Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Algebra often looks like a list of symbols, but one of the first skills you need is to simplify expressions by collecting like terms. This topic shows how to group terms that have exactly the same variable part and combine their coefficients.

代数常常看起来像一串符号，但你需要掌握的首要技能之一就是通过合并同类项来化简表达式。本主题将展示如何将变量部分完全相同的项分组，并合并它们的系数。

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1. What Are Like Terms? | 什么是同类项

Like terms are terms whose variable parts are exactly the same, including the powers of each variable. For example, 3x and 5x are like terms because both contain x to the power 1.

同类项是指变量部分完全相同的项，包括每个变量的幂。例如，3x 和 5x 是同类项，因为它们都含有 x 的一次幂。

Constants such as 4, -7 and 12 are also like terms with each other, because they have no variable part at all.

常数项如 4、-7 和 12 也互为同类项，因为它们根本没有变量部分。

The coefficient tells you how many of that variable part you have, so 5x means five x’s. When terms are like, only the coefficients need to be combined.

系数表示你有多少个该变量部分，因此 5x 表示五个 x。当项为同类项时，只需要合并系数。

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2. Why Collecting Matters | 为什么合并同类项重要

Collecting like terms makes an expression shorter and easier to use in solving equations, drawing graphs or substituting values. In IGCSE exams, simplification is often the first mark in a longer question.

合并同类项可以使表达式更短，在解方程、画图像或代入数值时更容易使用。在 IGCSE 考试中，化简往往是较长的题目中的第一分。

Think of like terms as the same kind of object: 3 apples plus 2 apples gives 5 apples, but 3 apples plus 2 bananas cannot be combined into a single fruit term.

可以把同类项想象成同一种物品：3 个苹果加 2 个苹果等于 5 个苹果，但 3 个苹果加 2 根香蕉不能合并成单一的水果项。

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3. Identifying Like Terms | 识别同类项

To decide whether two terms are like terms, ignore the coefficient and look only at the variable letters and their powers. 6xy and -2xy are like, but 6xy and 6x are not.

判断两项是否同类时，忽略系数，只看变量字母及其幂。6xy 和 -2xy 是同类项，但 6xy 和 6x 不是。

Be careful with powers: x and x² are different because the exponent changes the variable part. Likewise, xy and x²y are not like terms.

注意幂：x 和 x² 不同，因为指数改变了变量部分。同样，xy 和 x²y 不是同类项。

Remember that xy and yx are like terms because multiplication is commutative, but it is often clearer to keep letters in a consistent order.

记住 xy 和 yx 是同类项，因为乘法满足交换律，但保持字母顺序一致通常更清晰。

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4. Single-Variable Examples | 单变量示例

For a simple expression such as 4x + 7x – 2x, add and subtract the coefficients while keeping the variable part unchanged. This gives (4 + 7 – 2)x = 9x.

对于像 4x + 7x – 2x 这样的简单表达式，加减系数，保持变量部分不变。得到 (4 + 7 – 2)x = 9x。

Another example is 2a – 9a + 5a. The coefficients 2, -9 and 5 add to -2, so the simplified result is -2a.

另一个例子是 2a – 9a + 5a。系数 2、-9 和 5 相加等于 -2，因此化简结果为 -2a。

4x + 7x – 2x = 9x

2a – 9a + 5a = -2a

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5. Multiple Variables and Powers | 多变量与幂

In expressions with more than one variable, only terms with the same combination of letters and the same powers can be combined. For example, 3xy + 5xy = 8xy, but 3xy + 5x cannot be simplified further.

在含有多个变量的表达式中，只有字母组合和幂都相同的项才能合并。例如，3xy + 5xy = 8xy，但 3xy + 5x 不能进一步化简。

With powers, 2x² + 3x² = 5x², while 2x² + 3x stays as it is because the variable parts differ. The same rule applies to terms like 4ab² and -ab².

对于幂，2x² + 3x² = 5x²，而 2x² + 3x 保持不变，因为变量部分不同。同样的规则适用于 4ab² 和 -ab²。

3xy + 5xy = 8xy

2x² + 3x² = 5x²

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6. Collecting with Negative Coefficients | 含负系数的合并

Negative coefficients are a common source of errors. Treat the sign in front of a term as part of its coefficient, then add all coefficients carefully.

负系数是常见的错误来源。把项前面的符号视为其系数的一部分，然后仔细相加所有系数。

For 5x – 3x – 4x, the coefficients are 5, -3 and -4. Their sum is 5 – 3 – 4 = -2, so the expression simplifies to -2x.

对于 5x – 3x – 4x，系数是 5、-3 和 -4。它们的和是 5 – 3 – 4 = -2，因此表达式化简为 -2x。

Another example: -7y + 2y – y means -7 + 2 – 1 = -6, so the answer is -6y. Remember that a lone y has coefficient 1.

另一个例子：-7y + 2y – y 意味着 -7 + 2 – 1 = -6，所以答案是 -6y。记住单独的 y 系数为 1。

A useful check is to substitute a small value for x before and after simplifying; both expressions should give the same result.

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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