Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Algebra is the language of mathematics, and at the heart of algebraic manipulation lies a fundamental skill: collecting like terms. This process allows us to simplify expressions, making them easier to understand, evaluate, and solve. Whether you are solving equations, rearranging formulas, or working with functions, mastering this skill is essential for success in IGCSE Mathematics.

代数是数学的语言，而代数运算的核心是一项基本技能：合并同类项。这一过程使我们能够简化表达式，使其更易于理解、计算和求解。无论你是解方程、重排公式，还是处理函数，掌握这项技能对于在IGCSE数学中取得成功都至关重要。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variables raised to exactly the same powers. The numerical coefficients may be different, but the variable parts must match precisely. For example, 3x and 5x are like terms because both contain the variable x raised to the power 1. Similarly, 4x² and -2x² are like terms, but 4x² and 4x are not like terms because the powers of x differ.

同类项是指具有完全相同的变量且变量的指数完全相同的项。数字系数可以不同，但变量部分必须精确匹配。例如，3x 和 5x 是同类项，因为两者都包含一次方的变量 x。类似地，4x² 和 -2x² 是同类项，但 4x² 和 4x 不是同类项，因为 x 的指数不同。

Like terms: 3x and 5x | 4x² and -2x² | 7xy and 3xy

Unlike terms: 3x and 3x² | 4xy and 4x | 2a and 2b

2. Identifying the Coefficient and Variable | 识别系数和变量

Every algebraic term consists of two parts: the coefficient (the numerical factor) and the variable part (the letters). In the term 7x, 7 is the coefficient and x is the variable. When the coefficient is 1, it is often omitted, so x means 1x. When the coefficient is -1, it is written as -x. Understanding this notation is crucial when collecting like terms, as you must identify which parts of each term are the same.

每个代数项由两部分组成：系数（数字因子）和变量部分（字母）。在项 7x 中，7 是系数，x 是变量。当系数为 1 时，通常省略不写，因此 x 表示 1x。当系数为 -1 时，写作 -x。理解这种记法在合并同类项时至关重要，因为你需要识别每一项中哪些部分相同。

Coefficient: the numerical multiplier of the variable | 系数：变量的数字乘数
Variable: the letter(s) representing an unknown quantity | 变量：表示未知量的字母
Constant term: a term with no variable, e.g., 5 or -3 | 常数项：没有变量的项，例如 5 或 -3

3. The Rule of Collecting Like Terms | 合并同类项的规则

The rule for collecting like terms is simple: add or subtract the coefficients while keeping the variable part unchanged. This works because like terms represent the same quantity, so they can be combined using the distributive property of multiplication over addition. For instance, 3x + 5x can be thought of as (3 + 5)x = 8x.

合并同类项的规则很简单：对系数进行加或减，同时保持变量部分不变。这是因为同类项代表相同的量，所以可以利用乘法对加法的分配律进行合并。例如，3x + 5x 可以看作 (3 + 5)x = 8x。

3x + 5x = 8x | 7y – 2y = 5y | 4a + a = 5a

4. Collecting Like Terms with Positive and Negative Coefficients | 处理正负系数的合并

Expressions often contain both positive and negative coefficients, which requires careful attention to signs. When collecting such terms, treat the sign before each term as part of that term’s coefficient. For example, in the expression 5x – 3x + 2x, the coefficients are +5, -3, and +2. Adding these gives 5 – 3 + 2 = 4, so the result is 4x.

表达式通常同时包含正系数和负系数，这需要特别注意符号。在合并此类项时，应将每项前面的符号视为该系数的一部分。例如，在表达式 5x – 3x + 2x 中，系数分别是 +5、-3 和 +2。将它们相加得到 5 – 3 + 2 = 4，所以结果是 4x。

Consider a more complex example: 4x² – 2x + 3x² + 5x. First, identify the like terms: 4x² and 3x² are like terms, and -2x and 5x are like terms. Combining the x² terms gives 4 + 3 = 7, so 7x². Combining the x terms gives -2 + 5 = 3, so 3x. The simplified expression is 7x² + 3x.

考虑一个更复杂的例子：4x² – 2x + 3x² + 5x。首先，识别同类项：4x² 和 3x² 是同类项，-2x 和 5x 是同类项。合并 x² 项得到 4 + 3 = 7，即 7x²。合并 x 项得到 -2 + 5 = 3，即 3x。简化后的表达式为 7x² + 3x。

5. Working with Multiple Variables | 处理多个变量

Expressions can contain more than one variable, such as x and y, or x² and xy. When collecting like terms, you must group terms that have identical variable combinations. For example, 6x + 3y – 2x + y can be simplified by collecting the x terms: 6x – 2x = 4x, and the y terms: 3y + y = 4y. The simplified expression is 4x + 4y.

表达式可以包含多个变量，例如 x 和 y，或 x² 和 xy。在合并同类项时，必须将具有相同变量组合的项分组。例如，6x + 3y – 2x + y 可以通过合并 x 项：6x – 2x = 4x，以及 y 项：3y + y = 4y 来简化。简化后的表达式为 4x + 4y。

It is important to note that terms like xy and yx are the same because multiplication is commutative. Therefore, 3xy and -5yx are like terms and can be combined: 3xy – 5xy = -2xy.

需要注意的是，xy 和 yx 这样的项是相同的，因为乘法满足交换律。因此，3xy 和 -5yx 是同类项，可以合并：3xy – 5xy = -2xy。

6. Collecting Like Terms with Powers | 处理含幂的同类项

When terms involve powers (indices), the exponents must match exactly for the terms to be considered “like.” The term x² is different from x³ because they represent different quantities. Similarly, x² and xy are not like terms because the variable parts are different. Only terms with identical variable parts and identical exponents can be combined.

当项涉及幂（指数）时，指数必须完全匹配才能被视为同类项。项 x² 与 x³ 不同，因为它们代表不同的量。类似地，x² 和 xy 不是同类项，因为变量部分不同。只有变量部分和指数完全相同的项才能合并。

5x² + 2x² = 7x² (valid) | 5x² + 2x³ = 5x² + 2x³ (cannot be simplified)

In the second case above, 5x² and 2x³ are not like terms because the exponents of x are different. The expression must be left as it is, which is acceptable and correct.

在上面的第二个例子中，5x² 和 2x³ 不是同类项，因为 x 的指数不同。表达式必须保持原样，这是可接受且正确的。

7. Constants Are Like Terms | 常数项也是同类项

Constant terms — numbers without any variables — are also considered like terms with each other. This means you can collect all the constant terms together. For example, in the expression 5x + 3 – 2x + 7, the constants 3 and 7 are like terms. Combining them gives 10, and combining the x terms gives 3x, resulting in 3x + 10.

常数项——不含任何变量的数字——彼此之间也被视为同类项。这意味着你可以将所有常数项合并在一起。例如，在表达式 5x + 3 – 2x + 7 中，常数 3 和 7 是同类项。将它们合并得到 10，合并 x 项得到 3x，结果为 3x + 10。

When simplifying expressions, it is conventional to write the terms in a standard order: first the variable terms (often in descending powers), then the constant term at the end. This makes the expression easier to read and compare.

在简化表达式时，通常按标准顺序书写各项：先是变量项（通常按降幂排列），最后是常数项。这使表达式更易于阅读和比较。

8. Common Mistakes to Avoid | 需要避免的常见错误

Even experienced students can make mistakes when collecting like terms. Being aware of these common pitfalls will help you avoid them in your exams. The most frequent errors include confusing x with x², forgetting to include the sign before a term, and attempting to combine unlike terms.

即使是经验丰富的学生在合并同类项时也会犯错。了解这些常见陷阱将帮助你在考试中避免它们。最常见的错误包括混淆 x 和 x²、忘记项前的符号，以及试图合并非同类的项。

Incorrect \| 错误	Reason \| 原因	Correct \| 正确
3x + 2x² = 5x³	x and x² are unlike terms \| x 和 x² 不是同类项	3x + 2x² (leave unchanged)
7x – 3x = 4	The variable x must be kept \| 必须保留变量 x	7x – 3x = 4x
2x + 3y = 5xy	Different variables cannot be combined \| 不同变量不能合并	2x + 3y (leave unchanged)

9. Worked Examples | 例题精讲

Let us work through several examples step by step to reinforce the methods discussed. Each example demonstrates a different aspect of collecting like terms, from simple to more complex expressions.

让我们逐步完成几个例题，以加深对上述方法的理解。每个例题展示了合并同类项的不同方面，从简单到复杂的表达式。

Example 1 | 例 1: Simplify 8a + 3b – 5a + 2b | 化简 8a + 3b – 5a + 2b

Step 1: Identify like terms. The a terms are 8a and -5a. The b terms are 3b and 2b.
Step 2: Combine the a terms: 8 – 5 = 3, so 3a.
Step 3: Combine the b terms: 3 + 2 = 5, so 5b.
Answer: 3a + 5b

步骤1：识别同类项。a 项是 8a 和 -5a。b 项是 3b 和 2b。
步骤2：合并 a 项：8 – 5 = 3，即 3a。
步骤3：合并 b 项：3 + 2 = 5，即 5b。
答案：3a + 5b

Example 2 | 例 2: Simplify 4x² + 6x – 2x² + 3x – 1 | 化简 4x² + 6x – 2x² + 3x – 1

Step 1: Group x² terms: 4x² – 2x² = 2x².
Step 2: Group x terms: 6x + 3x = 9x.
Step 3: The constant term is -1.
Answer: 2x² + 9x – 1

步骤1：合并 x² 项：4x² – 2x² = 2x²。
步骤2：合并 x 项：6x + 3x = 9x。
步骤3：常数项为 -1。
答案：2x² + 9x – 1

Example 3 | 例 3: Simplify 5xy + 2x – 3xy + x | 化简 5xy + 2x – 3xy + x

Step 1: Group xy terms: 5xy – 3xy = 2xy.
Step 2: Group x terms: 2x + x = 3x.
Answer: 2xy + 3x

步骤1：合并 xy 项：5xy – 3xy = 2xy。
步骤2：合并 x 项：2x + x = 3x。
答案：2xy + 3x

10. Applications in Solving Equations | 在解方程中的应用

Collecting like terms is not only useful for simplifying expressions — it is a critical step in solving equations. When solving an equation like 3x + 5 = 2x + 9, you must first collect the x terms on one side and the constants on the other. This is a form of collecting like terms applied to equation solving.

合并同类项不仅对简化表达式有用——它是解方程的关键步骤。在解方程 3x + 5 = 2x + 9 时，你必须先将 x 项集中到一边，常数项集中到另一边。这就是合并同类项在解方程中的一种应用。

3x + 5 = 2x + 9

3x – 2x = 9 – 5

x = 4

The ability to quickly and accurately collect like terms directly impacts your efficiency in solving equations, inequalities, and simultaneous equations throughout the IGCSE course.

快速准确地合并同类项的能力直接影响你在整个IGCSE课程中解方程、不等式和联立方程的效率。

11. Practice Questions | 练习题目

Mastery comes with practice. The following questions cover a range of difficulties. Attempt each question before checking the solution provided below.

熟能生巧。以下题目涵盖了不同难度。请先尝试作答，再对照下方答案。

Question 1 | 题目 1: Simplify 7x + 2x – 5x | 化简 7x + 2x – 5x

Question 2 | 题目 2: Simplify 3a + 4b – a + 6b | 化简 3a + 4b – a + 6b

Question 3 | 题目 3: Simplify 5x² + 3x – 2x² + x – 4 | 化简 5x² + 3x – 2x² + x – 4

Question 4 | 题目 4: Simplify 8m – 3n + 2m – 5m + n | 化简 8m – 3n + 2m – 5m + n

Solutions | 答案：

Q1: 7 + 2 – 5 = 4, so 4x | Q1: 7 + 2 – 5 = 4，即 4x

Q2: a terms: 3 – 1 = 2, so 2a; b terms: 4 + 6 = 10, so 10b. Answer: 2a + 10b | Q2：a 项：3 – 1 = 2，即 2a；b 项：4 + 6 = 10，即 10b。答案：2a + 10b

Q3: x² terms: 5 – 2 = 3, so 3x²; x terms: 3 + 1 = 4, so 4x; constant: -4. Answer: 3x² + 4x – 4 | Q3：x² 项：5 – 2 = 3，即 3x²；x 项：3 + 1 = 4，即 4x；常数：-4。答案：3x² + 4x – 4

Q4: m terms: 8 + 2 – 5 = 5, so 5m; n terms: -3 + 1 = -2, so -2n. Answer: 5m – 2n | Q4：m 项：8 + 2 – 5 = 5，即 5m；n 项：-3 + 1 = -2，即 -2n。答案：5m – 2n

12. Summary and Key Takeaways | 总结与要点回顾

Collecting like terms is a cornerstone of algebraic manipulation. It enables you to transform complex, unwieldy expressions into concise, manageable forms. Remember the fundamental rule: combine only the coefficients of identical variable parts, and preserve the variable part exactly. Pay careful attention to signs, exponents, and multiple variables to avoid common errors.

合并同类项是代数运算的基石。它使你能够将复杂、冗长的表达式转化为简洁、易处理的形式。记住基本规则：只合并相同变量部分的系数，并保持变量部分不变。要特别注意符号、指数和多个变量，以避免常见错误。

With regular practice, collecting like terms will become second nature, and you will find that this skill underpins nearly every topic in IGCSE Mathematics — from solving equations to graphing functions. Keep practising, and you will build a solid foundation for all your future mathematical studies.

通过定期练习，合并同类项将成为你的自然反应，你会发现这项技能支撑着IGCSE数学几乎每一个主题——从解方程到绘制函数图像。持续练习，你将为未来所有的数学学习打下坚实的基础。

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Collecting Like Terms | 合并同类项