Algebra Basics: Variables, Operations, Properties

Basic Concepts

• Variables: Symbols representing unknown values (e.g., x, y).
• Constants: Fixed values that do not change (e.g., 2, -5).
• Expressions: Combinations of variables and constants using operations (e.g., 3x + 2).
• Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).

Operations

• Addition & Subtraction: Combine or remove quantities.
• Multiplication: Repeated addition of the same number.
• Division: Splitting into equal parts; inverse of multiplication.

Properties

• Commutative Property:
  • Addition: a + b = b + a
  • Multiplication: ab = ba
• Associative Property:
  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (ab)c = a(bc)
• Distributive Property: a(b + c) = ab + ac

Solving Equations

1. Isolate the Variable: Get the variable on one side of the equation.
2. Perform Inverse Operations: Use addition/subtraction or multiplication/division to simplify.
3. Check Your Solution: Substitute back to ensure both sides are equal.

Types of Equations

• Linear Equations: Form ax + b = 0; graph is a straight line.
• Quadratic Equations: Form ax² + bx + c = 0; graph is a parabola.
• Polynomial Equations: Involves terms in the form of ax^n where n is a non-negative integer.

Functions

• Definition: A relation between a set of inputs and a set of possible outputs, defined by a rule.
• Notation: f(x) represents the function value at x.
• Types:
  • Linear Functions: f(x) = mx + b
  • Quadratic Functions: f(x) = ax² + bx + c

Graphing

• Coordinate Plane: Consists of a horizontal (x-axis) and vertical (y-axis) axis.
• Plotting Points: (x, y) represents a specific location on the graph.
• Slope: Measure of steepness; calculated as rise/run (change in y/change in x).
• Intercepts:
  • Y-intercept: Point where the graph crosses the y-axis (x=0).
  • X-intercepts: Points where the graph crosses the x-axis (y=0).

Inequalities

• Definition: Expressions showing the relationship of one expression being greater or less than another (e.g., x + 3 > 5).
• Solving: Similar to equations, but remember to reverse inequality when multiplying/dividing by a negative number.

Systems of Equations

• Definition: A set of equations with the same variables.
• Methods of Solution:
  • Graphing: Plotting each equation and finding intersection points.
  • Substitution: Solving one equation for a variable and substituting into another.
  • Elimination: Adding or subtracting equations to eliminate a variable.

Exponents and Polynomials

• Exponent Rules:
  • a^m × a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(m*n)
• Polynomial Operations: Addition, subtraction, multiplication, and division of polynomials.

Factoring

• Definition: Expressing a polynomial as a product of its factors.
• Common Techniques:
  • Factoring out the greatest common factor (GCF).
  • Using special products (e.g., difference of squares, perfect square trinomials).

Applications

• Real-world Problems: Algebra is used in various fields such as physics, engineering, economics, and statistics for modeling relationships and solving problems.

Basic Concepts

• Variables are symbols representing unknown values, commonly denoted as x or y.
• Constants are fixed numerical values that remain unchanged, such as 2 or -5.
• Expressions consist of variables and constants combined using mathematical operations, for example, 3x + 2.
• Equations are statements asserting the equality of two expressions, such as 2x + 3 = 7.

Operations

• Addition and subtraction involve combining or removing quantities, respectively.
• Multiplication can be seen as repeated addition of the same number.
• Division serves to split quantities into equal parts; it is the inverse operation of multiplication.

Properties

• Commutative Property applies to addition (a + b = b + a) and multiplication (ab = ba).
• Associative Property states that for addition: (a + b) + c = a + (b + c), and for multiplication: (ab)c = a(bc).
• Distributive Property allows for distributing multiplication over addition: a(b + c) = ab + ac.

Solving Equations

• Isolate the variable by maneuvering it to one side of the equation.
• Use inverse operations, such as addition/subtraction or multiplication/division, to simplify the equation.
• Verify your solution by substituting back into the original equation to check for equality.

Types of Equations

• Linear Equations follow the format ax + b = 0 and their graphs form straight lines.
• Quadratic Equations take the shape ax² + bx + c = 0, and their graphs yield parabolas.
• Polynomial Equations comprise terms of the form ax^n, where n is a non-negative integer.

Functions

• A function represents a relationship between input sets and output sets defined by specific rules.
• The notation f(x) indicates the value of the function at a given input x.
• Linear Functions are expressed as f(x) = mx + b, while Quadratic Functions are f(x) = ax² + bx + c.

Graphing

• The coordinate plane consists of a horizontal x-axis and a vertical y-axis.
• Points are plotted using (x, y) coordinates to represent specific locations on the graph.
• Slope is calculated as the ratio of rise to run (change in y/change in x), indicating the steepness of a line.
• Intercepts refer to points where the graph crosses the axes: y-intercept occurs when x=0, and x-intercept when y=0.

Inequalities

• An inequality shows the relationship of one expression being either greater than or less than another, like x + 3 > 5.
• Solving inequalities is similar to solving equations, but the inequality sign must be reversed when multiplying or dividing by a negative number.

Systems of Equations

• A system of equations comprises multiple equations that share the same variables.
• Solutions methods include graphing to find intersection points, substitution to replace a variable, and elimination to cancel out a variable through addition or subtraction.

Exponents and Polynomials

• Exponent rules include a^m × a^n = a^(m+n), a^m / a^n = a^(m-n), and (a^m)^n = a^(m*n).
• Polynomial operations involve addition, subtraction, multiplication, and division among polynomials.

Factoring

• Factoring means expressing a polynomial as a product of its factors.
• Common techniques include extracting the greatest common factor (GCF) and utilizing special products like the difference of squares or perfect square trinomials.

Applications

• Algebra serves as a powerful tool in diverse fields such as physics, engineering, economics, and statistics, aiding in modeling relationships and solving real-world problems.