What is an Identity

1Summary 重點摘要

Learn what an identity is, know how it differs from normal equations.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What an identity isAn identity is an equation that holds for every number: whatever value is substituted, the two sides remain equal. For example, 2(x+1) = 2x+2 — substituting x = 1 gives left = 2 × 2 = 4 and right = 2 + 2 = 4, while x = 5 gives left = 2 × 6 = 12 and right = 10 + 2 = 12. The symbol ≡ is often used to stress 'always equal', as in 2(x+1) ≡ 2x+2. Both sides of an identity are simply two ways of writing the same expression, which is the key difference from an ordinary equation.
2. Step 2: Identity versus equationAn ordinary equation is true only for particular values: x + 1 = 3 holds only when x = 2 and fails for anything else, and x is an unknown waiting to be solved. An identity, by contrast, is true for all values, and the idea of a 'solution' does not arise. When judging, ask whether the statement is meant to solve for a specific x, or to assert that two expressions are permanently equal — the former is an equation, the latter an identity. Examinations often set x + 1 = 3 against 2(x+1) = 2x+2 for exactly this comparison.
3. Step 3: How to verify an identityThe quickest check is to substitute two or three different numbers, ideally including positives, negatives and zero; if both sides match every time, the statement is very likely an identity. For example, 3(x+2) = 3x+6 gives 9 = 9 at x = 1 and 12 = 12 at x = 2. Keep in mind, however, that substitution is only a check, not a proof, since it is impossible to test every number; a formal confirmation must expand and simplify algebraically, which is also the only method accepted when an examination demands a proof.
4. Step 4: How to prove formallyThe standard format for proving an identity: start from one side, expand the brackets, combine like terms and simplify step by step until the other side appears, then state the conclusion. To prove 2(x+1) = 2x+2: left = 2(x+1) = 2x+2 = right, so it is proven. Change one thing at a time and keep the equals signs and written reasons clear; do not manipulate both sides at once, or the reasoning becomes tangled. The worked example 3(x+2) = 3x+6 finishes in one line: expanding the left gives 3x+6, matching the right, so it is an identity.
5. Step 5: Common examples of identitiesMany identities were already met in primary school: the distributive law a(b+c) = ab+ac holds for all a, b and c, and so does the associative law (a+b)+c = a+(b+c). The multiplication formulas coming up, such as the difference of squares (a+b)(a-b) = a²-b² and the perfect square (a+b)² = a²+2ab+b², are all identities, which is exactly why they may be applied confidently in any question. Remember: anything called a 'formula' must first be proven to be an identity before using it has any justification.
6. Step 6: Why learn identitiesLearning identities lays the groundwork for the multiplication formulas: the difference of squares and perfect squares are identities at heart and will be used constantly in expansion and factorisation. Understanding that an identity holds for every value explains why any numbers may be substituted for a and b in a formula. Proving identities also trains step-by-step reasoning and clear presentation — the same skills that geometry proofs and harder algebra demand — so mastering this lesson makes the following topics much smoother.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Identity true for any number
• Equation only correct for certain numbers
• Verify by substituting numbers
• Formal proof uses algebra
• Later multiplication formulas are identities