Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Welcome to this comprehensive revision guide on collecting like terms, one of the fundamental skills in IGCSE Mathematics. This topic forms the building block for algebra, equation solving, and higher-level problem solving. In this guide, we will explore the concept step by step, from basic definitions to exam-style questions.

欢迎阅读本篇关于合并同类项的综合复习指南，这是IGCSE数学中最基础的技能之一。该知识点是代数、方程求解及更高层次问题解决的基础。在本指南中，我们将从基本定义到考试题型，一步步探索这一概念。

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1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable parts, including the same powers (exponents). The coefficients of the terms can be different, but the variable components must match identically. For example, 3x and 5x are like terms because both contain the variable x raised to the first power. Similarly, 4x² and -2x² are like terms because both contain x². However, 3x and 3x² are NOT like terms because the powers of x are different.

同类项是指变量部分完全相同（包括指数相同）的项。各项的系数可以不同，但变量部分必须完全匹配。例如，3x 和 5x 是同类项，因为两者都含有一次幂的变量 x。同样，4x² 和 -2x² 是同类项，因为两者都含有 x²。然而，3x 和 3x² 不是同类项，因为 x 的幂不同。

The key point to remember is that only like terms can be combined through addition or subtraction. Terms with different variable parts must remain separate in your final answer.

需要记住的关键点是：只有同类项才能通过加法或减法进行合并。变量部分不同的项必须在最终答案中保持分开。

3x + 5x = 8x (合并同类项 ✓)

3x + 5x² = 3x + 5x² (无法合并 ✗)

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2. Identifying Like Terms | 识别同类项

To identify like terms, you must carefully examine the variable part of each term. Look at both the letter(s) and the exponent(s) attached to them. The coefficient (the number in front) is irrelevant when determining whether terms are like. For instance, 7a and -2a are like terms because both contain the variable a. The terms 7ab and 7a are not like terms because the first contains both a and b, while the second contains only a.

要识别同类项，必须仔细检查每一项的变量部分。观察字母及其上标指数。判断是否为同类项时，系数（前面的数字）无关紧要。例如，7a 和 -2a 是同类项，因为两者都含有变量 a。而 7ab 和 7a 不是同类项，因为前者含有 a 和 b，后者只含有 a。

Here is a quick reference table to help you classify terms:

下面是一个帮助你快速分类的参考表格：

Terms 项	Like? 是否为同类项	Reason 原因
2x, 5x, -x	✓ Yes 是	Same variable x 相同变量 x
2x, 2x²	✗ No 否	Different powers 指数不同
3ab, 3ab²	✗ No 否	b vs b² 指数不同
4x²y, -x²y	✓ Yes 是	Same x²y 相同变量 x²y

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3. Adding and Subtracting Like Terms | 同类项的加减法

When combining like terms, you only add or subtract the coefficients. The variable part remains unchanged in the result. Think of it as counting objects: if you have 3 apples and you get 5 more apples, you now have 8 apples. The “apple” (the variable) stays the same; only the count changes.

合并同类项时，只需对系数进行加法或减法运算，变量部分保持不变。可以把它想成数数：如果你有3个苹果，又得到5个苹果，那么你就有8个苹果。”苹果”（即变量）不变，变化的只是数量。

Let us look at some worked examples:

让我们看一些示例：

Example 1 例1:

6a + 3a = (6 + 3)a = 9a

Example 2 例2:

7b – 4b = (7 – 4)b = 3b

Example 3 例3:

-2c + 5c = (-2 + 5)c = 3c

Notice that when the coefficient is 1, we often do not write it. For example, 1x is written simply as x. When the coefficient is -1, we write -x.

注意：当系数为1时，我们通常不写出来。例如，1x 简写为 x。当系数为-1时，我们写作 -x。

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4. Combining Multiple Types of Terms | 合并多种不同类型的项

In many expressions, you will encounter several different types of like terms at once. The most reliable strategy is to identify and group like terms first, then combine each group separately. You can use highlighting or circling to visually separate different types of terms on the page.

在很多表达式中，你会同时遇到几种不同类型的同类项。最可靠的策略是先将同类项分组，然后分别合并每一组。你可以在纸上用不同颜色高亮或画圈来区分不同类型的项。

Let us work through a detailed example:

让我们详细看一个例子：

4x + 3y – 2x + 5y

Step 1: Group x terms together and y terms together.

第一步：将 x 项和 y 项分别组合。

= (4x – 2x) + (3y + 5y)

Step 2: Combine each group.

第二步：分别合并各组。

= 2x + 8y

This cannot be simplified further because 2x and 8y are not like terms. The final answer is 2x + 8y.

由于 2x 和 8y 不是同类项，表达式无法再进一步化简。最终答案为 2x + 8y。

Let us try another example involving squares:

再看一个含平方项的例子：

5x² + 2x – 3x² + 7x

= (5x² – 3x²) + (2x + 7x)

= 2x² + 9x

Note that 2x² and 9x cannot be added because they have different powers.

注意：2x² 和 9x 的指数不同，不能相加。

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5. Simplifying Expressions with Brackets | 含括号表达式的化简

Sometimes you will need to expand brackets before collecting like terms. This is a two-step process: first, remove the brackets by multiplying out; second, collect and combine like terms. The distributive law states that a(b + c) = ab + ac.

有时需要先展开括号，再合并同类项。这是一个两步过程：首先通过乘法去除括号；其次再合并同类项。分配律指出 a(b + c) = ab + ac。

Let us examine a typical example:

让我们看一个典型例子：

3(2x + 4) + 2(5x – 1)

Step 1: Expand both brackets.

第一步：展开两个括号。

= 6x + 12 + 10x – 2

Step 2: Collect like terms.

第二步：合并同类项。

= (6x + 10x) + (12 – 2)

= 16x + 10

Be especially careful with minus signs before a bracket. For example, 4a – (2a + 3) means 4a – 2a – 3. The minus sign outside the bracket changes the sign of every term inside.

要特别注意括号前的负号。例如，4a – (2a + 3) 表示 4a – 2a – 3。括号外的负号会改变括号内每一项的符号。

Another example with a negative multiplier:

另一个含负数乘数的例子：

5(x + 2) – 2(x – 3)

= 5x + 10 – 2x + 6

= 3x + 16

Remember: -2 × (-3) = +6, so the +6 appears at the end.

记住：-2 × (-3) = +6，所以最后出现 +6。

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6. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Students frequently make several predictable errors when collecting like terms. Understanding these errors in advance will help you avoid them in the exam.

学生在合并同类项时经常犯几种可预见的错误。提前了解这些错误有助于你在考试中避免它们。

• Error 1 错误1: Adding unlike terms. For example, writing 3x + 5y = 8xy. This is incorrect. Since x and y are different variables, they cannot be added. Correct answer: 3x + 5y remains unchanged.

  错误1: 将非同类项相加。例如，错误地写 3x + 5y = 8xy。因为 x 和 y 是不同变量，不能相加。正确答案：3x + 5y 保持不变。

• Error 2 错误2: Confusing x with x². For instance, 2x + 3x² = 5x³ or 5x is wrong. The terms 2x and 3x² cannot be combined at all.

  错误2: 混淆 x 与 x²。例如，错误地认为 2x + 3x² = 5x³ 或 = 5x。2x 和 3x² 完全不能合并。

• Error 3 错误3: Dropping coefficients. When writing the final answer, some students forget to keep the coefficient 1 or -1. For example, x – 3x = 2x is wrong; the correct answer is -2x.

  错误3: 遗漏系数。有些学生在写最终答案时，忘记保留系数1或-1。例如，x – 3x = 2x 是错的；正确答案是 -2x。

• Error 4 错误4: Sign errors when removing brackets. For example, 2 – (3x + 1) should equal 2 – 3x – 1 = 1 – 3x, not 2 – 3x + 1.

  错误4: 去括号时符号错误。例如，2 – (3x + 1) 应等于 2 – 3x – 1 = 1 – 3x，而不是 2 – 3x + 1。

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7. Collecting Like Terms in Practical Contexts | 合并同类项在实际情境中的应用

Collecting like terms is not just a classroom exercise. It appears in perimeter calculations, formulas in physics, and many other real-world applications. For example, if a rectangle has length 3x + 2 and width 2x – 1, its perimeter can be found by adding all four sides.

合并同类项不只是课堂练习。它出现在周长计算、物理公式以及许多其他实际应用中。例如，如果一个矩形的长为 3x + 2，宽为 2x – 1，可以通过将四条边相加求出其周长。

Let us solve this problem:

让我们解决这个问题：

Perimeter = (3x + 2) + (2x – 1) + (3x + 2) + (2x – 1)

Group like terms:

合并同类项：

= (3x + 3x + 2x + 2x) + (2 – 1 + 2 – 1)

= 10x + 2

So the perimeter of the rectangle is 10x + 2 units. Notice how we applied the same collection rules within a real geometric context.

因此，该矩形的周长为 10x + 2 个单位。注意我们如何在真实的几何情境中应用相同的合并规则。

Another practical example: suppose you earn 5x + 3 dollars on Monday and 2x + 7 dollars on Tuesday. Your total earnings are (5x + 3) + (2x + 7) = 7x + 10 dollars.

另一个实际例子：假设你周一赚了 5x + 3 美元，周二赚了 2x + 7 美元。你的总收入为 (5x + 3) + (2x + 7) = 7x + 10 美元。

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8. Exam-Style Questions | 考试典型题型

The following are examples of questions that commonly appear on IGCSE papers. Try each one yourself before reading the solution.

以下例题常见于IGCSE试卷。请先自行尝试，再阅读答案。

Question 1 题目1: Simplify 7a + 3b – 2a + 4b.

题目1: 化简 7a + 3b – 2a + 4b。

Solution 解答: (7a – 2a) + (3b + 4b) = 5a + 7b

Question 2 题目2: Simplify 4(2x + 1) – 3(x – 2).

题目2: 化简 4(2x + 1) – 3(x – 2)。

Solution 解答: 8x + 4 – 3x + 6 = 5x + 10

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

题目3: 化简 3x² + 2x – x² + 5x – 4。

Solution 解答: (3x² – x²) + (2x + 5x) – 4 = 2x² + 7x – 4

Question 4 题目4: Simplify 2(3a + 4b) + 5(2a – b).

题目4: 化简 2(3a + 4b) + 5(2a – b)。

Solution 解答: 6a + 8b + 10a – 5b = 16a + 3b

Question 5 题目5: A triangle has side lengths 2x + 1, 3x – 2, and 4x + 3. Find its perimeter.

题目5: 一个三角形的三边长为 2x + 1、3x – 2 和 4x + 3。求其周长。

Solution 解答: (2x + 1) + (3x – 2) + (4x + 3) = 9x + 2

When tackling exam questions, always show your steps clearly. Write down the grouping stage before combining, as this demonstrates your method and earns method marks from examiners.

应对考试题目时，始终要清晰地写出步骤。先写出分组阶段再合并，这能展示你的解题方法并从考官处获得方法分。

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9. Quick Reference Summary | 快速参考总结

Here are the essential rules for collecting like terms that you should memorise before your exam.

以下是你在考试前应该牢记的合并同类项基本规则。

• Like terms have identical variable parts with identical exponents. Only the coefficients may differ.

  同类项具有相同的变量部分和指数。只有系数可以不同。

• To combine like terms, add or subtract the coefficients and keep the variable part unchanged.

  合并同类项时，对系数进行加减运算，变量部分保持不变。

• Terms with different variables or different powers cannot be combined.

  变量不同或指数不同的项不能合并。

• Always expand brackets first, then collect like terms.

  始终先展开括号，再合并同类项。

• Be careful with negative signs outside brackets: they reverse the sign of every term inside.

  注意括号外的负号：它会改变括号内每一项的符号。

• In your final answer, write terms in a logical order: often alphabetical or by descending powers (x², then x, then constants).

  在最终答案中，按合理顺序书写各项：通常按字母顺序或降幂排列（先 x²，再 x，最后常数项）。

Mastering the skill of collecting like terms will make all subsequent algebra topics much easier, including solving equations, factorising, and working with algebraic fractions. Make sure you practise enough problems to become fully confident.

掌握合并同类项的技能将让后续所有代数专题变得更加容易，包括解方程、因式分解和代数分数运算。请确保通过足够的练习来建立充分的信心。

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