Learn & Review: Basic Algebra Lessons for Beginners

Learning Algebra from Scratch: Addition, Subtraction, Multiplication, and Division

This summary covers fundamental algebraic operations: addition, subtraction, multiplication, and division, focusing on how letters (variables) and their exponents affect these operations.

Addition and Subtraction in Algebra

• Rule: Terms can only be added or subtracted if they have the exact same letters and exponents.

• Process: If terms are the same, perform the addition or subtraction on the numerical coefficients and keep the common letters and exponents.

  • Example 1: 2x + 3x can be added because both terms have x. The result is (2+3)x = 5x.
  • Example 2: 2x + 3y cannot be added because the letters (x and y) are different.
  • Example 3: 2x² + 3x³ cannot be added because the exponents are different (2 and 3).
  • Example 4: 3ab + 2ac cannot be added because although both have a, the remaining letters (b and c) are different.
  • Example 5: 5ab - 2ab can be subtracted because both terms have ab. The result is (5-2)ab = 3ab.

• Multiple Terms: Combine like terms by grouping them together.

  • Example: 2ab + 5bc + 3ab - 2bc
    • Combine ab terms: 2ab + 3ab = 5ab
    • Combine bc terms: 5bc - 2bc = 3bc
    • The final answer is 5ab + 3bc because ab and bc are different.

• The Invisible One: If a variable has no visible coefficient, it is assumed to be 1.

  • Example: ab is the same as 1ab.
  • Example: 2ab + ab = 2ab + 1ab = 3ab.
  • Example: 3ac + ac = 3ac + 1ac = 4ac.

Multiplication and Division of Negative Numbers

• Rule for Multiplication:

  • Negative × Positive = Negative
    • Example: -2 × 3 = -6
    • Example: 5 × -6 = -30
  • Negative × Negative = Positive
    • Example: -2 × -3 = 6
    • Example: -5 × -6 = 30

• Rule for Division:

  • Negative ÷ Negative = Positive
    • Example: -6 ÷ -2 = 3
  • Negative ÷ Positive = Negative
    • Example: -6 ÷ 2 = -3
  • Positive ÷ Negative = Negative
    • Example: 6 ÷ -2 = -3

Multiplication and Division in Algebra

• Key Difference from Addition/Subtraction: For multiplication and division, the letters and exponents do not need to be exactly the same.

• Multiplication Process:

  1. Multiply the numerical coefficients.
  2. Combine the letters by adding their exponents if the same letter appears multiple times.
  • Example 1: 6ab × 2ac
    • Multiply numbers: 6 × 2 = 12
    • Combine letters: a appears twice (a¹ × a¹ = a²), b appears once, c appears once.
    • Result: 12a²bc
  • Example 2: -3ab × -2bc
    • Multiply numbers: -3 × -2 = 6 (negative times negative is positive)
    • Combine letters: a appears once, b appears twice (b¹ × b¹ = b²), c appears once.
    • Result: 6ab²c
  • Order of Operations (for consistency): When multiplying multiple terms, work from left to right.
    • Example: -2a × 3ab × 2b
      • Step 1: -2a × 3ab = -6a²b
      • Step 2: -6a²b × 2b = -12a²b²

• Division Process:

  1. Divide the numerical coefficients.
  2. For each letter, subtract the exponent in the denominator from the exponent in the numerator. If a letter remains in the denominator, its exponent will be positive there.
  • Example 1: 6a²b ÷ -2ab²
    • Write as a fraction: (6a²b) / (-2ab²)
    • Divide numbers: 6 ÷ -2 = -3
    • Handle a: a² / a¹ = a⁽²⁻¹⁾ = a¹ (one a left in the numerator)
    • Handle b: b¹ / b² = b⁽¹⁻²⁾ = b⁻¹ (one b left in the denominator)
    • Result: -3a / b
  • Example 2: 15x²y⁴ ÷ 5xy⁶
    • Write as a fraction: (15x²y⁴) / (5xy⁶)
    • Divide numbers: 15 ÷ 5 = 3
    • Handle x: x² / x¹ = x⁽²⁻¹⁾ = x¹ (one x left in the numerator)
    • Handle y: y⁴ / y⁶ = y⁽⁴⁻⁶⁾ = y⁻² (two y's left in the denominator)
    • Result: 3x / y²