Algebra Basics

Algebra

Definition

• Algebra is a branch of mathematics that deals with the study of variables and their relationships.
• It involves the use of symbols, equations, and formulas to solve problems.

Key Concepts

• Variables: Letters or symbols that represent unknown values or quantities.
• Constants: Numbers that do not change in value.
• Expressions: Combinations of variables, constants, and mathematical operations.
• Equations: Statements that express the equality of two mathematical expressions.
• Inequalities: Statements that express the relationship between two mathematical expressions using greater than, less than, or equal to.

Types of Algebra

• Elementary Algebra: Deals with the basic concepts of algebra, including linear equations, quadratic equations, and functions.
• Intermediate Algebra: Covers more advanced topics, such as systems of equations, polynomial equations, and rational expressions.
• College Algebra: Focuses on advanced topics, including linear algebra, vector spaces, and eigenvalues.

Algebraic Operations

• Addition: Combining two or more expressions by adding corresponding terms.
• Subtraction: Combining two or more expressions by subtracting corresponding terms.
• Multiplication: Combining two or more expressions by multiplying corresponding terms.
• Division: Combining two or more expressions by dividing corresponding terms.

Solving Equations

• Linear Equations: Equations in which the highest power of the variable is 1.
• Quadratic Equations: Equations in which the highest power of the variable is 2.
• Systems of Equations: Sets of two or more equations that must be solved simultaneously.

Graphing

• Coordinate Plane: A two-dimensional plane with x and y axes used to graph equations.
• Graph of an Equation: A visual representation of the solutions to an equation.
• X-Intercept: The point at which the graph intersects the x-axis.
• Y-Intercept: The point at which the graph intersects the y-axis.

Algebra

Definition

• Algebra studies variables and their relationships using symbols, equations, and formulas to solve problems.

Key Concepts

• Variables represent unknown values or quantities, often denoted by letters or symbols.
• Constants are numbers that do not change in value.
• Expressions combine variables, constants, and mathematical operations.
• Equations state the equality of two mathematical expressions, often denoted by the equal sign (=).
• Inequalities express the relationship between two mathematical expressions using greater than (>), less than (<), or equal to (=).

Types of Algebra

• Elementary Algebra covers basic concepts, including linear equations, quadratic equations, and functions.
• Intermediate Algebra explores more advanced topics, including systems of equations, polynomial equations, and rational expressions.
• College Algebra focuses on advanced topics, including linear algebra, vector spaces, and eigenvalues.

Algebraic Operations

• Addition combines two or more expressions by adding corresponding terms.
• Subtraction combines two or more expressions by subtracting corresponding terms.
• Multiplication combines two or more expressions by multiplying corresponding terms.
• Division combines two or more expressions by dividing corresponding terms.

Solving Equations

• Linear Equations have the highest power of the variable as 1, e.g., 2x + 3 = 5.
• Quadratic Equations have the highest power of the variable as 2, e.g., x^2 + 4x + 4 = 0.
• Systems of Equations are sets of two or more equations that must be solved simultaneously, e.g., 2x + 3y = 7 and x - 2y = -3.

Graphing

• The Coordinate Plane is a two-dimensional plane with x and y axes used to graph equations.
• The Graph of an Equation is a visual representation of the solutions to an equation.
• The X-Intercept is the point at which the graph intersects the x-axis, often denoted by (a, 0).
• The Y-Intercept is the point at which the graph intersects the y-axis, often denoted by (0, b).