Mathematics (AKU-MTH)
Topic 2 of 7Aga Khan Board

Algebra & Equations

Linear, simultaneous and quadratic equations with real-world applications

Algebra uses letters to represent unknown numbers so we can solve problems without knowing every value upfront.


Linear equations have one unknown, raised to the power 1. Solve by isolating the variable:

  • 4x + 9 = 25 → 4x = 16 → x = 4

    • Simultaneous equations — two equations, two unknowns. Use substitution or elimination:

  • Elimination: add/subtract equations to cancel one variable
  • Substitution: express x in terms of y, then substitute

    • Quadratic equations (ax² + bx + c = 0) — three methods:

  • Factorising: x² + 7x + 12 = 0 → (x + 3)(x + 4) = 0 → x = -3 or x = -4
  • Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
  • Completing the square: useful when factorising is difficult

    • Key rule for inequalities: When you multiply or divide both sides by a **negative number**, flip the inequality sign:

  • −3x > 9 → x < −3

    • Key Points to Remember

    • 1Isolate the variable to solve linear equations
    • 2Simultaneous: use elimination or substitution
    • 3Quadratic: factorise, formula, or complete the square
    • 4Flip inequality sign when dividing by a negative

    Pakistan Example

    Mobile Data Plans — Simultaneous Equations in Karachi

    Jazz offers Rs. 60 base + Rs. 3/GB. Ufone offers Rs. 30 base + Rs. 5/GB. Setting up simultaneous equations: 60 + 3x = 30 + 5x → 2x = 30 → x = 15 GB. Below 15 GB, Ufone is cheaper. Above 15 GB, Jazz wins. AKU-EB students see this kind of everyday algebra on their papers.

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