[discia] GCSE Maths › Algebra › Expressions, Equations and Identities › Practise
  1. Either paper 3 marks

    Here is an expression.

    \(8t^2 - 4t + 2\)

    (a)

    State how many terms the expression has.

    (b)

    State the coefficient of \(t^2\).

    (c)

    State the constant term.

    Show answer
    (a) \(3\)
    (b) \(8\)
    (c) \(2\)

    Terms are the parts separated by \(+\) and \(-\) signs: \(8t^2\), \(-4t\) and \(2\).

    That is 3 terms.

    The coefficient is the number multiplying the letter part, so for \(8t^2\) it is 8.

    The constant term is the one with no letter in it: 2.

  2. Either paper 1 mark
    Here are four statements.

    A  \(3x - 9\)

    B  \(3x - 9 = 27\)

    C  \(y = 3x - 9\)

    D  \(5(x + 5) \equiv 5x + 25\)

    Write down the letter of the statement that is a formula.
    Show answer
    C

    The answer is C.

    It uses letters to give a rule for working one quantity out from another.

    The four differ only in what the equals sign is doing. A has none at all, so it is an expression. B has one and is true for a single value of \(x\), so it is an equation. C gives a rule for working \(y\) out from \(x\), so it is a formula. D uses \(\equiv\), which claims both sides are equal whatever \(x\) is.

  3. Either paper 3 marks

    Tomas charges for repairing a bike using the formula

    \(C = 9n + 17\)

    where \(C\) is the total charge in pounds and \(n\) is the number of parts. The call-out fee is included in the formula.

    (a)

    Work out the total charge when \(n = 7\).

    (b)

    Explain why \(C = 9n + 17\) is a formula and not an equation.

    Show answer
    (a) \(\text{£}80\)
    (b) It is a rule for working the charge out from the number of parts, and it holds for any number of parts. An equation such as \(9n + 17 = 80\) is true only when \(n = 7\).

    Substitute \(n = 7\) into the formula.

    \(C = 9 \times 7 + 17\)

    \(C = 63 + 17 = 80\)

    The total charge is £80.

    A formula connects two or more quantities and lets you work one out from the other, for any value you put in.

    An equation such as \(9n + 17 = 80\) is only true for one value of \(n\), and solving it means finding that value.

    • (a) M1substitutes \(n = 7\): \(9 \times 7 + 17\)
    • (a) A1total charge £80
  4. Either paper 3 marks
    Match each box on the left to the box on the right with the same value. One has been done for you.
    LeftRight
    A  \(x · x · x\)P  \(x^2\)
    B  \(3x - x\)Q  \(x^3\)
    C  \(x · x\)R  \(x^4\)
    D  \(5x - 2x\)S  \(2x\)
     T  \(3x\)
    Box A on the left has been linked to box Q on the right. Link the other three boxes on the left. Give your answer as pairs of letters, for example A to P.
    Show answer
    B to S; C to P; D to T

    Simplify each box on the left, then find the box on the right with that value.

    Box A: \(x · x · x = x^3\), so it links to box Q.

    Box B: \(3x - x = 2x\), so it links to box S.

    Box C: \(x · x = x^2\), so it links to box P.

    Box D: \(5x - 2x = 3x\), so it links to box T.

    The link A to Q was the one done for you, so the other three are B to S; C to P; D to T.

    Box R (\(x^4\)) matches none of the boxes on the left. Take care with \(x^2\) and \(2x\), and with \(x^3\) and \(3x\): \(x \times x\) is \(x^2\), not \(2x\).

    • B1simplifies \(3x - x\) to \(2x\) and links B to S
    • B1simplifies \(x · x\) to \(x^2\) and links C to P
    • B1simplifies \(5x - 2x\) to \(3x\) and links D to T
  5. Either paper 1 mark
    One of these four statements is an identity.

    A  \(6x + 8 \equiv 50\)

    B  \(4x \equiv 8\)

    C  \(3x - 4x \equiv x\)

    D  \(3x - 4x \equiv -x\)

    Circle the identity.
    Show answer
    \(3x - 4x \equiv -x\)

    An identity must be true for every value of \(x\). Each of the other three is only true for particular values, so test them.

    The identity is \(3x - 4x \equiv -x\). Collecting the x terms gives the same expression on both sides, whatever x is.

    \(4x \equiv 8\) is only true when \(x = 2\).

    \(6x + 8 \equiv 50\) is only true when \(x = 7\).

    \(3x - 4x \equiv x\) is not an identity: when \(x = 1\) the two sides are \(-1\) and \(1\), so it is only true when \(x = 0\).

  6. Either paper 1 mark
    Here is a statement about \(x\). \(6(x - 2) = 8x - 15\) Tick one box.

    ☐  It is true for all values of \(x\)

    ☐  It is true for some values of \(x\)

    ☐  It is true for no values of \(x\)

    Show answer
    It is true for some values of \(x\)

    Expand the bracket on the left-hand side:

    \(6(x - 2) = 6x - 12\)

    Compare it with the right-hand side, \(8x - 15\).

    The x terms are different, so the two sides are equal for exactly one value of x, which solving the equation would find, and for no other.

    So the box to tick is true for some values of \(x\).

  7. Either paper 2 marks

    \(8(x + c) \equiv 8x - 32\)

    Find the value of \(c\).

    Show answer
    \(-4\)

    The symbol \(\equiv\) means the two sides are equal for every value of \(x\), so the numbers on each side must match exactly.

    Expanding the left-hand side gives \(8x + 8c\).

    Comparing the number terms: \(8c = -32\)

    \(c = -32 \div 8 = -4\)

    • M1expands the bracket: \(8x + 8c\), and compares number terms \(8c = -32\)
    • A1\(c = -4\)
  8. Either paper 2 marks

    Show that

    \(6(x + 2) + 2 \equiv 6x + 14\)

    Show answer
    Expanding gives \(6x + 12 + 2 = 6x + 14\), which is the right-hand side.

    To show a statement is an identity, work on one side until it matches the other. Start with the left.

    Expand the bracket: \(6(x + 2) = 6x + 12\)

    Now add the 2: \(6x + 12 + 2\)

    The two number terms combine: \(12 + 2 = 14\)

    \(6x + 14\), which is exactly the right-hand side, so the two sides are equal for every value of \(x\).

    • M1expands the bracket: \(6x + 12\)
    • A1adds \(2\) and combines the numbers to \(14\), reaching the right-hand side
  9. Either paper 3 marks

    \(m\) and \(n\) are numbers.

    \(m - n = 16\)

    (a)

    Find the value of \(2(m - n)\)

    (b)

    Find the value of \(n - m\)

    (c)

    Find the value of \(3m - 3n\)

    Show answer
    (a) \(32\)
    (b) \(-16\)
    (c) \(48\)

    \(2(m - n)\) is the block \(m - n\) multiplied by \(2\), and \(m - n = 16\).

    \(2(16) = 32\)

    \(n - m\) is the block \(m - n\) multiplied by \(-1\), and \(m - n = 16\).

    \(-(16) = -16\)

    \(3m - 3n\) is the block \(m - n\) multiplied by \(3\), and \(m - n = 16\).

    \(3(16) = 48\)

  10. Either paper 4 marks

    Work out the values of \(a\) and \(b\) in the identity

    \(3(9x + 3) + 4(9x + b) \equiv ax - 3\)

    Give your answer as \(a = \ldots\), \(b = \ldots\)

    Show answer
    \(a = 63\), \(b = -3\)

    The two sides of an identity are equal for every value of \(x\), so expand the brackets on the left and compare.

    \(3(9x + 3) = 27x + 9\)

    \(4(9x + b) = 36x + 4b\)

    The left-hand side is \(27x + 36x + 9 + 4b = 63x + 9 + 4b\)

    Compare the \(x\) terms: \(a = 63\)

    Compare the numbers: \(9 + 4b = -3\)

    Subtract 9 from both sides: \(4b = -12\)

    Divide both sides by 4: \(b = -3\)

    • M1both brackets expanded: \(27x + 9\) and \(36x + 4b\)
    • A1x terms matched: \(a = 63\)
    • M1constants matched: \(9 + 4b = -3\)
    • A1\(b = -3\)