Introduction to Algebra Concepts

Introduction to Algebra

• Algebra is a branch of mathematics that uses symbols, often letters, to represent numbers and quantities in mathematical expressions and equations.
• It deals with finding unknown values, often represented by variables.
• It extends arithmetic by introducing symbols to represent unknowns and generalizing arithmetic operations and relationships.

Basic Algebraic Concepts

• Variables: Symbols (usually letters) that represent unknown numerical values. Examples: x, y, z.
• Constants: Fixed numerical values. Examples: 2, 5, -10.
• Expressions: Combinations of variables, constants, and mathematical operations. Examples: 2x + 3, y² - 4.
• Equations: Statements that show the equality of two expressions. Examples: 2x + 5 = 11, y²-4 = 0.
• Inequalities: Statements that show the relationship between two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to). Examples: x < 5, y ≥ -2.
• Terms: Parts of an expression separated by addition or subtraction signs. Examples: In the expression 3x² - 2x + 5, the terms are 3x², -2x, and 5.

Properties of Algebra

• Commutative Property: The order of addition or multiplication of numbers does not change the result. Examples: a + b = b + a and ab = ba
• Associative Property: The grouping of numbers in addition or multiplication does not change the result. Examples: (a + b) + c = a + (b + c) and (ab)c = a(bc)
• Distributive Property: Allows expanding expressions by multiplying a factor into each term within parentheses. Example: a(b + c) = ab + ac

Solving Equations

• Simplifying expressions: Using the commutative, associative, and distributive properties to rewrite expressions in a more manageable form.
• Isolating the variable: Manipulating equations to put the variable on one side of the equal sign and numerical values on the other side. Often involves addition, subtraction, multiplication, and division.
• Methods for solving:
  • Applying the properties of equality (adding, subtracting, multiplying, or dividing both sides of the equation by the same value).
  • Using inverse operations to cancel terms.

Linear Equations

• Equations that represent a straight line on a graph.
• General form: Ax + By = C
• Contain variables raised to the power of 1.

Solving Systems of Equations

• A set of two or more equations with the same variables.
• Methods for solving:
  • Graphing: Plotting each equation and finding the intersection point.
  • Substitution: Solving one equation for a variable and substituting the value into another equation.
  • Elimination (addition/subtraction): Using addition or subtraction to eliminate a variable from the equations.

Polynomials

• Expressions consisting of variables and constants connected by addition, subtraction, and multiplication. Examples: 2x² + 3x - 1.
• Types of polynomials:
  • Monomials: One-term expressions (e.g., 3x, 5y²).
  • Binomials: Two-term expressions (e.g., x + 2, y² - 3).
  • Trinomials: Three-term expressions (e.g., x² + 2x + 1).

Factoring Polynomials

• Expressing a polynomial as a product of simpler polynomials.

Quadratic Equations

• Equations with a variable raised to the power of 2.
• General form: ax² + bx + c = 0. Example: 2x² + 5x - 3 = 0
• Methods for solving:
  • Factoring.
  • Quadratic formula

Functions

• A relationship between inputs (independent variables) and outputs (dependent variables) where each input has exactly one output.

Further Algebraic Concepts

• Exponents and radicals.
• Logarithms.
• Matrices and determinants.
• Polynomials and their graph theory.