Simultaneous Equations

    OCR
    GCSE

    Simultaneous equations require the determination of coordinate sets that satisfy multiple algebraic constraints concurrently, extending beyond linear systems to include non-linear functions such as circles and parabolas. Mastery involves the rigorous application of algebraic substitution and elimination to solve for two or more unknowns, often resulting in quadratic equations requiring factorization or the formula. Candidates must also interpret solutions geometrically as points of intersection or tangency, and in advanced contexts, utilize matrix methods or Gaussian elimination for systems involving three variables.

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    Objectives
    4
    Exam Tips
    4
    Pitfalls
    4
    Key Terms
    5
    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award M1 for correct rearrangement of the linear equation to make one variable the subject (e.g., x = 5 - 2y)
    • Award M1 for correct substitution of the linear expression into the non-linear equation
    • Award M1 for expanding brackets correctly and simplifying to a 3-term quadratic equation (=0)
    • Award A1 for correct method to solve the quadratic (factorisation or formula) leading to critical values
    • Award A1 for both correct pairs of coordinates (x, y), clearly matched

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You correctly rearranged the linear equation, but check your expansion of the squared bracket — remember the middle term"
    • "You found the x-values correctly. To gain full marks, you must substitute these back to find the corresponding y-values"
    • "Good algebraic process. Now, explicitly state which y-value belongs to which x-value to ensure the pairs are clear"
    • "Excellent solution. For the next step, explain what the discriminant of your quadratic tells you about the relationship between the line and the curve"

    Marking Points

    Key points examiners look for in your answers

    • Award M1 for correct rearrangement of the linear equation to make one variable the subject (e.g., x = 5 - 2y)
    • Award M1 for correct substitution of the linear expression into the non-linear equation
    • Award M1 for expanding brackets correctly and simplifying to a 3-term quadratic equation (=0)
    • Award A1 for correct method to solve the quadratic (factorisation or formula) leading to critical values
    • Award A1 for both correct pairs of coordinates (x, y), clearly matched

    Examiner Tips

    Expert advice for maximising your marks

    • 💡Always rearrange the linear equation to make the variable with a coefficient of 1 the subject to avoid working with fractions
    • 💡If the question asks for 'coordinates', write your final answer as (x, y) pairs; if it asks to 'solve', x=... and y=... is sufficient
    • 💡Check your solutions by substituting both pairs back into the non-linear equation (the one you didn't use for the initial substitution)
    • 💡Recognise that if the resulting quadratic has equal roots (discriminant = 0), the line is a tangent to the curve

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Incorrect expansion of squared binomials during substitution (e.g., expanding (3-2y)² as 9 - 4y² instead of 9 - 12y + 4y²)
    • Stopping after finding values for the first variable and failing to substitute back to find the second variable
    • Mismatching x and y values in the final answer (e.g., pairing x₁ with y₂)
    • Algebraic errors when rearranging the linear equation, particularly involving sign changes

    Key Terminology

    Essential terms to know

    Algebraic substitution for linear and non-linear systems
    Geometric interpretation of intersection and tangency
    Systems of equations with three variables (3x3)
    Use of the discriminant to determine solution distinctness

    Likely Command Words

    How questions on this topic are typically asked

    Solve
    Find the coordinates
    Determine
    Show that
    Calculate

    Practical Links

    Related required practicals

    • •{"code":"Kinematics","title":"Collision detection","relevance":"Determining if and where two particle trajectories intersect"}

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