Mathematics Study Guide: Simultaneous Equations and Factorisation

Solving Simultaneous Equations

  • Definition of Simultaneous Equations
      - Simultaneous equations consist of two or more equations that are solved at the same time.
      - The solution to these equations is the point where the equations' graphs intersect, indicating a common solution for both equations.

  • Possible Outcomes
      - Two linear equations can yield:
        - One solution: The lines intersect at a single point.
        - No solution: The lines are parallel and never intersect.
        - Infinite solutions: The lines coincide, meaning they lie on top of each other, sharing all points along the line.

  • Methods of Solving Simultaneous Equations
      1. Substitution Method
         - Select one equation; solve for one variable.
         - Substitute this expression in the other equation to find the value of the second variable.
         - Back-substitute to find the value of the first variable.

      2. Elimination Method
         - Align the equations and manipulate them (addition or subtraction) to eliminate one variable, creating a single-variable equation.
         - Solve for that variable, then substitute back to find the other variable.

  • Word Problems
      - Approach: Define variables for the quantities involved; establish equations based on the relationships stated in the problem; and then employ either the substitution or elimination method to solve.

Factorising Quadratic Expressions

  • Expansion and Factorisation
      - Expansion is the process of distributing multiplication over addition to create a polynomial.
      - Factorisation is the reverse process, decomposing a polynomial into its constituent factors. It is important to understand that factorisation is effectively the opposite of expansion.

  • Technique of Factorisation
      - For algebraic expressions, factorisation can be performed by identifying and extracting a common factor from all terms in the expression (this may involve numerical coefficients or variables).

  • Example
      - To factorise the expression x2+5x+6x^2 + 5x + 6, you can express it as:
    x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)
      - Here, both terms in the brackets multiply to give the original quadratic expression, showcasing the process of expressing an expanded form through factorisation.