Equations: Definition, Types, and Solving

Defining Equations: An equation is a statement that says two expressions are equal.
Types of Equations:
- Linear Equations: ax + by = c, where a, b, and c are constants.
- Quadratic Equations: ax^2 + bx + c = 0, where a, b, and c are constants.
- Exponential Equations: a^x = b, where a is a constant and x is the variable.
Solving Equations:
- Addition and Subtraction Properties: Add or subtract the same value to both sides of the equation.
- Multiplication and Division Properties: Multiply or divide both sides of the equation by the same non-zero value.

Defining Functions: A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
Types of Functions:
- Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
- Quadratic Functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Exponential Functions: f(x) = a^x, where a is a constant.
Function Operations:
- Composition: (f ∘ g)(x) = f(g(x))
- Inverse: f^(-1)(x) is the function that reverses f(x)

Defining Systems of Equations: A system of equations is a set of two or more equations with variables.
Methods for Solving Systems:
- Substitution Method: Solve one equation for one variable and substitute into the other equation.
- Elimination Method: Add or subtract the equations to eliminate one variable.
- Graphing Method: Graph the equations on the same coordinate plane and find the point of intersection.

Coordinate Plane: A graph with an x-axis and a y-axis that intersect at the origin (0,0).
Graphing Linear Equations:
- Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
- Standard Form: Ax + By = C, where A, B, and C are constants.
Graphing Quadratic Equations:
- Vertex Form: f(x) = a(x - h)^2 + k, where (h,k) is the vertex.
- Factored Form: f(x) = a(x - r)(x - s), where r and s are the x-intercepts.

Defining Quadratic Equations: A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
Methods for Solving Quadratic Equations:
- Factoring: Factor the equation into the product of two binomials.
- Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
- Graphing: Graph the equation and find the x-intercepts.

Algebraic Properties:
- Commutative Property: a + b = b + a
- Associative Property: (a + b) + c = a + (b + c)
- Distributive Property: a(b + c) = ab + ac
Inequalities:
- Linear Inequalities: ax + by >, <, ≥, or ≤ c
- Quadratic Inequalities: ax^2 + bx + c >, <, ≥, or ≤ 0
Functions and Relations:
- Domain and Range: The set of inputs and possible outputs of a function.
- Function Operations: Composition, Inverse, and Identity.
Systems of Equations and Inequalities:
- Substitution and Elimination Methods for solving systems of equations.
- Graphing systems of inequalities on a coordinate plane.

Addition and Subtraction Properties: Add or subtract the same value to both sides of the equation.
Multiplication and Division Properties: Multiply or divide both sides of the equation by the same non-zero value.

A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
Quadratic Functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
Exponential Functions: f(x) = a^x, where a is a constant.
Function Operations:
- Composition: (f ∘ g)(x) = f(g(x))
- Inverse: f^(-1)(x) is the function that reverses f(x)

A system of equations is a set of two or more equations with variables.
Methods for Solving Systems:
- Substitution Method: Solve one equation for one variable and substitute into the other equation.
- Elimination Method: Add or subtract the equations to eliminate one variable.
- Graphing Method: Graph the equations on the same coordinate plane and find the point of intersection.

Coordinate Plane: A graph with an x-axis and a y-axis that intersect at the origin (0,0).
Graphing Linear Equations:
- Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
- Standard Form: Ax + By = C, where A, B, and C are constants.
Graphing Quadratic Equations:
- Vertex Form: f(x) = a(x - h)^2 + k, where (h,k) is the vertex.
- Factored Form: f(x) = a(x - r)(x - s), where r and s are the x-intercepts.

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
Methods for Solving Quadratic Equations:
- Factoring: Factor the equation into the product of two binomials.
- Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
- Graphing: Graph the equation and find the x-intercepts.