Algebra Basics

What is Algebra?

• 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 and model real-world situations.

Key Concepts

Variables and Expressions

• A variable is a symbol that represents a value that can change.
• An expression is a combination of variables, numbers, and operations.
• Examples: 2x, 3x + 5, x^2 - 4

Equations and Inequalities

• An equation is a statement that says two expressions are equal. (e.g. 2x + 3 = 5)
• An inequality is a statement that says one expression is greater than, less than, or equal to another. (e.g. 2x + 3 > 5)
• Equations and inequalities can be solved using various methods, such as addition, subtraction, multiplication, and division.

Functions

• A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
• Functions can be represented algebraically, graphically, or numerically.
• Examples: f(x) = 2x, f(x) = x^2 + 3

Algebraic Operations

Addition and Subtraction

• To add or subtract algebraic expressions, combine like terms.
• Example: (2x + 3) + (4x - 2) = 6x + 1

Multiplication

• To multiply algebraic expressions, use the distributive property.
• Example: (2x + 3) × (x + 2) = 2x^2 + 7x + 6

Division

• To divide algebraic expressions, use the inverse operation of multiplication.
• Example: (2x^2 + 5x + 3) ÷ (x + 1) = 2x + 3

Solving Equations and Inequalities

Linear Equations

• A linear equation is an equation in which the highest power of the variable is 1.
• Examples: 2x + 3 = 5, x - 2 = 3
• Linear equations can be solved using substitution, elimination, or graphing.

Quadratic Equations

• A quadratic equation is an equation in which the highest power of the variable is 2.
• Examples: x^2 + 4x + 4 = 0, x^2 - 4x - 3 = 0
• Quadratic equations can be solved using factoring, the quadratic formula, or graphing.

Graphing

• Graphing is a way to visually represent algebraic equations and functions.
• Graphs can be used to identify key features, such as intercepts, asymptotes, and maxima/minima.

Applications of Algebra

• Algebra is used in many real-world applications, such as:
  • Physics and engineering to model and solve problems
  • Computer science to write algorithms and code
  • Economics to model and analyze economic systems
  • Data analysis to understand and interpret data

What is Algebra?

• 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 and model real-world situations.

Variables and Expressions

• A variable is a symbol that represents a value that can change.
• An expression is a combination of variables, numbers, and operations.
• Examples of expressions include 2x, 3x + 5, and x^2 - 4.

Equations and Inequalities

• An equation is a statement that says two expressions are equal, such as 2x + 3 = 5.
• An inequality is a statement that says one expression is greater than, less than, or equal to another, such as 2x + 3 > 5.
• Equations and inequalities can be solved using various methods, such as addition, subtraction, multiplication, and division.

Functions

• A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
• Functions can be represented algebraically, graphically, or numerically.
• Examples of functions include f(x) = 2x and f(x) = x^2 + 3.

Algebraic Operations

• To add or subtract algebraic expressions, combine like terms, such as (2x + 3) + (4x - 2) = 6x + 1.
• To multiply algebraic expressions, use the distributive property, such as (2x + 3) × (x + 2) = 2x^2 + 7x + 6.
• To divide algebraic expressions, use the inverse operation of multiplication, such as (2x^2 + 5x + 3) ÷ (x + 1) = 2x + 3.

Solving Equations and Inequalities

• Linear equations are equations in which the highest power of the variable is 1, such as 2x + 3 = 5 and x - 2 = 3.
• Linear equations can be solved using substitution, elimination, or graphing.
• Quadratic equations are equations in which the highest power of the variable is 2, such as x^2 + 4x + 4 = 0 and x^2 - 4x - 3 = 0.
• Quadratic equations can be solved using factoring, the quadratic formula, or graphing.

Graphing

• Graphing is a way to visually represent algebraic equations and functions.
• Graphs can be used to identify key features, such as intercepts, asymptotes, and maxima/minima.

Applications of Algebra

• Algebra is used in many real-world applications, such as:
  • Physics and engineering to model and solve problems.
  • Computer science to write algorithms and code.
  • Economics to model and analyze economic systems.
  • Data analysis to understand and interpret data.