Notation, vocabulary and manipulation: to include identities

The idea

This row extends the vocabulary to identities. An identity is an equation that is true for every value of the variable, so it cannot be solved to give a single value. The identity symbol ≡ is used to show this, for example 2(x + 3) ≡ 2x + 6 is true whatever number replaces x. In contrast, 2x + 6 = 10 is an equation with the single solution x = 2. Also check whether the option can be rearranged to 0 = 0, which confirms an identity, or to a single value of x, which confirms an equation.

What earns the marks

• Defines an identity as an algebraic statement that is true for all values of the variable.
• Recognises the identity symbol ≡ and distinguishes it from the equals sign =.
• Tests a candidate identity by substituting values, for example x = 0 and x = 1, and checking that both sides agree.
• Distinguishes an identity such as 2(x + 3) ≡ 2x + 6 from an equation such as 2x + 6 = 10, which has a single solution.
• Explains that an identity cannot be solved for one value of the variable because it holds for every value.

Common mistakes and how to put them right

• Treating an identity as an equation to solve.

  Put it right: An identity is true for all values, so there is no single solution to find.

• Assuming every statement with an equals sign is an identity.

  Put it right: Only statements true for all values qualify, so 2x + 6 = 10 is an equation, not an identity.

• Ignoring the identity symbol ≡.

  Put it right: It signals that the statement holds for all values, whereas = in an equation usually signals a condition to solve.

• Testing only one value and concluding too soon.

  Put it right: One agreeing value is not enough to prove an identity, so check several values or rearrange both sides.

Exam tips

• Substitute x = 0 and x = 1 into both sides; if the results differ, the statement is not an identity.
• Look for the ≡ symbol, which marks an identity, and do not confuse it with the = sign of an equation.
• Try to simplify both sides; if they become identical expressions, the statement is an identity.