Algebra Basics

Algebra

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations.
Key Concepts:
- Variables: Symbols (often letters) used to represent unknown values.
- Constants: Fixed values represented by numbers.
- Expressions: Combinations of variables, constants, and operators (e.g., 2x + 3).
- Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
Types of Algebra:
- Elementary Algebra: Basic operations and concepts including solving for unknowns.
- Abstract Algebra: Studies algebraic structures like groups, rings, and fields.
- Linear Algebra: Focuses on vector spaces and linear transformations.
Operations:
- Addition and Subtraction: Combining or removing values.
- Multiplication and Division: Scaling values and grouping.
- Factoring: Expressing a polynomial as a product of its factors (e.g., x^2 - 5x + 6 = (x - 2)(x - 3)).
- Exponents: Representing repeated multiplication (e.g., x^n).
Solving Equations:
- Linear Equations: Form: ax + b = c; solve for x.
- Quadratic Equations: Form: ax^2 + bx + c = 0; solutions via factoring, completing the square, or the quadratic formula.
- Inequalities: Statements that express a relationship of greater or lesser value (e.g., x + 3 > 5).
Functions:
- Definition: A relation that assigns exactly one output for each input.
- Notation: f(x), where f is the function name and x is the input variable.
- Types: Linear, quadratic, polynomial, exponential, logarithmic.
Graphing:
- Coordinate System: A two-dimensional space defined by x (horizontal) and y (vertical) axes.
- Plotting Points: Each point is represented as (x, y).
- Graphing Functions: Visual representation of functions on the coordinate plane.
Polynomials:
- Definition: An expression involving a sum of powers in one or more variables multiplied by coefficients.
- Degree: The highest power of the variable (e.g., 3x^4 + 2x^3 + 1; degree is 4).
Systems of Equations:
- Definition: A set of equations with the same variables.
- Methods of Solution: Substitution, elimination, and graphing.
Key Formulas:
- Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a.
- Slope of a Line: m = (y2 - y1) / (x2 - x1).
- Point-Slope Form: y - y1 = m(x - x1), where m is the slope.
Applications: Algebra is foundational for advanced mathematics, physics, engineering, economics, and various fields that involve quantitative analysis.

Algebra: A Basic Introduction

Definition: Algebra is a branch of math concerned with symbols and manipulating them to solve equations. It's used in various fields like physics, engineering, and economics.
Variables: Symbols (often letters) that represent unknown values in equations, like "x" or "y."
Constants: Numbers that have fixed values within equations, for example, 2, 3, or -5.
Expressions: Combinations of variables, constants, and mathematical operations. Examples include things like: 2x + 3 or 5y^2 - 1.
Equations: Two expressions set equal to each other, like 2x + 3 = 7. Solving an equation means finding the value of the unknown variable.
Types of Algebra:
- Elementary Algebra: Focuses on basic operations like adding, subtracting, multiplying, and dividing.
- Abstract Algebra: Deals with more complex algebraic structures, focusing on concepts like groups, rings, and fields.
- Linear Algebra: Focuses on vector spaces and linear transformations, important in various fields like physics and computer science.
Operations:
- Addition and Subtraction: Used to combine or decrease values within expressions.
- Multiplication and Division: Used to scale values and group them.
- Factoring: Expressing a polynomial as a product of its factors. For example, x^2 - 5x + 6 = (x - 2)(x - 3).
- Exponents: Represent repeated multiplication, like x^n which means multiplying x by itself "n" times.
Solving Equations:
- Linear Equations: Equations in the form ax + b = c, where 'a' and 'b' are constants and 'x' is the unknown variable. These can be solved by isolating the unknown variable (x) using algebraic manipulations.
- Quadratic Equations: Equations in the form ax^2 + bx + c = 0. These can be solved using different methods like:
  - Factoring: Breaking down the equation into its factors.
  - Completing the Square: Transforming the equation to a perfect square.
  - Quadratic Formula: A general formula to solve for the solutions given the coefficients (a, b, c) of the equation.
- Inequalities: Expressions that indicate a relationship between two values. For example, x + 3 > 5 means that the value of x plus 3 is greater than 5.
Functions:
- Definition: A relationship that assigns a specific output for each input.
- Notation: Represented as f(x), where "f" is the function name and "x" is the input variable.
- Types: Many types of functions exist, including linear functions, quadratic functions, polynomial functions, exponential functions, and logarithmic functions.
Graphing:
- Coordinate System: A two-dimensional space with a horizontal (x) axis and a vertical (y) axis.
- Plotting Points: Each point is represented by its coordinates (x, y).
- Graphing Functions: A visual representation of a function, showing how the output (y) changes with the input (x) on the coordinate plane.
Polynomials:
- Definition: An expression involving a sum of terms where each term is a product of a coefficient and one or more variables raised to non-negative integer powers.
- Degree: The highest power of the variable in a polynomial. For example, in the polynomial 3x^4 + 2x^3 + 1, the degree is 4.
Systems of Equations:
- Definition: A set of two or more equations with the same variables. Solving a system of equations means finding the values of the variables that satisfy all the equations in the system.
- Solution Methods: There are various methods for solving systems of equations, including:
  - Substitution: Solving one equation for a variable and substituting it into the other equation.
  - Elimination: Adding or subtracting equations to cancel out a variable.
  - Graphing: Plotting the equations on a coordinate plane to determine their intersection point.
Key Formulas:
- Quadratic Formula: A formula used to solve for the solutions (x) of a quadratic equation (ax^2 + bx + c = 0), given by: x = [-b ± √(b² - 4ac)] / 2a.
- Slope of a Line: The rate of change of a line, calculated as the ratio of the vertical change to the horizontal change between two points on the line: m = (y2 - y1) / (x2 - x1).
- Point-Slope Form: An equation for a line given a point (x1, y1) on the line and its slope (m): y - y1 = m(x - x1).