Introduction to Algebra (Copy)

Algebra Basics Cheat Sheet (IGCSE Mathematics 0580)
Topic: Generalised Numbers & Substitution

1. Letters Represent Generalised Numbers

• In algebra, letters (variables) are used to represent unknown or general values.
  Examples:
  • Let x represent a number.
  • 2x means 2 multiplied by x
  • a + b means the sum of two values, a and b.
• Why Use Letters?
  • To form expressions, equations, and formulas
  • To generalise rules and relationships in maths
  • To describe patterns and solve problems for any value
• Examples of Generalised Expressions:
  • Perimeter of a square = 4 × s → 4s
  • Area of rectangle = l × w
  • Total cost = number of items × price per item = n × p

2. Substitute Numbers into Expressions and Formulas

Substitution means replacing letters with actual values.

Basic Substitution Examples

• If x = 3, then:
  2x + 5 = 2(3) + 5 = 6 + 5 = 11
• If a = 4, b = 7, then:
  ab + b = (4)(7) + 7 = 28 + 7 = 35

Formula Substitution Examples

• Formula: A = l × w
  Given l = 8, w = 5
  → A = 8 × 5 = 40
• Formula: C = 2πr
  If r = 7, and use π ≈ 3.14
  → C = 2 × 3.14 × 7 = 43.96
• Formula: v = u + at
  If u = 10, a = 2, t = 5
  → v = 10 + 2×5 = 10 + 10 = 20

Tips for Substitution

• Use brackets when substituting negative numbers:
  If x = –2, 3x² becomes: 3(–2)² = 3(4) = 12
• Always follow order of operations (BIDMAS/BODMAS):
  • Brackets
  • Indices (powers)
  • Division & Multiplication
  • Addition & Subtraction

Common Expressions and Their Meanings

Application in Word Problems

• Example 1:
  “The cost, C, of buying n pens at ₨25 each”
  → C = 25n
• Example 2:
  “A car travels at speed s for time t. Distance = ?”
  → Distance = s × t

Key Exam Tip:
Always show substitution step clearly before final calculation.
Example:
If A = bh ÷ 2, b = 6, h = 10
→ A = (6 × 10) ÷ 2 = 60 ÷ 2 = 30

This topic is the foundation of algebra. It links directly to solving equations, simplifying expressions, and applying formulas in geometry, physics, and finance.