What is Algebra?

Study Notes

Algebra is a branch of mathematics that deals with the study of variables and their relationships, often expressed through symbols, equations, and functions.
It involves the use of mathematical operations to solve equations, manipulate expressions, and model real-world problems.

Variables and Constants:
- Variables: letters or symbols that represent unknown values or quantities
- Constants: numbers or values that do not change
Algebraic Expressions:
- Simplified expressions consisting of variables, constants, and mathematical operations (e.g., 2x + 3)
Equations and Inequalities:
- Equations: statements that two algebraic expressions are equal (e.g., 2x + 3 = 5)
- Inequalities: statements that one algebraic expression is greater than, less than, or equal to another (e.g., 2x + 3 > 5)

Addition and Subtraction:
- Commutative Property: a + b = b + a
- Associative Property: (a + b) + c = a + (b + c)
Multiplication and Division:
- Commutative Property: a × b = b × a
- Associative Property: (a × b) × c = a × (b × c)
- Distributive Property: a × (b + c) = a × b + a × c
Exponents and Roots:
- Exponents: a^2 means "a squared"
- Roots: √a means "square root of a"

Linear Equations:
- One variable, degree 1 (e.g., 2x + 3 = 5)
- Solution methods: addition, subtraction, multiplication, and division
Quadratic Equations:
- One variable, degree 2 (e.g., x^2 + 4x + 4 = 0)
- Solution methods: factoring, quadratic formula
Systems of Equations:
- Two or more equations with multiple variables
- Solution methods: substitution, elimination, and graphing

Graphs:
- Visual representations of equations and functions on a coordinate plane
Functions:
- Relations between a set of inputs (domain) and a set of possible outputs (range)
- Notation: f(x) = output value for input x

Algebra is a branch of mathematics that deals with the study of variables and their relationships.
It involves the use of mathematical operations to solve equations, manipulate expressions, and model real-world problems.

Variables and Constants: • Variables represent unknown values or quantities and are expressed through letters or symbols. • Constants are numbers or values that do not change.
Algebraic Expressions: • Simplified expressions consist of variables, constants, and mathematical operations (e.g., 2x + 3).
Equations and Inequalities: • Equations are statements that two algebraic expressions are equal (e.g., 2x + 3 = 5). • Inequalities are statements that one algebraic expression is greater than, less than, or equal to another (e.g., 2x + 3 > 5).

Addition and Subtraction: • The Commutative Property of addition states that a + b = b + a. • The Associative Property of addition states that (a + b) + c = a + (b + c).
Multiplication and Division: • The Commutative Property of multiplication states that a × b = b × a. • The Associative Property of multiplication states that (a × b) × c = a × (b × c). • The Distributive Property states that a × (b + c) = a × b + a × c.
Exponents and Roots: • Exponents are used to represent repeated multiplication (e.g., a^2 means "a squared"). • Roots are used to represent the inverse operation of exponents (e.g., √a means "square root of a").

Linear Equations: • Linear equations have one variable and a degree of 1 (e.g., 2x + 3 = 5). • Solution methods for linear equations include addition, subtraction, multiplication, and division.
Quadratic Equations: • Quadratic equations have one variable and a degree of 2 (e.g., x^2 + 4x + 4 = 0). • Solution methods for quadratic equations include factoring and the quadratic formula.
Systems of Equations: • Systems of equations consist of two or more equations with multiple variables. • Solution methods for systems of equations include substitution, elimination, and graphing.

Graphs: • Graphs are visual representations of equations and functions on a coordinate plane.
Functions: • Functions are relations between a set of inputs (domain) and a set of possible outputs (range). • Functions are denoted using notation f(x) = output value for input x.