📚 Identities and Equations | 恒等式与方程
In algebra, the concepts of identities and equations form the backbone of mathematical reasoning. An identity is a statement that is true for all permissible values of the variables involved, while an equation is a statement that is only true for certain specific values, known as solutions. Mastering this distinction is crucial for solving problems ranging from polynomial manipulations to trigonometric proofs in the IB Mathematics curriculum. This article explores the definitions, notation, types, and common techniques associated with identities and equations.
在代数中，恒等式与方程的概念构成了数学推理的支柱。恒等式是对所涉及变量的所有允许值都成立的陈述，而方程是仅对某些特定值（即解）才成立的陈述。掌握这种区别对于解决IB数学课程中从多项式运算到三角证明的各类问题至关重要。本文探讨与恒等式和方程相关的定义、符号、类型和常用技巧。
1. Defining Identities and Equations | 定义恒等式与方程
An identity is an equality that holds true for every permissible value of its variables. For example, the expansion (x+2)² = x² + 4x + 4 is actually an identity because if you evaluate both sides for any real number x, they will always be equal. In IB Mathematics, we often use the identity symbol ≡ to emphasise this fact, but it is also common to see a standard equals sign.
恒等式是一种等式，对于其变量的每一个允许值都成立。例如，展开式 (x+2)² = x² + 4x + 4 实际上是一个恒等式，因为对于任意实数 x，计算两边始终相等。在IB数学中，我们常用恒等符号 ≡ 来强调这一点，但也经常看到普通的等号。
An equation, on the other hand, is a statement that is true only for specific values of the variable(s). For example, 2x + 5 = 15 is an equation because it is only true when x = 5. Equations are solved to find these unknown values, and the solutions may be checked by substitution.
另一方面，方程是仅对变量的特定值才成立的陈述。例如，2x + 5 = 15 是一个方程，因为只有当 x = 5 时它才成立。求解方程就是为了找出这些未知值，并且可以通过代入来检验解。
| Feature | Identity | Equation |
| Truth | For all variable values | For specific solutions |
| Symbol | ≡ (often =) | = |
| Example | (a+b)² ≡ a²+2ab+b² | 2x+3=7 |
The table above summarises the key differences: an identity is universally true, while an equation requires solving to find the restricted set of values that satisfy it.
上表总结了关键区别：恒等式普遍成立，而方程需要求解以找到满足它的有限值集。
2. The Identity Symbol (≡) and Its Usage | 恒等符号及其使用
The symbol ≡ (read as “is identically equal to”) is used to express that two expressions are equivalent for all values of the variables within their common domain. In IB exam papers, you may be asked to determine whether a given
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