Expressions and Formulae | Year 9 Mathematics (Cambridge) PDF Download

Introduction to Algebraic Expressions

• Definition: An algebraic expression is a mathematical phrase that can include numbers, variables (letters that represent numbers), and operations (such as addition, subtraction, multiplication, and division).
• Example: The expression "three times a number increased by five" is written as 3x + 5.

Simplifying Expressions

• Combining Like Terms: To simplify an expression, combine like terms (terms with the same variable raised to the same power).
• Example: Simplify 2x + 3x − 4:
  2x + 3x = 5x
  So, 2x + 3x − 4 = 5x − 4.

Formulas and Perimeters

• Perimeter of a Rectangle: The perimeter (P) of a rectangle with length (l) and width (w) is given by: P = 2(l + w)
• Example: l = 5 and w = 3:
  P = 2(5 + 3) = 2 × 8 = 16

Solving Equations

Steps to Solve Linear Equations:

• Isolate the variable on one side of the equation.
• Simplify both sides if necessary.

Example: Solve 3x − 7 = 14:

• Add 7 to both sides:
  3x − 7 + 7 = 14 + 7
  3x = 21
• Divide by 3: 
  x = 3/21
  x = 7

Factoring Expressions

Factoring Difference of Squares: An expression of the form a² − b² can be factored as (a − b)(a + b).
Example:
Factor 4x² − 9:
4x² − 9 = (2x)² − 3²
= (2x − 3)(2x + 3)

Solving Quadratic Equations by Factoring

Steps:

• Set the equation to 0.
• Factor the quadratic expression.
• Solve for the variable by setting each factor equal to 0.

Example:
Solve x² − 16 = 0: 
Factor: x² − 16 = (x − 4)(x + 4)
Set each factor to 0: 
x − 4 = 0 or x + 4 = 0
x = 4 or x = −4

Distributive Property

• Definition: The distributive property states that a(b + c) = ab + ac.
• Example: Simplify 3(2x − 5) + 4(3x + 2):
  Distribute: 3(2x) − 3(5) + 4(3x) + 4(2)
  6x − 15 + 12x + 8
  Combine like terms:
  6x + 12x = 18x
  −15 + 8 = −7
  18x − 7

Expressing Verbal Phrases Algebraically

• Translating Phrases: To express "the sum of twice a number and seven" algebraically: 2x + 7
• Example: If the number is 4:
  2(4) + 7 = 8 + 7 = 15

Solving Proportions

• Definition: A proportion is an equation that states that two ratios are equal.
• Example: Solve 2/x = 5
  Cross-multiply:
  2 = 5x
  Solve for x:
  x = 2/5 = 0.4