Equations

OCR

GCSE

Equations represent the fundamental mathematical assertion of equality between two algebraic expressions, requiring the isolation of unknown variables through precise inverse operations. The study progresses from linear equations involving a single variable to complex quadratic and simultaneous systems requiring factorization, substitution, or algorithmic approaches like the quadratic formula. Mastery encompasses both the procedural fluency to manipulate algebraic terms and the interpretative skill to formulate mathematical models from physical or contextual scenarios.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Subtopics in this area

Equations

What You Need to Demonstrate

Key skills and knowledge for this topic

Award M1 for a correct first step in isolating the variable, such as subtracting the constant term from both sides or expanding brackets correctly
Award A1 for the correct final value, accepting fractional or decimal equivalents unless 'exact form' is specified
In simultaneous equations, award M1 for a complete method to eliminate one variable (equating coefficients and adding/subtracting) or substitution
For quadratic equations using the formula, award M1 for correct substitution of a, b, and c, paying strict attention to negative signs
Award B1 for correctly setting up an equation from a text-based problem before attempting to solve it
Award M1 (Method Mark) for a correct first step in isolation, such as expanding brackets correctly or collecting variable terms on one side of the equation
Award A1 (Accuracy Mark) only for the explicit statement of the solution (e.g., 'x = 3'); embedded answers in working are often not credited without the final statement
In 'Show that' questions, credit requires a complete, logical chain of reasoning where the derived equation matches the target exactly, with no steps skipped

Example Examiner Feedback

Real feedback patterns examiners use when marking

"You have correctly expanded the brackets, but check your signs when collecting like terms—this is a preventable error."
"For simultaneous equations, you found x correctly. Remember to substitute this back to find y to get full marks."
"You used trial and error here, which scores zero. You must use an algebraic method (balancing) to solve."
"Excellent use of the quadratic formula. Ensure you show the substitution step clearly to secure the method mark even if the calculation goes wrong."

Marking Points

Key points examiners look for in your answers

Award M1 for a correct first step in isolating the variable, such as subtracting the constant term from both sides or expanding brackets correctly
Award A1 for the correct final value, accepting fractional or decimal equivalents unless 'exact form' is specified
In simultaneous equations, award M1 for a complete method to eliminate one variable (equating coefficients and adding/subtracting) or substitution
For quadratic equations using the formula, award M1 for correct substitution of a, b, and c, paying strict attention to negative signs
Award B1 for correctly setting up an equation from a text-based problem before attempting to solve it
Award M1 (Method Mark) for a correct first step in isolation, such as expanding brackets correctly or collecting variable terms on one side of the equation
Award A1 (Accuracy Mark) only for the explicit statement of the solution (e.g., 'x = 3'); embedded answers in working are often not credited without the final statement
In 'Show that' questions, credit requires a complete, logical chain of reasoning where the derived equation matches the target exactly, with no steps skipped
For simultaneous equations, award M1 for a valid method to eliminate one variable (equating coefficients and subtracting/adding) or substitution
When 'Solve algebraically' is specified, award 0 marks for correct answers obtained via trial and improvement

Examiner Tips

Expert advice for maximising your marks

💡When solving quadratics, always rearrange the equation to equal zero (ax^2 + bx + c = 0) before attempting to factorise or use the formula
💡For 'Show that' questions involving forming equations, explicitly state the geometric or algebraic property being used (e.g., 'Area = ...') before substituting expressions
💡If the question asks for exact solutions or surds, do not convert your answer to a decimal, as this will lose the final accuracy mark
💡Check your solution by substituting the value back into the original equation—this is a quick way to verify accuracy
💡If a quadratic equation asks for answers to '2 decimal places' or '3 significant figures', this is a trigger to use the Quadratic Formula, as it will not factorise
💡When solving equations involving algebraic fractions, multiply every term by the common denominator immediately to eliminate fractions in the first step
💡Check your solution by substituting the value back into the original equation; if it does not balance, check your sign changes first
💡For 'Form and solve' questions, define your variable explicitly (e.g., 'Let x be the width') to ensure the examiner can follow your modelling logic

Common Mistakes

Pitfalls to avoid in your exam answers

Incorrectly expanding brackets, such as expanding 4(x - 3) as 4x - 3 instead of 4x - 12
Failing to find both roots when solving quadratic equations, particularly when solving x^2 = k
In simultaneous equations, adding equations when subtraction is required to eliminate a variable with matching positive coefficients
Using trial and improvement where an algebraic solution is explicitly required, resulting in zero marks
Failure to invert operations correctly when rearranging formulae, particularly when the subject appears in a denominator or within a square root
In simultaneous equations, adding equations when subtraction is required (or vice versa) to eliminate a variable with matching coefficients
Not setting a quadratic equation to equal zero before attempting to factorise or apply the quadratic formula
Sign errors when expanding brackets with a negative multiplier, leading to incorrect linear terms

Study Guide Available

Comprehensive revision notes & examples

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Key Terminology

Essential terms to know

Linear equations with unknowns on one or both sides

Quadratic equations (factorisation, formula, completing the square)

Simultaneous equations (linear/linear and linear/quadratic)

Rearranging formulae and changing the subject

Forming equations from real-world contexts

Solving linear equations (unknowns on both sides, brackets, fractions)

Quadratic equations (factorisation, formula, completing the square)

Simultaneous equations (linear/linear and linear/quadratic)

Rearranging formulae and changing the subject

Forming algebraic equations from geometric or real-world contexts

Likely Command Words

How questions on this topic are typically asked

Solve

Rearrange

Form

Show that

Find

Calculate

Practical Links

Related required practicals

•{"code":"Physics Eq","title":"Rearranging kinematic equations","relevance":"Direct application of rearranging formulae (e.g., v = u + at)"}

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