GCSE · Algebraic manipulation

Identities and the ≡ signExam-style questions

An identity is true for every value of its letter; an equation is true only for some. How to tell whether a statement is true for all, some or no values, why one number can disprove an identity but never prove one, how to show an identity by expanding, and how to find unknown constants by matching coefficients or by choosing values of x.

Start practising Read the lesson

17 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4
  1. 1
    Amir writes the expression

    4(3x−2)−5(x−3)4(3x - 2) - 5(x - 3)

    Bella writes the expression

    3(2x+5)+x−83(2x + 5) + x - 8

    Show that the two expressions are equivalent.

    (3 marks)
    Level 3 · 3 marksTry it
  2. 2
    For each statement, tick one box.
    StatementTrue for all values of xxTrue for some values of xxTrue for no values of xx
    (x−4)2=x2+16(x - 4)^2 = x^2 + 16☐☐☐
    2(3x−1)−6x=−22(3x - 1) - 6x = -2☐☐☐
    4(x+1)−x=3x+14(x + 1) - x = 3x + 1☐☐☐
    (3 marks)
    Level 3 · 3 marksTry it
  3. 3
    Here is a question and an incorrect solution.

    Question: Find the values of aa and bb such that 3(x+a)+2x≡bx+123(x + a) + 2x \equiv bx + 12.

    Solution: Comparing the two sides, b=3b = 3 and a=12a = 12.
    (a) Describe the error made.

    (1 mark)
    (b) Find the correct values of aa and bb.

    (2 marks)
    Level 3 · 3 marksTry it
  4. 4
    (a) Show that (x+4)2−(x−2)(x+10)≡36(x + 4)^2 - (x - 2)(x + 10) \equiv 36

    (3 marks)

    Mira says,

    "So (x+4)2−(x−2)(x+10)(x + 4)^2 - (x - 2)(x + 10) is equal to 3636 only when x=0x = 0."
    (b) Is Mira correct?

    Give a reason for your answer.

    (1 mark)
    Level 3 · 4 marksTry it
  5. 5
    Here is a statement.

    3(2x−k)=6x+123(2x - k) = 6x + 12

    kk is a number.
    (a) Work out the value of kk for which the statement is true for all values of xx.

    (2 marks)
    (b) For any other value of kk, the statement is true for no values of xx.

    Explain why.

    (1 mark)
    Level 3 · 3 marksTry it

2 more questions on identities and the ≡ sign

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free