Algebra Fundamentals and Operations

Fundamental Concepts

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities.
• These symbols, often letters, allow for generalizing mathematical relationships and solving problems without specifying specific numerical values.
• Algebra provides a powerful tool for modeling real-world situations mathematically.

Basic Algebraic Operations

• Addition: Commutative (a + b = b + a). Associative ((a + b) + c = a + (b + c)).
• Subtraction: Inverse of addition (a - b = a + (-b)).
• Multiplication: Commutative (a × b = b × a). Associative ((a × b) × c = a × (b × c)). Distributive (a × (b + c) = a × b + a × c).
• Division: Inverse of multiplication. (a ÷ b = a × (1/b), assuming b ≠ 0).

Variables and Expressions

• Variables: Letters or symbols that represent unknown or changeable quantities. They allow for generalizations.
• Expressions: Combinations of variables, numbers, and operational symbols (e.g., +, -, ×, ÷).
  • Examples: 2x + 3, y - 5, 4ab.
  • Expressions do not include equal signs.

Equations and Inequalities

• Equations: Statements that show that two expressions are equal (e.g., 2x + 3 = 7).
• Solutions to Equations: The values of the variables that make the equation true.
• Inequalities: Statements that show that two expressions are not equal (e.g., x > 5, y ≤ 10).
• Solving Equations and Inequalities: Methods to isolate the variable and find the values that satisfy the equation or inequality.

Linear Equations

• Linear equations represent relationships that form straight lines on a graph.
• They typically take the form ax + by = c, where a, b, and c are constants.
• Solving linear equations typically involves isolating the variable using inverse operations.

Polynomials

• Polynomials are algebraic expressions that consist of variables and coefficients.
• They often involve addition, subtraction, and multiplication, but not division by a variable.
• Types of polynomials include monomials (one term), binomials (two terms), trinomials (three terms), and so on.
• Polynomial operations involve combining like terms.

Systems of Equations

• A system of equations consists of two or more equations involving the same variables.
• Solving a system of equations involves finding values for the variables that satisfy all the equations simultaneously.
• Methods for solving include graphing, substitution, and elimination.

Factoring

• Factoring is a process of expressing an expression as a product of simpler expressions.
• Factoring can be used to solve equations and simplify expressions.
• Common factoring involves finding common factors in expressions.

Exponents and Radicals

• Exponents represent repeated multiplication, and radicals represent roots (e.g., square roots, cube roots).
• Rules exist for working with exponents and simplifying expressions with exponents and radicals.

Word Problems

• Word problems involve translating real-world situations into algebraic expressions, equations, or systems of equations to solve.
• Understanding the key relationships and translating them into mathematical terms is critical to correctly solving word problems.