalent Expressions: Two or more expressions that yield equal results when evaluated for any real number for which they are defined.

The Distributive Law and Opposites

• Distributive Law: $a(b + c) = ab + ac$
• Negation Rule: $-1a = -a$, meaning $-1$ times a number is equal to the opposite of that number.

Simplifying Expressions

• Combining Like Terms: Add the coefficients of the terms and keep the variable factors the same.
• Procedure: Expressions are simplified by removing parentheses (distributing) and combining like terms.