1.2 Use the Language of Algebra

Algebraic expressions are the building blocks of algebra. They use variables, symbols, and numbers to represent mathematical relationships. Understanding how to work with these expressions is crucial for solving equations and modeling real-world scenarios.

Order of operations, substitution, and combining like terms are key skills for manipulating algebraic expressions. These techniques allow you to simplify complex expressions and evaluate them for specific values, forming the foundation for more advanced algebraic concepts.

Algebraic Expressions and Variables

Variables and algebraic symbols

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• Variables represent unknown or changing quantities in mathematical expressions and equations (x, y, z)
• Algebraic symbols represent mathematical operations such as addition (+), subtraction (-), multiplication (⋅ or × or parentheses), division (÷ or fraction notation), and equality (=)
• Constants are fixed values in an expression that do not change (3 and 5 in the expression $3x + 5$)

Order of operations in expressions

• PEMDAS defines the order in which operations are performed in an algebraic expression
  • Parentheses: Simplify expressions inside parentheses first
  • Exponents: Evaluate exponents, powers, and roots next
  • Multiplication and Division: Perform from left to right
  • Addition and Subtraction: Perform from left to right
• Simplifying $3 + 2 ⋅ (4 - 1)^2$ involves first evaluating $(4 - 1)^2 = 3^2 = 9$, then $2 ⋅ 9 = 18$, and finally $3 + 18 = 21$

Substitution for expression evaluation

• Evaluate expressions by replacing variables with given values and simplifying the resulting expression
• Evaluating $2x - 3$ when $x = 5$ involves substituting 5 for x to get $2(5) - 3$, then simplifying to $10 - 3 = 7$

Like terms in expressions

• Like terms have the same variables raised to the same exponents ($3x$ and $5x$ are like terms, but $3x$ and $5x^2$ are not)
• Combine like terms by adding or subtracting their coefficients ($3x + 5x = 8x$)
• The distributive property allows multiplying each term inside parentheses by the factor outside ($2(3x + 4) = 6x + 8$)

Verbal to algebraic conversion

• Convert verbal statements into algebraic expressions by identifying the unknown quantity and assigning a variable to represent it
• Translate words into mathematical symbols:
  • "sum" or "plus" → addition (+)
  • "difference" or "minus" → subtraction (-)
  • "product" or "times" → multiplication (⋅ or ×)
  • "quotient" or "divided by" → division (÷ or fraction notation)
• "The sum of three times a number and five" translates to $3x + 5$, where $x$ represents the unknown number
• This process of converting words to mathematical expressions is known as algebraic translation

Algebraic notation and simplification

• Algebraic notation is a system of symbols used to represent mathematical concepts and operations
• Mathematical operations are represented using specific symbols in algebraic expressions
• Simplification involves reducing an expression to its most concise form by combining like terms and performing operations
• A numerical coefficient is a number that multiplies a variable in an algebraic expression (e.g., 5 in 5x)