Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, expressions often contain terms that share the same variable part. Combining like terms is one of the most fundamental skills for IGCSE Mathematics, appearing in nearly every algebraic topic — from solving equations to simplifying rational expressions.

在代数中，表达式通常包含具有相同变量部分的项。合并同类项是IGCSE数学中最基础的技能之一，它几乎出现在所有代数主题中——从解方程到化简有理式。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable part, with the same variables raised to the same powers. For example, 3x and 5x are like terms because both contain x to the first power. The coefficients can be different; only the variable part must match.

同类项是指具有完全相同变量部分、且变量的幂次也完全相同的项。例如，3x 和 5x 是同类项，因为它们都包含一次幂的 x。系数可以不同；只有变量部分必须完全一致。

Consider the table below:

请观察下表：

Notice that a²b and ab² are not like terms, because the powers of a and b are distributed differently: a²b = a·a·b, while ab² = a·b·b.

注意 a²b 和 ab² 不是同类项，因为 a 和 b 的幂次分布不同：a²b = a·a·b，而 ab² = a·b·b。

2. Why Can We Combine? The Distributive Property | 为什么可以合并？分配律

Combining like terms is actually an application of the distributive property in reverse. Since a·c + b·c = (a + b)·c, we can factor out the common variable part.

合并同类项实际上是分配律的逆向运用。因为 a·c + b·c = (a + b)·c，我们可以提取公共的变量部分。

3x + 5x = (3 + 5)x = 8x

Here, the variable part x is the common factor, and the coefficients 3 and 5 are what we add together. This is why we only add the coefficients and keep the variable part unchanged.

这里的变量部分 x 是公共因子，系数 3 和 5 是我们加在一起的数。这就是为什么我们只将系数相加，而保持变量部分不变。

3. Step-by-Step Method | 逐步合并方法

Follow these steps to simplify any algebraic expression by combining like terms:

按照以下步骤，通过合并同类项来化简任何代数表达式：

• Step 1: Identify all like terms. Group terms that have identical variable parts.

  第一步：识别所有同类项。将具有相同变量部分的项归为一组。

• Step 2: Add or subtract the coefficients. Keep the variable part exactly the same.

  第二步：对系数进行加减。变量部分保持不变。

• Step 3: Write the simplified expression. List the results in a clear order, usually in descending powers.

  第三步：写出化简后的表达式。按清晰的顺序列出结果，通常按降幂排列。

Example: Simplify 4x + 3y − 2x + 7y.

示例：化简 4x + 3y − 2x + 7y。

Group the x-terms: 4x − 2x = 2x. Group the y-terms: 3y + 7y = 10y. The simplified expression is 2x + 10y.

将 x 项分组：4x − 2x = 2x。将 y 项分组：3y + 7y = 10y。化简结果为 2x + 10y。

4x + 3y − 2x + 7y = 2x + 10y

4. Working with Negative Coefficients | 处理负系数

When combining terms with negative coefficients, use

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