Mathematics (4024)
Topic 6 of 8Cambridge O Levels

Equations & Inequalities

Solving linear, simultaneous and quadratic equations; inequalities

Linear equations: Isolate the variable. Example: 3x + 7 = 22 → 3x = 15 → x = 5.


Simultaneous equations: Two equations, two unknowns. **Elimination:** Make coefficients equal, add/subtract. **Substitution:** Express one variable in terms of the other.

Example: 2x + y = 7, x - y = 2. Adding: 3x = 9, x = 3, y = 1.


Quadratic equations (ax² + bx + c = 0):

  • Factorising: x² + 5x + 6 = 0 → (x+2)(x+3) = 0 → x = -2 or -3
  • Quadratic formula: x = (-b ± √(b²-4ac)) / 2a
  • Completing the square: x² + 6x = (x+3)² - 9

    • Inequalities: Solve like equations BUT flip the sign when multiplying/dividing by a negative.

    Example: -2x > 6 → x < -3 (sign flips!)

    Key Points to Remember

    • 1Linear: isolate the variable
    • 2Simultaneous: elimination or substitution
    • 3Quadratic: factorise, formula, or complete the square
    • 4Inequalities: flip sign when multiplying by negative

    Pakistan Example

    Mobile Data Pricing — Simultaneous Equations in Real Life

    Jazz charges Rs. 50 base + Rs. 2/GB. Telenor charges Rs. 20 base + Rs. 4/GB. Setting up simultaneous equations: at what GB usage do both cost the same? 50 + 2x = 20 + 4x → 2x = 30 → x = 15 GB. Below 15 GB, Telenor is cheaper.

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