Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Algebra is a language of symbols, and just as you group similar words in a sentence, you group similar terms in an expression. Collecting like terms is the process of simplifying an algebraic expression by adding or subtracting terms that share the same variable part. This skill underpins everything from solving linear equations to manipulating quadratic expressions, and it appears in nearly every IGCSE Mathematics examination.

代数是符号的语言，正如在一句话中把相似的词语归类一样，你也可以在表达式中将相似的项分组。合并同类项就是通过相加或相减具有相同变量部分的项来化简代数表达式的过程。这一技能支撑着从解线性方程到处理二次表达式的方方面面，几乎出现在每一场IGCSE数学考试中。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part, including the same powers. For example, 3x and 5x are like terms because both contain the variable x raised to the power 1. The coefficients (the numbers in front of the variables) do not need to be the same — only the variable part must match exactly.

同类项是指具有完全相同变量部分（包括相同的幂）的项。例如，3x 和 5x 是同类项，因为它们都包含一次幂的变量 x。系数（变量前面的数字）不需要相同——只有变量部分必须完全匹配。

Consider the following pairs of terms. Are they like terms?

考虑以下几组项。它们是同类项吗？

4a and 9a → Yes, both contain exactly the variable a.
4a 和 9a → 是，两者都恰好只含变量 a。
2b and 2c → No, the variables are different letters.
2b 和 2c → 不是，变量是不同的字母。
5x² and 5x → No, the powers of x are different.
5x² 和 5x → 不是，x 的幂不同。
−3xy and 7xy → Yes, both contain the product xy.
−3xy 和 7xy → 是，两者都包含乘积 xy。

2. Identifying Like Terms | 识别同类项

To identify like terms quickly, temporarily ignore the coefficient and focus on the variable part. Ask yourself: “Are the letters and their exponents identical?” If the answer is yes, the terms are like terms.

要快速识别同类项，请暂时忽略系数，专注于变量部分。问自己：”字母及其指数是否完全相同？”如果答案是肯定的，那么这些项就是同类项。

Expression	Variable Part	Like Terms
7p and −2p	p	Yes
3m² and 3m	m² vs m	No
4ab and −5ba	ab = ba	Yes
6 and 9	(no variable)	Yes

Notice that constants (numbers without variables) are all like terms with each other. Also, because multiplication is commutative, 4ab and −5ba are like terms — the order of variables does not matter.

注意，常数（不含变量的数字）彼此之间都是同类项。此外，由于乘法满足交换律，4ab 和 −5ba 是同类项——变量的顺序无关紧要。

3. The Rule for Combining | 合并规则

The rule for combining like terms is simple: add or subtract the coefficients, and keep the variable part exactly the same. You never change the variables or their exponents when collecting like terms.

合并同类项的规则很简单：对系数进行加减，并保持变量部分完全不变。合并时你绝不改变变量或其指数。

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

Here, the coefficient 3 and the coefficient 5 are added, while x remains unchanged. Similarly, subtraction works the same way.

在这里，系数 3 和系数 5 相加，而 x 保持不变。同样，减法也是如此运作。

12y − 4y = (12 − 4)y = 8y

What about negative coefficients? Treat them carefully, remembering that subtracting a negative is the same as adding a positive.

那负系数呢？小心处理它们，记住减去一个负数等于加上一个正数。

2x − (−7x) = 2x + 7x = 9x

4. Working with Different Variables | 处理不同变量

When an expression contains several different variables, you must group each variable separately. Terms with different variables cannot be combined — they are like apples and oranges.

当一个表达式包含多个不同的变量时，你必须分别对每个变量进行分组。具有不同变量的项无法合并——它们就像苹果和橙子一样。

Consider the expression:

考虑表达式：

3x + 2y − x + 5y

Group the x terms together and the y terms together:

将 x 项和 y 项分别归组：

(3x − x) + (2y + 5y) = 2x + 7y

The final simplified expression is 2x + 7y. Notice that 2x and 7y are not like terms, so they remain as separate terms in the answer.

最终化简结果为 2x + 7y。注意 2x 和 7y 不是同类项，因此它们作为独立的项保留在答案中。

Always include the sign of each term when grouping. A common strategy is to rewrite the expression with all plus signs before grouping:

分组时始终包含每项的正负号。一个常见策略是先将表达式改写为全部用加号连接的形式再进行分组：

3x + 2y − x + 5y = 3x + 2y + (−x) + 5y

This makes it easier to see which terms to add and which to subtract.

这使得看清哪些项要加、哪些项要减变得更加容易。

5. Combining More Complex Terms | 合并更复杂的项

Some expressions contain terms with multiple variables, such as 2xy or 3ab. These terms follow the same rule: combine only if the variable part matches exactly.

某些表达式包含多变量项，如 2xy 或 3ab。这些项遵循同样的规则：只有当变量部分完全匹配时才合并。

Simplify the following expression:

化简以下表达式：

5xy + 3x − 2xy + 4y

Here, 5xy and −2xy are like terms, but 3x and 4y are different from them and from each other. Combine only the xy terms:

这里，5xy 和 −2xy 是同类项，但 3x 和 4y 与它们不同，彼此之间也不同。只合并 xy 项：

(5xy − 2xy) + 3x + 4y = 3xy + 3x + 4y

Now consider an expression with constants:

现在考虑一个包含常数的表达式：

6a + 7 − 2a + 3

Combine the a terms and the constant terms separately:

分别合并 a 项和常数项：

(6a − 2a) + (7 + 3) = 4a + 10

Remember: constants are like terms with each other, but a constant and a variable term are never like terms.

请记住：常数彼此互为同类项，但常数和变量项永远不是同类项。

6. Handling Squares and Higher Powers | 处理平方与高次幂

A classic trap in IGCSE examinations is trying to combine x² with x. These are not like terms because the exponents differ. The variable part must match in both letter and power.

IGCSE考试中的一个经典陷阱是试图将 x² 与 x 合并。它们不是同类项，因为指数不同。变量部分必须在字母和幂上都匹配。

x² + x ≠ 2x or x³ or x²x

The expression x² + x is already in its simplest form. However, you can combine x² with other x² terms:

表达式 x² + x 已经是最简形式。然而，你可以将 x² 与其他 x² 项合并：

4x² + 3x² = 7x²

Higher powers follow the same principle. Terms with x³ only combine with other x³ terms; terms with x⁴ only combine with other x⁴ terms, and so on.

高次幂遵循相同的原则。含 x³ 的项只能与其他 x³ 项合并；含 x⁴ 的项只能与其他 x⁴ 项合并，依此类推。

Simplify the following expression:

化简以下表达式：

2x² + 3x − 5x² + x

Group x² terms and x terms separately:

分别对 x² 项和 x 项进行分组：

(2x² − 5x²) + (3x + x) = −3x² + 4x

Notice that −3x² and 4x remain separate in the final answer because they are not like terms.

注意 −3x² 和 4x 在最终答案中保持分离，因为它们不是同类项。

7. Simplifying with Brackets | 含括号的化简

Before collecting like terms, you may need to expand brackets using the distributive law. This is a two-step process: first expand, then collect like terms.

在合并同类项之前，你可能需要使用分配律展开括号。这是一个两步过程：先展开，再合并同类项。

3(2x + 4) + 5x

Step 1: Expand the brackets — multiply each term inside by 3:

第1步：展开括号——将括号内每一项乘以 3：

3(2x + 4) = 6x + 12

Step 2: Collect like terms — combine 6x and 5x:

第2步：合并同类项——合并 6x 和 5x：

6x + 12 + 5x = 11x + 12

When brackets are preceded by a negative sign, be especially careful. Distribute the negative sign to every term inside the brackets.

当括号前有负号时，要格外小心。将负号分配到括号内的每一项。

4x − (2x + 3) = 4x − 2x − 3 = 2x − 3

A frequent error is writing 4x − 2x + 3, which incorrectly treats the 3 as positive. Remember: the minus sign applies to the entire bracket.

一个常见错误是写成 4x − 2x + 3，这错误地将 3 当成了正数。请记住：负号适用于整个括号。

8. Common Mistakes | 常见错误

Students often lose marks on this topic for avoidable reasons. Here are the most common mistakes, along with corrections:

学生常常因可避免的原因在此主题上失分。以下是最常见的错误及其纠正方法：

Mistake	Correct	Explanation
x + x = x²	x + x = 2x	Adding like terms adds coefficients, not powers.
x² + x² = 2x⁴	x² + x² = 2x²	The exponent never changes when combining like terms.
3x + 4y = 7xy	3x + 4y cannot be simplified	Different variables cannot be combined.
5 − 2(x + 3) = 3(x + 3)	5 − 2x − 6 = −2x − 1	Distribute −2, not subtraction of 2 from 5.

Also, be careful with the invisible coefficient. A term like x has a coefficient of 1, not 0. Therefore:

同时，注意隐形系数。像 x 这样的项系数为 1，而不是 0。因此：

x − 3x = (1 − 3)x = −2x

9. Worked Examples | 例题详解

Let us work through three full examples step by step, exactly as you would in an examination.

让我们逐步完成三个完整例题，就像你在考试中做的那样。

Example 1: Simplify 7a + 3b − 2a + b.

例1：化简 7a + 3b − 2a + b。

Group like terms: (7a − 2a) + (3b + b).

归并同类项：(7a − 2a) + (3b + b)。

= 5a + 4b

The answer cannot be simplified further because 5a and 4b are unlike terms.

答案无法进一步化简，因为 5a 和 4b 是不同类项。

Example 2: Simplify 4(2x − 1) − 3x.

例2：化简 4(2x − 1) − 3x。

First expand: 8x − 4 − 3x.

先展开：8x − 4 − 3x。

Then collect: (8x − 3x) − 4 = 5x − 4.

再合并：(8x − 3x) − 4 = 5x − 4。

Example 3: Simplify 2x² + 3x − x² + 5 − 2x.

例3：化简 2x² + 3x − x² + 5 − 2x。

Group each type of term:

对每类项进行分组：

(2x² − x²) + (3x − 2x) + 5 = x² + x + 5

Always write the final answer in standard form: terms in descending order of powers (x² first, then x, then constants).

始终以标准形式书写最终答案：按幂次降序排列各项（先是 x²，然后是 x，最后是常数）。

10. Practice Problems | 练习题

Attempt the following problems on your own before checking the answers below. These are modelled on typical IGCSE questions.

先独立尝试以下问题，再对照下面的答案。这些题目以典型IGCSE试题为模型。

Problem set:

练习组：

Simplify 5x + 8x − 3x.
化简 5x + 8x − 3x。
Simplify 6p − 2q + 3p + 5q.
化简 6p − 2q + 3p + 5q。
Simplify 4r² − 2r + r² + 7r.
化简 4r² − 2r + r² + 7r。
Simplify 3(2a + 1) + 4a.
化简 3(2a + 1) + 4a。
Simplify 8 − 3(t − 2) + 5t.
化简 8 − 3(t − 2) + 5t。

Answers:

参考答案：

1. 10x (add 5 + 8 − 3 = 10).
1. 10x（5 + 8 − 3 = 10）。
2. 9p + 3q (combine p terms: 6 + 3; q terms: −2 + 5).
2. 9p + 3q（合并 p 项：6 + 3；q 项：−2 + 5）。
3. 5r² + 5r (combine r² terms: 4 + 1; r terms: −2 + 7).
3. 5r² + 5r（合并 r² 项：4 + 1；r 项：−2 + 7）。
4. 10a + 3 (expand: 6a + 3 + 4a = 10a + 3).
4. 10a + 3（展开：6a + 3 + 4a = 10a + 3）。
5. 2t + 14 (expand: 8 − 3t + 6 + 5t = (5t − 3t) + (8 + 6)).
5. 2t + 14（展开：8 − 3t + 6 + 5t = (5t − 3t) + (8 + 6)）。

11. Exam Tips | 考试技巧

To maximise your marks on this topic, follow these examination strategies.

要在这个主题上最大化你的分数，请遵循以下考试策略。

Underline or circle like terms in the expression before combining them. This helps you track which terms have been used.
在合并前，在表达式中给同类项画下划线或圈出来。这有助于你跟踪哪些项已被使用。
Always carry the sign that appears immediately before each term. A minus sign belongs to the term that follows it.
始终携带紧邻每项前面的符号。负号属于其后跟随的项。
Write one step per line in your working. Avoid doing multiple operations in your head at once.
在解题过程中每行只写一个步骤。避免同时在心中进行多个运算。
If a term has no written coefficient, remember it is 1 or −1. For example, −y means −1y.
如果一项没有写出系数，请记住它是 1 或 −1。例如，−y 表示 −1y。
After simplifying, check whether the answer makes sense by substituting a small value, such as x = 1, into both the original and simplified expressions. They should give the same numerical result.
化简后，代入一个小值（如 x = 1）检验答案是否合理，原始表达式和化简后的表达式应给出相同的数值结果。

Collecting like terms is a quick win in the IGCSE examination. With practice, you will recognise patterns instantly and complete these questions in seconds. Master this skill, and you will build a solid foundation for every algebra topic that follows.

合并同类项在IGCSE考试中是稳拿分的题型。通过练习，你将能迅速识别模式，在几秒内完成这些问题。掌握这一技能，你将为后续每一个代数主题打下坚实的基础。

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Collecting Like Terms | 合并同类项