Introduction to Algebra Concepts

Introduction to Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and variables.
• It deals with relationships between quantities and their operations like addition, subtraction, multiplication, and division.
• Algebra allows for generalizing arithmetic operations and solving for unknown values.

Basic Algebraic Concepts

• Variables: Symbols (like (x, y, z)) that represent unknown or varying quantities.
• Constants: Fixed values (like (2, 5, \pi)).
• Expressions: Combinations of variables, constants, and operators.
• Examples: (3x + 2), (y^2 - 4), (5a + b - c)
• Equations: Statements that show the equality of two expressions.
• Examples: (x + 5 = 10), (2y - 3 = 7), (a^2 + b^2 = c^2) (Pythagorean theorem)
• Inequalities: Statements that show the relationship between two expressions using symbols like >, <, ≥, ≤.
• Examples: (x > 5), (y ≤ 10), (a + b < 2c)

Solving Equations

• The goal is to isolate the variable using inverse operations.
• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.
• Exponents and roots are inverse operations.

Simplifying Expressions

• Combining like terms: Combine terms with the same variable and exponent.

Linear Equations

• Linear equations have the form (ax + b = 0), where (a) and (b) are constants and (x) is the variable.
• They represent a straight line on a graph.
• Solutions can be found by isolating the variable.

Quadratic Equations

• Quadratic equations have the form (ax^2 + bx + c = 0), where (a), (b), and (c) are constants.
• Can be solved by factoring, using the quadratic formula, or completing the square.

Polynomials

• Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
• Degree refers to the highest power of the variable in the expression.
• Examples: (x^2 + 2x + 1), (3y^3 - 4y + 5)

Factoring

• Factoring is the process of writing an expression as a product of simpler expressions.

Systems of Equations

• A set of two or more equations with multiple variables.

Introduction to Functions

• A function assigns exactly one output value to each input value.
• Notation: Usually written as (f(x)).
• Input values are often referred to as 'domain'.
• Output values are often referred to as 'range'.

Graphing

• Illustrating relationships between variables on a coordinate system.
• Plotting points using coordinates (x, y).

Exponents and Radicals

• Rules of exponents (e.g., product rule, quotient rule, power rule).
• Properties of radicals and simplifying radicals.

Word Problems

• Applying algebraic concepts to solve real-world situations.
• Translating written descriptions into equations or inequalities.