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Simplifying Algebraic Expressions

  • a² is short for a × a.
  • When simplifying arithmetic or algebraic expressions, the order of operations is important.
    • If there are no brackets, do multiplication or division before addition and subtraction.
    • If there are brackets, first simplify the terms inside them.
  • When removing brackets:
    • A positive sign outside the bracket (on the left hand side) does not change the sign of each term inside the bracket. For example:
      • 2 + (x + 2) = 2 + x + 2 = x + 4
      • 3x + (2y – 4) = 3x + 2y – 4
    • A negative sign outside the bracket (on the left hand side) does change the sign of each term inside the bracket. For example:
      • 2 – (x + 2) = 2 – x – 2 = -x
      • 3x – (2y – 4) = 3x – 2y + 4
Examples
  1. 2a × 3
    • = 2 × a × 3
    • = 2 × 3 × a
    • = 6 × a
    • = 6a
  2. 6b × 4b
    • = 6 × b × 4 × b
    • = 6 × 4 × b × b
    • = 24 × b²
    • = 24b²
  3. 8ab × 7a
    • = 8 × a × b × 7 × a
    • = 8 × 7 × a × a × b
    • = 56 × a² × b
    • = 56a²b
  4. 14a/7
    • = 14 × a/7
    • = 7 × 2a/7
    • = 1 × 2a/1
    • = 2a
  5. 1/3 of 36c
    • = 36 × c/3
    • = 3 × 12c/3
    • = 1 × 12c/1
    • = 12c
  6. 24d²e ÷ 3de
    • = 24 × d² × e/3 × d × e
    • = 3 × 8 × d × d × e/3 × d × e
    • = 1 × 8 × 1 × d × 1/1 × 1 × 1
    • = 8d
  7. Find the value of 16 × 2 – 3 + 14 ÷ 7.
    • = (16 × 2) – 3 + (14 ÷ 7)
    • = 32 – 3 + 2
    • = 32 + 2 – 3
    • = 34 – 3
    • = 31
  8. Simplify 7 × 3a – (3a + 5a) × 2.
    • = 7 × 3a – 8a × 2
    • = (7 × 3a) – (8a × 2)
    • = 21a – 16a
    • = 5a
  9. Simplify (6x – 5y) + (3y + 4x)
    • = 6x – 5y + 3y + 4x
    • = 6x + 4x + 3y – 5y
    • = 10x – 2y
    • NOTE: There is no sign before the first bracket; we take it to be positive.
  10. (6x – y) – (7x – 2y)
    • = 6x – y – 7x + 2y
    • = 6x – 7x + 2y – y
    • = -x + y or y – x
  11. The greater of two consecutive numbers is x. (NOTE: Two whole numbers are consecutive when their difference is 1. For example, 5 and 6 are consecutive numbers; 49 and 50 are also consecutive. In algebra, x and x + 1 or y – 1 and y are consecutive, provided x and y are whole numbers)
    1. Find the sum of the two numbers.
      • If x is the greater of the two numbers, the lower number is x – 1.
      • ∴ The sum of the two numbers = x + (x – 1)
      •                                                            = x + x – 1
      • ∴ The sum of the two numbers = 2x – 1
    2. Subtract the sum of the two numbers from 5x.
      • = 5x – (2x – 1)
      • = 5x – 2x + 1
      • = 3x + 1
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