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Algebra Toolkit

Algebra and Sequences

Replacing letters with numbers

  • Substitution means putting given numbers in place of the letters, then evaluating
  • Write the expression or formula down first, then replace each letter, then calculate
  • Follow the order of operations exactly as with any numerical calculation

Negative values need brackets

  • A negative value substituted for a letter is written in brackets, which keeps the sign attached through powers and multiplications
    • Where x = −4, x² is (−4)² = 16, not −16
  • Without brackets a calculator applies the power before the sign, giving the wrong result
Exam tip

Bracket every negative value you substitute — that is where signs are lost, especially under a square. Only genuinely like terms combine: x and x² never merge. And when a question says "in terms of x", the answer must contain x and no other letter.

Worked example

Substituting a negative value

Given a = 5 and b = −3, find the value of (i) 3a − b² (ii) 2b(a + b).

Solution:

  • Substitute a negative value inside brackets every time, so the sign cannot be separated from the number
  • (i) The expression becomes 3 × 5 − (−3)²
  • BIDMAS puts the index before the subtraction, so square first
  • (−3)² = (−3) × (−3) = 9, positive because two negatives multiply to a positive
  • 15 − 9 = 6
  • Without the bracket this reads 3 × 5 − 3², which is 15 − 9 by accident here but goes wrong the moment the sign matters
  • (ii) The expression becomes 2 × (−3) × (5 + (−3))
  • Brackets first: 5 + (−3) is 5 − 3 = 2
  • Then left to right: 2 × (−3) = −6, and −6 × 2 = −12
  • The answer is negative because exactly one of the factors is negative