Equations and Identities | 方程与恒等式

📚 Equations and Identities | 方程与恒等式

In Edexcel A-Level Mathematics, the distinction between equations and identities is fundamental across algebra, proof, and problem-solving. An equation is a statement that is true only for particular values of the unknown, while an identity is true for all allowable values of the variables. This article builds the key concepts, techniques, and exam strategies needed for equations and identities.

在Edexcel A-Level数学中，方程与恒等式的区别是代数、证明和解题的基础。方程只对未知量的特定值成立，而恒等式对所有允许的变量值都成立。本文构建方程与恒等式的核心概念、技巧和考试策略。

1. What is an Equation? | 什么是方程

An equation links two algebraic expressions with an equals sign and is only true for certain values of the unknown. For example, x + 3 = 7 is true only when x = 4. Solving an equation means finding the set of values, called the solution set, that makes the statement true.

方程用等号连接两个代数表达式，只对未知量的某些特定值成立。例如，x + 3 = 7 仅在 x = 4 时成立。解方程意味着找出使等式成立的值的集合，称为解集。

Equations can be linear, quadratic, simultaneous, or involve more complicated functions. In each case, we apply inverse operations or factorisation to isolate the unknown while keeping the equation balanced.

方程可以是线性的、二次的、联立的，或涉及更复杂的函数。无论哪种情况，我们都应用逆运算或因式分解来隔离未知量，同时保持方程平衡。

2. What is an Identity? | 什么是恒等式

An identity is an equation that is true for all values of the variable for which both sides are defined. It is often written with the symbol ≡, although the equals sign is also used when the context is clear. For example, x² − 1 ≡ (x − 1)(x + 1) is an identity.

恒等式是对所有使两边都有定义的变量值都成立的等式。它常用符号 ≡ 表示，不过当上下文清楚时也可用等号。例如，x² − 1 ≡ (x − 1)(x + 1) 是一个恒等式。

Identities are powerful tools in algebra because they allow one expression to be replaced by another equivalent form. Edexcel questions often ask you to decide whether a given statement is an equation or an identity, or to find unknown constants in an identity.

恒等式是代数中的强大工具，因为它们允许用一个等价形式替换另一个表达式。Edexcel 考题经常要求你判断给定命题是方程还是恒等式，或者在恒等式中求出未知常数。

3. Key Differences Between Equations and Identities | 方程与恒等式的关键区别

The main difference is the set of values for which the statement is true. An equation is conditional, while an identity is universal. For instance, 2x + 1 = 5 is true only at x = 2, but 2(x + 1) ≡ 2x + 2 is true for all real x.

主要区别在于命题成立的取值范围。方程是条件等式，而恒等式是普遍成立。例如，2x + 1 = 5 仅在 x = 2 时成立，但 2(x + 1) ≡ 2x + 2 对所有实数 x 都成立。

We can include a comparison table to clarify the distinction.

我们可以用一个对比表来澄清区别。

In an identity, you can choose any value of x to substitute, which is the basis of the coefficient comparison method.

在恒等式中，你可以代入任意 x 值，这是比较系数法的基础。

4. Polynomial Identities and Equating Coefficients | 多项式恒等式与比较系数

If two polynomials are identically equal, then the coefficients of corresponding powers of x must be the same. For example, if ax² + bx + c ≡ 3x² − 5x + 2, then a = 3, b = −5, and c = 2.

如果两个多项式恒等，那么 x 的同次幂的系数必须相同。例如，若 ax² + bx + c ≡ 3x² − 5x + 2，则 a = 3，b = −5，c = 2。

This method is extremely helpful for finding unknown constants in partial fractions, in factorised forms, or when proving an identity. You can also substitute convenient values, such as x = 0 or the roots, to generate simultaneous equations for the unknown coefficients.

这种方法在部分分式、因式分解形式或证明恒等式时对求未知常数极为有用。你也可以代入方便的值，如 x = 0 或根，来生成关于未知系数的联立方程。

Worked example: If (x + 2)(x + k) ≡ x² + 5x + 6, find k. Expanding the left side gives x² + (2 + k)x + 2k. Comparing coefficients: 2 + k = 5 and 2k = 6, so k = 3.

例题：若 (x + 2)(x + k) ≡ x² + 5x + 6，求 k。展开左边得 x² + (2 + k)x + 2k。比较系数：2 + k = 5 且 2k = 6，所以 k = 3。

5. Solving Linear and Quadratic Equations | 解线性方程和二次方程

For linear equations, use inverse operations to isolate x: ax + b = c gives x = (c − b) ÷ a. Always check your answer by substitution.

对于线性方程，使用

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