Equating Coefficients (AQA GCSE Further Maths)

Equating Coefficients

What is an identity?

• An identity is an algebraic statement with an identity sign, ≡, between a left-hand side and a right-hand side that is true for all values of x

  • E.g. x + x ≡ 2x

  • This means x + x is identical to 2x, or that x + x "can also be written as" 2x

• An identity cannot be solved

• All numbers can be substituted into an identity and it will remain true

  • E.g. x + x ≡ 2x is true for x = 1, x = 2, x = 3 … (even x = -0.01, x = π etc)

  • Unlike with equations, where only the solutions work

  • E.g. 2x = 10 is not true for x = 1, x = 2, x = 3 …  only for x = 5

How do I use the method of equating coefficients?

• Identities can be used to write algebraic expressions in different forms

• For example, find p and q if 3(x + y) + 2y ≡ px + qy

  • 3(x + y) + 2y expands to 3x + 5y

  • The coefficient of x on the left is 3 and on the right is p, so p = 3

  • The coefficient of y on the left is 5 and on the right is q, so q = 5

  • Therefore 3(x + y) + 2y is identical to 3x + 5y

  • This method is called equating coefficients