Algebra Fundamentals

Algebra

• Definition: Branch of mathematics dealing with symbols and the rules for manipulating those symbols.

• Key Concepts:

  • Variables: Symbols (often x, y, z) representing unknown values.
  • Constants: Fixed values (e.g., numbers like 2, -5).
  • Expressions: Combinations of variables, constants, and operations (e.g., 3x + 2).
  • Equations: Mathematical statements asserting the equality of two expressions (e.g., 2x + 3 = 7).

• Operations:

  • Addition and Subtraction: Combine or remove terms.
  • Multiplication and Division: Scale terms or split them.
  • Exponents: Indicates repeated multiplication of a base (e.g., x² means x multiplied by itself).

• Types of Algebra:

  • Elementary Algebra: Basics of algebraic equations and operations.
  • Abstract Algebra: Studies algebraic structures like groups, rings, and fields.
  • Linear Algebra: Focuses on vector spaces and linear mappings.

Linear Equations

• Definition: An equation in which the highest exponent of the variable is one, forming a straight line when graphed.

• Standard Form: Ax + By = C, where A, B, and C are constants.

• Slope-Intercept Form: y = mx + b, where:

  • m = slope of the line (rise over run).
  • b = y-intercept (point where the line crosses the y-axis).

• Graphing Steps:

  1. Identify slope (m) and y-intercept (b).
  2. Plot the y-intercept on the graph.
  3. Use the slope to find another point.
  4. Draw a straight line through the points.

• Solutions:

  • A linear equation can have:
    • One solution: Intersects the line at one point (consistent).
    • No solution: Parallel lines (inconsistent).
    • Infinitely many solutions: Same line represented in different forms (dependent).

• Systems of Linear Equations:

  • Set of two or more linear equations with the same variables.
  • Can be solved using:
    • Graphical Method: Graph each equation and identify intersection.
    • Substitution Method: Solve one equation for a variable and substitute into the other.
    • Elimination Method: Add or subtract equations to eliminate a variable.

Algebra

• A branch of mathematics focusing on symbols and their manipulation rules.
• Variables are symbols like x, y, and z that represent unknown values.
• Constants are fixed values, such as numbers like 2 or -5, that do not change.
• Expressions combine variables, constants, and operations, for example, 3x + 2.
• Equations are mathematical assertions of equality between two expressions, such as 2x + 3 = 7.
• Operations include:
  • Addition and Subtraction: Combine or remove terms from expressions.
  • Multiplication and Division: Scale terms or divide them.
  • Exponents: Represent repeated multiplication of a base, e.g., x² is x multiplied by itself.
• Types of Algebra encompass:
  • Elementary Algebra: Covers the fundamentals of equations and operations.
  • Abstract Algebra: Studies structures such as groups, rings, and fields.
  • Linear Algebra: Focused on vector spaces and linear mappings.

Linear Equations

• Defined as equations where the highest variable exponent is one, creating a straight line when graphed.
• Standard Form is expressed as Ax + By = C, with A, B, and C as constants.
• Slope-Intercept Form is y = mx + b, with:
  • m indicating the slope of the line (rise/run).
  • b representing the y-intercept (where the line crosses the y-axis).
• Graphing Steps involve:
  • Identifying the slope (m) and y-intercept (b).
  • Plotting the y-intercept on a graph.
  • Using the slope to determine another point on the line.
  • Drawing a straight line through the identified points.
• Solutions for linear equations can be:
  • One solution: The line intersects at a single point (consistent).
  • No solution: The lines are parallel (inconsistent).
  • Infinitely many solutions: The same line represented in multiple forms (dependent).
• Systems of Linear Equations consist of two or more linear equations sharing variables, solvable by:
  • Graphical Method: Graphing each equation to find intersections.
  • Substitution Method: Solving one equation for a variable and substituting into another.
  • Elimination Method: Adding or subtracting equations to eradicate a variable.

Geometry Overview

• Geometry is a mathematical discipline focused on the characteristics and relationships of points, lines, surfaces, and solids.
• Key elements include points (dimensionless locations), lines (one-dimensional figures extending infinitely), and planes (two-dimensional surfaces that extend infinitely).

Two-Dimensional Shapes

• Two-dimensional shapes are flat figures characterized solely by length and width, lacking depth.

Common 2-D Shapes

• Circle:

  • Defined by all points being equidistant from a central point.
  • Important properties include radius, diameter, and circumference.

• Triangle:

  • A three-sided polygon with three fundamental types:
    • Equilateral (equal sides)
    • Isosceles (two equal sides)
    • Scalene (all sides different)
  • The sum of interior angles in a triangle is always 180 degrees.

• Square:

  • A four-sided polygon (quadrilateral) with all sides equal and angles of 90 degrees.
  • Area calculated as side squared (Area = side²) and perimeter as four times the side (Perimeter = 4 × side).

• Rectangle:

  • A quadrilateral with opposite sides equal and all angles measuring 90 degrees.
  • Area determined by length times width (Area = length × width) and perimeter by doubling the sum of length and width (Perimeter = 2(length + width)).

• Polygon:

  • A closed figure formed by three or more straight sides; examples include:
    • Pentagon (5 sides)
    • Hexagon (6 sides)
    • Heptagon (7 sides)

• Trapezoid (US)/Trapezium (UK):

  • A quadrilateral possessing at least one pair of parallel sides.

• Rhombus:

  • A quadrilateral with equal-length sides but not required to have right angles.

• Parallelogram:

  • A quadrilateral with both pairs of opposite sides parallel and equal in length.

Properties of 2-D Shapes

• Area: Represents the total space enclosed within a shape.
• Perimeter: The total distance surrounding a shape.
• Symmetry: A shape is symmetrical if it can be split into two parts that mirror each other.

Applications

• The principles of geometry find practical use in diverse fields such as architecture, engineering, graphic design, and everyday problem-solving tasks.