Algebraic expressions are the building blocks of math equations. They combine numbers, variables, and operations to represent real-world situations. By learning to evaluate and simplify these expressions, you'll unlock the power to solve complex problems efficiently.

Translating word problems into algebraic expressions is a crucial skill. It bridges the gap between everyday language and mathematical notation, allowing you to tackle real-life scenarios using the tools of algebra. Mastering this skill opens doors to advanced problem-solving.

Evaluating and Simplifying Expressions

Evaluation of algebraic expressions

• Substitute the given values for the variables in the expression (x = 3, y = -2)
• Follow the order of operations (PEMDAS) to simplify the expression:
  • Simplify expressions within parentheses first
  • Evaluate exponents (powers, roots, etc.)
  • Perform multiplication and division from left to right
  • Perform addition and subtraction from left to right
• Simplify any remaining operations to obtain the final value (-7)

Components of algebraic expressions

• Terms are the parts of an expression separated by addition or subtraction
  • In the expression $3x + 2y - 5$, the terms are $3x$, $2y$, and $-5$
• Coefficients are the numerical factors multiplied by variables in a term
  • In the term $3x$, the coefficient is 3
  • If a term has no numerical factor, the coefficient is understood to be 1 ($x$ is the same as $1x$)
• Like terms are terms that have the same variables raised to the same powers
  • $3x$ and $5x$ are like terms, but $3x$ and $5x^2$ are not

Simplification by combining terms

• Identify like terms in the expression ($3x$ and $2x$, $2y$ and $-3y$)
• Add or subtract the coefficients of like terms to combine them ($3x + 2x = 5x$, $2y - 3y = -y$)
• Keep the same variable and exponent for the combined term ($5x$, $-y$)
• Simplify the expression by combining like terms and keeping unlike terms separate ($3x + 2y - 5 + 2x - 3y = 5x - y - 5$)
• The associative property allows regrouping of terms without changing the result (e.g., $(3x + 2x) + y = 3x + (2x + y)$)

Properties and Notation in Algebra

• Inverse operations are used to undo each other (e.g., addition and subtraction, multiplication and division)
• Algebraic notation uses letters to represent unknown quantities and numbers to represent known quantities
• Mathematical symbols are used to represent operations and relationships between quantities (e.g., +, -, ×, ÷, =, <, >)

Translating Expressions

Word phrases to algebraic expressions

• Identify the unknown quantity and assign it a variable (let x represent the unknown number)
• Translate words into mathematical operations:
  • "sum" or "plus" indicates addition ($5 + x$)
  • "difference" or "minus" indicates subtraction ($x - 3$)
  • "product" or "times" indicates multiplication ($6x$)
  • "quotient" or "divided by" indicates division ($x ÷ 4$ or $\frac{x}{4}$)
• Use the correct order of operations when translating phrases
• Examples:
  • "the sum of five and a number" translates to $5 + x$
  • "the difference between a number and three" translates to $x - 3$
  • "the product of six and a number" translates to $6x$
  • "the quotient of a number and four" translates to $x ÷ 4$ or $\frac{x}{4}$