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Rearranging formulae and algebraic proof

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Notes de leçon

Key Vocabulary and Core Ideas

  • A formula shows a relationship between variables, e.g. v = u + at (v = final velocity, u = initial velocity, a = acceleration, t = time).
  • The subject of a formula is the variable on its own on one side, e.g. in A = \frac{1}{2} b h, A is the subject.
  • Rearranging (or changing the subject) means rewriting the formula so a different variable becomes the subject.
  • An equation states equality; an identity is an equation true for every permitted value. 3x + 2 = 11 has the particular solution x = 3.
  • In an identity, use the symbol \equiv to show it is always true.
  • Algebraic proof uses algebra to show a statement is always true, not just for a few examples.

Rearranging Formulae: The Golden Rule

  • Whatever you do to one side of the formula, do exactly the same to the other side.
  • Use inverse operations, taking care with restrictions: division needs a nonzero divisor, and reversing a square generally introduces positive and negative roots.
  • Work in the reverse order of operations (like undoing a function machine).
  • Example: make r the subject of C = 2\π r: divide both sides by 2\π to get r = \frac{C}{2\π}.
  • Example: make h the subject of A = \frac{1}{2} b h: multiply both sides by 2, then divide by b: h = \frac{2A}{b}.
  • Always state the new subject clearly, e.g. t = \frac{v - u}{a} from v = u + at.

Dealing with Roots and Powers

  • If x² = A for A ≥ 0, then x = ±√A. A length, radius or nonnegative speed uses the nonnegative branch.
  • Example: A = \π r2 → r = \√{\frac{A}{\π}} (take the positive root unless context says otherwise).
  • To remove a cube, take the cube root: V = x3 → x = \√[3]{V}.
  • If the subject is inside a fraction, first multiply to clear the denominator.
  • Example: P = \frac{F}{A} → F = PA and A = \frac{F}{P}.
  • Be careful: \√{a + b} is not the same as \√{a} + \√{b}.

Subject Appears More Than Once (Higher Tier)

  • If the new subject appears in more than one term, collect those terms on one side of the formula.
  • Then factorise to take the subject out as a common factor.
  • Example: make x the subject of ax + b = cx + d: bring cx to the left: ax - cx = d - b.
  • Factorise: x(a - c) = d - b, then divide: x = \frac{d - b}{a - c}.
  • Example: make t the subject of s = ut + \frac{1}{2} a t2: this is quadratic in t, so use factorising or the quadratic formula (Higher).
  • Always check that you have not divided by zero (e.g. a - c \neq 0).

Equations vs Identities

  • An equation may hold for particular values, no values, or every permitted value. An identity is the case that holds for every permitted value.
  • An identity is true for every value of the variable (e.g. 3(x + 2) \equiv 3x + 6).
  • The symbol ≡ emphasises an identity. An equals sign can also appear in a true identity.
  • To prove an identity, expand and simplify one side until it matches the other side.
  • Example: show (x + 3)2 - 9 \equiv x(x + 6): expand left to x2 + 6x + 9 - 9 = x2 + 6x = x(x + 6).
  • You cannot prove an identity by substituting numbers; you must use algebra.

Algebraic Proof: Representing Numbers

  • Use 2n to represent an even number (n is an integer).
  • Use 2n + 1 (or 2n - 1) for an odd number.
  • Consecutive integers can be written as n, n+1, n+2 (or n, n+1, n+2, \ldots).
  • Consecutive even numbers: 2n, 2n+2, 2n+4.
  • Consecutive odd numbers: 2n+1, 2n+3, 2n+5.
  • When proving statements about odd/even numbers, always state that n is an integer.

Constructing Algebraic Proofs

  • Start with a general form (e.g. 2n+1 for an odd number) and manipulate algebraically.
  • To prove a result is a multiple of 3, show it can be written as 3 \times (some expression).
  • To prove a result is even, show it can be written as 2 \times (some integer expression).
  • Example: prove the sum of three consecutive integers is a multiple of 3: n + (n+1) + (n+2) = 3n + 3 = 3(n+1).
  • Example: prove the product of two consecutive even numbers is divisible by 8: let the numbers be 2n and 2n+2; product = 4n(n+1), and since n(n+1) is even, product is divisible by 8.
  • Always conclude with a clear statement, e.g. 'Therefore the sum is always a multiple of 3.'

Common Mistakes to Avoid

  • Do not 'move' terms without applying the inverse operation to the whole side.
  • When taking a root, take the root of the entire expression, not individual terms.
  • If the subject appears twice, do not stop after collecting terms; factorise to isolate it.
  • Do not use a few numerical examples as proof; algebra must be general.
  • Do not use the same letter for two different quantities in a proof.
  • Always check your final formula by substituting simple numbers.

Typical Exam Question Types

  • Make a stated variable the subject of a formula (e.g. make x the subject of y = \frac{ax + b}{c}).
  • Rearrange a scientific formula and then substitute values (e.g. F = ma to find a).
  • Prove that the sum of three consecutive integers is always a multiple of 3.
  • Show that a given expression is always even or always positive (e.g. (n+1)2 - n2 is always odd).
  • Prove an algebraic identity by expanding and simplifying.
  • For Higher tier, handle formulae where the subject appears more than once.

Diapos

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Questions d'entraînement

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  1. 1.Every algebraic equation is an identity.

    Easy

    True or false?

  2. 2.Rearrange the formula v = u + at to make t the subject. Assume a ≠ 0.

    Easy
    • At = (v - u) / a
    • Bt = (v + u) / a
    • Ct = v - u / a
    • Dt = (u - v) / a

  3. 3.Rearrange the formula A = 1/2 × b × h to make h the subject. Assume b ≠ 0.

    Medium
    • Ah = 2A / b
    • Bh = A / (2b)
    • Ch = 2A × b
    • Dh = (A - b) / 2

  4. 4.Rearrange the formula P = F / A to make A the subject. Assume P ≠ 0.

    Medium
    • AA = F / P
    • BA = P / F
    • CA = F × P
    • DA = F - P

  5. 5.Rearrange the formula y = (3x - 1) / 2 to make x the subject.

    Medium
    • Ax = (2y + 1) / 3
    • Bx = (2y - 1) / 3
    • Cx = (2y + 1) × 3
    • Dx = (2y - 1) × 3

  6. 6.Make t the subject of the formula s = ut + 1/2 a t2. Assume a ≠ 0 and real roots exist.

    Medium
    • At = (-u ± √(u2 + 2as)) / a
    • Bt = (-u ± √(u2 - 2as)) / a
    • Ct = (u ± √(u2 + 2as)) / a
    • Dt = (-u ± √(u2 + 2as)) / (2a)

  7. 7.Given the formula a = (b + c) / (b - c), make b the subject. Assume a ≠ 1 and b ≠ c.

    Medium
    • Ab = c(a + 1) / (a - 1)
    • Bb = c(a - 1) / (a + 1)
    • Cb = c(1 + a) / (1 - a)
    • Db = c(a + 1) / (1 - a)

  8. 8.Which of the following are algebraic identities? (select all that apply)

    Medium
    • Ax2 - y2 = (x + y)(x - y)
    • Bx2 + 2xy + y2 = (x + y)2
    • Cx2 + y2 = (x + y)2
    • D2(x + 3) = 2x + 6

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