Algebra Class: Variables and Equations

Variables and Expressions

• Variables are symbols (often letters) representing unknown or varying values.
• Example: x + 5 = 10, where x is unknown.
• An expression combines numbers, variables, and operations (e.g., 3x + 4).
• Simplifying expressions means reducing them to their simplest form (e.g., 2x + 3x = 5x).
• Commutative property: a + b = b + a and ab = ba

Solving Equations

• An equation states that two expressions are equal (e.g., 2x + 3 = 7).
• Goal: Find the variable value that makes the equation true.
• Steps:
  • Simplify both sides if needed.
  • Isolate the variable using inverse operations (addition, subtraction, multiplication, division).
  • Check the solution in the original equation.
• Example: Solve 3x − 4 = 8. Solution: x = 4

Inequalities

• Inequalities show relationships (greater than, less than, etc.) between expressions (e.g., x > 5, y ≤ 3).
• Solving inequalities is similar to equations, with one key difference: reversing the inequality sign when multiplying or dividing by a negative number.
• Example: Solve −2x ≥ 6. Solution: x ≤ −3

Functions

• A function assigns each input to exactly one output.
• Often written as f(x), where x is the input and f(x) is the output.
• Example: If f(x) = 2x + 1, then f(2) = 5 and f(0) = 1.
• Functions describe relationships between variables.

Linear Equations

• Linear equations graph as straight lines.
• Standard form: y = mx + b, where:
  • m is the slope (rate of change).
  • b is the y-intercept (where the line crosses the y-axis).
• Slope calculation: m = (y₂ - y₁) / (x₂ - x₁)
• Example: For points (1, 2) and (3, 6), m = 2.

Graphing Linear Equations

• Plot the y-intercept (b).
• Use the slope (m) to find other points (rise/run).
• Draw a line through the points.

Systems of Equations

• A system is two or more equations with the same variables.
• Goal: Find values that satisfy all equations.
• Solving methods:
  • Substitution: Solve one equation for one variable and substitute.
  • Elimination: Add or subtract equations to eliminate a variable.
  • Graphing: Find the intersection point.
• Example: Solve x + y = 5 and x − y = 1. Solution: x = 3, y = 2

Polynomials

• Polynomials are expressions with terms where variables have non-negative integer powers (e.g., 2x² + 3x + 4).
• Operations:
  • Addition/Subtraction: Combine like terms.
  • Multiplication: Use the distributive property or FOIL.

Factoring

• Factoring reverses multiplication (e.g., x² − 5x + 6 = (x − 2)(x − 3)).

Quadratic Equations

• Quadratic equations have a degree of 2 (e.g., ax² + bx + c = 0).
• Solving methods:
  • Factoring
  • Completing the square
  • Quadratic formula

Exponents and Radicals

• Rules of exponents (e.g., aᵐ ⋅ aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ).
• Radicals are the opposite of exponents (e.g., √25 = 5, √a² = a).

Word Problems

• Translate real-world situations into algebraic equations.
• Steps:
  • Define variables.
  • Write an equation.
  • Solve and interpret the result.
• Example: Cost of car rental: $50 + $0.20 per mile; total cost = $90. Find miles driven. Solution: 200 miles.