Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain many terms. Collecting like terms is the process of simplifying such expressions by adding or subtracting terms that share the same variable part. This skill is essential for solving equations, factorising, and working with formulas in IGCSE Mathematics.

在代数中，表达式常常包含许多项。合并同类项就是通过加上或减去具有相同变量部分的项来简化表达式的过程。这项技能对于IGCSE数学中解方程、因式分解以及处理公式都至关重要。

1. What Is a “Term”? | 什么是“项”？

A term is a single mathematical expression built from numbers, variables, and possibly powers. Terms are separated by addition or subtraction signs. For example, in the expression 3x + 5y – 2, the terms are 3x, 5y, and -2.

项是由数字、变量和可能的幂构成的单个数学表达式。项之间用加号或减号分隔。例如，在表达式 3x + 5y – 2 中，各项分别是 3x、5y 和 -2。

Each term has two parts: the coefficient (the number in front) and the variable part (the letter or power). In the term -7x², the coefficient is -7 and the variable part is x².

每一项由两部分组成：系数（前面的数字）和变量部分（字母或幂）。在项 -7x² 中，系数是 -7，变量部分是 x²。

A term can be a constant: 5, -3, ½, 0.8

项可以是常数：5、-3、½、0.8
A term can have one variable: 2x, -4y, 9z

项可以含有一个变量：2x、-4y、9z
A term can have powers: x², 3a³, -2b⁴

项可以含有幂：x²、3a³、-2b⁴
A term can have multiple variables: xy, 5ab, -3mn

项可以含有多个变量：xy、5ab、-3mn

2. What Are “Like Terms”? | 什么是“同类项”？

Like terms are terms whose variable parts are exactly the same. This means the same variable letters must appear, with the same powers, in the same order. Only the coefficients may differ.

同类项是指变量部分完全相同的项。这意味着相同的变量字母必须以相同的幂、相同的顺序出现。只有系数可以不同。

For example, 3x and 5x are like terms because both have the variable part x. Similarly, 4x² and -2x² are like terms because both have the variable part x². However, x and x² are NOT like terms because the powers are different.

例如，3x 和 5x 是同类项，因为它们都有变量部分 x。同样地，4x² 和 -2x² 是同类项，因为它们都有变量部分 x²。然而，x 和 x² 不是同类项，因为幂不同。

For terms with multiple variables, the order does not matter: xy and yx are like terms, because multiplication is commutative – xy = yx.

对于含有多个变量的项，顺序无关紧要：xy 和 yx 是同类项，因为乘法满足交换律——xy = yx。

Like Terms (can be collected) 同类项（可以合并）	Unlike Terms (cannot be collected) 非同类项（不能合并）
2x and -5x 2x 和 -5x	2x and 5x² 2x 和 5x²
3y² and 4y² 3y² 和 4y²	3y² and 4y³ 3y² 和 4y³
7ab and -2ab 7ab 和 -2ab	7ab and 2a 7ab 和 2a
-4 and 9 -4 和 9	-4 and 9x -4 和 9x

3. The Golden Rule of Collecting | 合并的基本原则

When collecting like terms, you only add or subtract the coefficients. The variable part stays exactly the same. You never change the exponents or the letters during the operation.

合并同类项时，你只对系数进行加法或减法运算。变量部分保持不变。在运算过程中绝不要改变指数或字母。

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

Here, the coefficient 3 and 5 were added, while x remained unchanged. Similarly:

这里，系数 3 和 5 相加，而 x 保持不变。类似地：

12y² – 7y² = (12 – 7)y² = 5y²

The subtraction of coefficients gives 5, and y² is simply carried through. This operation is a direct application of the distributive property: ax + bx = (a + b)x.

系数相减得到 5，y² 被直接保留。这个运算是分配律的直接应用：ax + bx = (a + b)x。

4. Collecting Terms with Positive and Negative Coefficients | 合并正负系数项

Care must be taken with signs. When a term is subtracted, the negative sign belongs to the coefficient of that term. Recall the rules of integer arithmetic: adding a negative number is the same as subtracting a positive number.

处理符号时需格外小心。当一个项被减去时，负号属于该项的系数。回顾整数运算规则：加一个负数等于减一个正数。

Consider the expression 7x – 3x + 2x. Group the coefficients: (7 – 3 + 2)x = 6x. The answer is 6x.

考虑表达式 7x – 3x + 2x。合并系数：(7 – 3 + 2)x = 6x。答案是 6x。

Now consider 4a – 6a. This gives (4 – 6)a = -2a. The result is negative because 6 is larger than 4.

现在考虑 4a – 6a。这给出 (4 – 6)a = -2a。结果是负数，因为 6 比 4 大。

When working with larger expressions, it can be helpful to rewrite subtraction as adding a negative. An expression like 5x – 3y + 2x + 7y can be rewritten as 5x + 2x + (-3y) + 7y. Then collect each variable group separately:

处理大表达式时，将减法改写为加负数会很有帮助。像 5x – 3y + 2x + 7y 这样的表达式可以改写为 5x + 2x + (-3y) + 7y。然后分别合并每组变量：

(5 + 2)x + (-3 + 7)y = 7x + 4y

5. Different Letters Cannot Be Collected | 不同字母不可合并

Unlike terms containing different variables must remain separate in the simplified expression. For instance, 2x + 3y cannot be simplified further because x and y are different variables.

含有不同变量的非同类项在简化表达式中必须保持分开。例如，2x + 3y 无法进一步简化，因为 x 和 y 是不同的变量。

This does not mean the expression is wrong; it is already as simple as possible. A common error is to write 2x + 3y = 5xy, which is incorrect because 2x and 3y are not like terms.

这并不意味着表达式是错的；它已经是最简形式了。一个常见错误是写出 2x + 3y = 5xy，这是不正确的，因为 2x 和 3y 不是同类项。

Likewise, different powers of the same letter cannot be combined. The expression 5x + 4x² cannot become 9x³ or 9x². It must remain 5x + 4x².

同样地，同一个字母的不同幂次也不能合并。表达式 5x + 4x² 不能变成 9x³ 或 9x²。它必须保持 5x + 4x²。

6. Constants Are Like Terms | 常数项是同类项

Constants – numbers without variables – are all like terms with each other. They can be collected just like variable terms. In the expression 8 + 3x – 5 + 2x, the constants are 8 and -5, so they combine to give 3.

常数——不含变量的数字——彼此之间都是同类项。它们可以像变量项一样合并。在表达式 8 + 3x – 5 + 2x 中，常数是 8 和 -5，它们合并为 3。

Combining the x terms gives 3x + 2x = 5x. Therefore the simplified expression is:

合并 x 项得到 3x + 2x = 5x。因此简化后的表达式是：

8 + 3x – 5 + 2x = 5x + 3

Notice the constant 3 is usually written after the variable term, but writing 3 + 5x is equally acceptable. By convention, we write terms in descending order of power: first x³ terms, then x² terms, then x terms, then constants.

注意常数 3 通常写在变量项之后，但写成 3 + 5x 也同样可以接受。按照惯例，我们按幂的降序排列各项：先是 x³ 项，然后是 x² 项、x 项，最后是常数项。

7. Collecting Terms with Powers of x | 合并含 x 的幂的项

When an expression contains x, x², and x³ terms, each group must be collected separately. The simplified expression will still contain every different power that appeared originally.

当表达式包含 x、x² 和 x³ 项时，每一组必须分别合并。简化后的表达式仍将包含原来出现的每一种不同幂次。

Example: Simplify 4x² + 3x – 2x² + 7 – x + 5.

示例：化简 4x² + 3x – 2x² + 7 – x + 5。

Step 1 – Identify groups: x² terms are 4x² and -2x²; x terms are 3x and -x; constants are 7 and 5.

第一步——识别各组：x² 项是 4x² 和 -2x²；x 项是 3x 和 -x；常数项是 7 和 5。

Step 2 – Collect each group:

第二步——合并每组：

(4 – 2)x² = 2x²
(3 – 1)x = 2x
7 + 5 = 12

Step 3 – Write the final answer:

第三步——写出最终答案：

4x² + 3x – 2x² + 7 – x + 5 = 2x² + 2x + 12

8. The Colour-Coding Trick | 彩色标记技巧

When an expression is long, a useful technique is to underline or circle each group of like terms with the same marking before collecting. For example:

当表达式较长时，一个有用的技巧是在合并之前，用相同的标记给每组同类项加下划线或画圈。例如：

Simplify 6a + 3b – 2a + 8b – a.

化简 6a + 3b – 2a + 8b – a。

Underline the a terms in the same way: 6a, -2a, and -a all share the variable a. Circle the b terms: 3b and 8b share the variable b.

用相同方式标记 a 项：6a、-2a 和 -a 都含有变量 a。圈出 b 项：3b 和 8b 含有变量 b。

Now collect: (6 – 2 – 1)a + (3 + 8)b = 3a + 11b.

现在合并：(6 – 2 – 1)a + (3 + 8)b = 3a + 11b。

Even with three identical letters like 5p, -3q, 8p, 2r, -p, marking groups helps avoid losing track. Here the p terms are 5p, 8p, and -p; q and r are on their own. The result is 12p – 3q + 2r.

即使遇到三个相同的字母，例如 5p、-3q、8p、2r、-p，标记各组有助于避免遗漏。这里 p 项是 5p、8p 和 -p；q 和 r 各自独立。结果是 12p – 3q + 2r。

9. Collecting After Removing Brackets | 去括号后合并

Sometimes like terms are hidden inside brackets. In such cases, you must first expand the brackets using the distributive property, and then collect like terms.

有时同类项隐藏在括号内。在这种情况下，你必须先用分配律去掉括号，然后再合并同类项。

Example: Simplify 2(x + 3) + 4x.

示例：化简 2(x + 3) + 4x。

First expand: 2(x + 3) = 2x + 6. The expression becomes 2x + 6 + 4x.

先展开：2(x + 3) = 2x + 6。表达式变为 2x + 6 + 4x。

Now collect: 2x + 4x = 6x, so the final answer is 6x + 6.

现在合并：2x + 4x = 6x，所以最终答案是 6x + 6。

When subtracting a bracket, distribute the negative sign to every term inside. For example: 5x – (2x – 3) becomes 5x – 2x + 3. Notice the subtract sign changes -3 to +3. Collecting gives 3x + 3.

当减去一个括号时，要将负号分配给括号内的每一项。例如：5x – (2x – 3) 变为 5x – 2x + 3。注意减号将 -3 变为 +3。合并得到 3x + 3。

10. Common Mistakes to Avoid | 常见错误提醒

Even advanced students make errors when collecting like terms. Here are the most frequent traps in IGCSE exams.

即使是程度较好的学生，在合并同类项时也会犯错误。以下是IGCSE考试中最常见的陷阱。

Mistake 错误	Why it is wrong 错误原因	Correct version 正确做法
3x + 4y = 7xy	Different variables are not like terms. 不同变量不是同类项。	3x + 4y (cannot simplify) 3x + 4y（不能简化）
5x + 2x² = 7x²	Powers differ: x and x² are not like terms. 幂不同：x 与 x² 不是同类项。	5x + 2x² (cannot simplify) 5x + 2x²（不能简化）
4 – 7 = 3 (with variable parts lost)	The variable must be preserved. 变量必须保留。	4x – 7x = -3x
Forgetting the invisible coefficient	A term like x has coefficient 1, not 0. 像 x 这样的项系数是 1，不是 0。	x + 3x = 4x, not 3x

Always double-check your signs after collecting, especially when dealing with subtraction. Write down each intermediate step instead of doing everything mentally.

合并之后始终要复核你的符号，尤其是在处理减法时。写下每个中间步骤，而不要全部心算。

11. Practice Questions | 自测练习

Try collecting like terms in each expression below. Answers are given afterwards.

请尝试合并下列每个表达式中的同类项。答案随后给出。

7x + 2x – 5x

7x + 2x – 5x
3a + 4b – a + 2b

3a + 4b – a + 2b
5y² + 2y – 3y² + 4 – y – 1

5y² + 2y – 3y² + 4 – y – 1
2(3x + 4) + 5x

2(3x + 4) + 5x
8 – 2(4x – 1) + 3x

8 – 2(4x – 1) + 3x

Answers:

答案：

1. 7x + 2x – 5x = 4x

2. 3a – a + 4b + 2b = 2a + 6b

3. 5y² – 3y² = 2y²; 2y – y = y; 4 – 1 = 3, so the answer is 2y² + y + 3.

3. 5y² – 3y² = 2y²；2y – y = y；4 – 1 = 3，所以答案是 2y² + y + 3。

4. 2(3x + 4) = 6x + 8, then 6x + 8 + 5x = 11x + 8.

4. 2(3x + 4) = 6x + 8，然后 6x + 8 + 5x = 11x + 8。

5. 8 – 2(4x – 1) = 8 – 8x + 2 = 10 – 8x; adding 3x gives 10 – 5x.

5. 8 – 2(4x – 1) = 8 – 8x + 2 = 10 – 8x；加上 3x 得到 10 – 5x。

12. Final Summary | 最终总结

Collecting like terms is the cornerstone of algebraic manipulation. Always identify which terms share the exact same variable part – letters and exponents together – then combine only their coefficients. Keep unlike terms separate. Simplify systematically by grouping, and you will avoid most common errors.

合并同类项是代数运算的基石。始终识别哪些项共享完全相同的变量部分——字母和指数必须一致——然后只合并它们的系数。保持非同类项分开。通过分组系统化地化简，你将避免大多数常见错误。

On a final note, collect the x² terms first, then x terms, then constants. This order makes your work clear and keeps your final answer in the standard descending-power format preferred by examiners.

最后请注意，先合并 x² 项，然后是 x 项，最后是常数项。这个顺序会让你的解答清晰明了，并使最终答案符合考官偏好的标准降幂格式。

With sufficient practice, collecting like terms will become automatic. Master this essential skill now, and every future algebraic topic – from solving equations to expanding brackets to quadratic manipulation – will become significantly easier.

经过足够的练习，合并同类项将变得自然而然。现在就掌握这项关键技能，未来所有代数主题——从解方程、展开括号到二次式运算——都会变得容易得多。

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Collecting Like Terms | 合并同类项