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 this language lies the process of simplifying expressions. Collecting like terms is the most essential skill in algebra, enabling you to transform long, messy expressions into concise, elegant forms. In this revision guide, you will learn exactly what like terms are, how to collect them correctly, and how to avoid common exam pitfalls.

代数是数学的语言，而代数语言的核心在于化简表达式。合并同类项是代数中最基础的技能，它能帮助你将冗长、杂乱的表达式转化为简洁、优雅的形式。在本复习指南中，你将会学到什么是同类项、如何正确合并它们，以及如何避免考试中常见的陷阱。

1. What is an Algebraic Expression | 什么是代数表达式

An algebraic expression is a combination of numbers, variables (letters), and operation signs such as +, -, × and ÷. It does not contain an equals sign, unlike an equation. For example, \(3x + 2y – 5\) and \(4a^2 – 7a + 1\) are both algebraic expressions.

代数表达式是数字、变量（字母）以及运算符号（如 +、-、×、÷）的组合。与方程不同，表达式中不包含等号。例如，3x + 2y − 5 和 4a² − 7a + 1 都是代数表达式。

In an expression, each part separated by a plus or minus sign is called a term. Understanding the structure of a term is the first step toward mastering the art of collecting like terms.

在表达式中，由加号或减号隔开的每一部分都称为一项。理解项的构成是掌握合并同类项艺术的第一步。

2. Understanding Terms | 理解”项”

A term is a product of a coefficient and a variable part. The coefficient is the numerical factor, and the variable part consists of letters that may carry exponents. For example, in the term 5x², the coefficient is 5 and the variable part is x². In the term −3xy, the coefficient is −3 and the variable part is xy.

一项由系数与变量部分相乘构成。系数是数字因数，变量部分则由可能带有指数的字母组成。例如，在项 5x² 中，系数是 5，变量部分是 x²；在项 −3xy 中，系数是 −3，变量部分是 xy。

Terms that contain no variable at all, such as 7 or −2, are called constant terms. Every term in an expression carries a sign, and that sign belongs to the term immediately following it.

完全不含变量的项，如 7 或 −2，称为常数项。表达式中的每一项都带有正负号，而该符号属于紧随其后的那一项。

Term = Coefficient × Variable Part

项 = 系数 × 变量部分

3. Defining Like Terms | 同类项的定义

Like terms are terms that have exactly the same variable part, including the same exponents. The coefficients can be different, but the letters and their powers must match perfectly. For instance, 3x and 5x are like terms, and 2xy and −7xy are like terms.

同类项是指变量部分完全相同的项，包括指数也必须相同。系数可以不同，但字母及其幂必须完全匹配。例如，3x 与 5x 是同类项，2xy 与 −7xy 也是同类项。

Critically, x and x² are not like terms, because their exponents differ. Similarly, xy and x²y are not like terms. Constant terms, such as 4 and −9, are all like terms with each other.

必须注意：x 与 x² 不是同类项，因为它们的指数不同；同理，xy 与 x²y 也不是同类项。所有常数项（如 4 和 −9）彼此互为同类项。

Like Terms \| 同类项	Not Like Terms \| 非同类项
4x, −2x	4x, 4x²
3a²b, −a²b	3a²b, 3ab²
7, −12	7x, 7

4. Rules for Collecting Like Terms | 合并同类项的规则

The rule is straightforward: to collect like terms, add or subtract their coefficients while keeping the variable part exactly unchanged. Each group of like terms is combined independently of the other groups.

合并规则非常简单：合并同类项时，只需对它们的系数进行加减运算，而变量部分保持不变。每一组同类项独立于其他组进行合并。

Always pay attention to the sign in front of each term. A common strategy is to rewrite the expression with positive and negative terms separated, then combine them carefully. For terms with different variables or different exponents, simply leave them untouched.

始终要留意每一项前面的正负号。一个常用的策略是先将正项与负项分开写，再仔细合并。对于变量不同或指数不同的项，直接保留原样即可。

a(m) + b(m) = (a + b)m

a(m) − b(m) = (a − b)m

5. Basic Examples | 基本示例

Let us apply the rule to simple one-variable expressions. Consider 3x + 5x. Both terms have the variable x, so we add the coefficients: 3 + 5 = 8. Thus, 3x + 5x = 8x.

让我们把规则应用到简单的单变量表达式中。考虑 3x + 5x：两项的变量都是 x，因此将系数相加：3 + 5 = 8。所以 3x + 5x = 8x。

For subtraction, take 7a − 2a. The coefficients are 7 and 2, and 7 − 2 = 5, so the result is 5a. Even with negative coefficients, the rule holds: −2b + 5b = 3b, since −2 + 5 = 3.

再看减法：7a − 2a，系数分别是 7 和 2，7 − 2 = 5，所以结果是 5a。即使系数为负数，规则依然成立：−2b + 5b = 3b，因为 −2 + 5 = 3。

Constant terms also combine: 4 + 9 − 3 = 10. Therefore, the expression 3x + 4 + 5x + 9 − 3 simplifies to 8x + 10.

常数项同样可以合并：4 + 9 − 3 = 10。因此，表达式 3x + 4 + 5x + 9 − 3 可化简为 8x + 10。

6. Expressions with Multiple Variables | 含多变量的表达式

When an expression contains more than one variable, group like terms according to their variable parts. For example, simplify 4xy + 3x − 2xy + 5x. Group the xy terms: 4xy − 2xy = 2xy. Group the x terms: 3x + 5x = 8x. The final answer is 2xy + 8x.

当表达式含多个变量时，按变量部分对同类项进行分组。例如，化简 4xy + 3x − 2xy + 5x：先合并 xy 项：4xy − 2xy = 2xy；再合并 x 项：3x + 5x = 8x。最终答案为 2xy + 8x。

It is vital not to combine terms that merely look similar. The terms 2xy and 2yx are in fact like terms, because multiplication is commutative (xy = yx). However, 2xy and 2x are not like terms, since one contains y and the other does not.

切勿将看起来相似但变量不同的项合并。项 2xy 与 2yx 实际上是同类项，因为乘法满足交换律（xy = yx）。但是，2xy 与 2x 不是同类项，因为前者含有 y，而后者不含 y。

Consider 6p²q − 3pq² + 2p²q + pq². Combine 6p²q + 2p²q = 8p²q, and −3pq² + pq² = −2pq². The simplified form is 8p²q − 2pq². Notice that p²q and pq² are different groups.

来看 6p²q − 3pq² + 2p²q + pq²：合并 6p²q + 2p²q = 8p²q，同时 −3pq² + pq² = −2pq²。化简结果为 8p²q − 2pq²。注意 p²q 与 pq² 是两组不同的项。

7. Expressions with Brackets | 含括号的表达式

When brackets appear in an expression, you must first expand them using the distributive law, then collect like terms. The distributive law states that a(b + c) = ab + ac. Take 2(3x + 4) − 3(x − 2).

当表达式中出现括号时，必须先利用分配律去括号，然后再合并同类项。分配律指出：a(b + c) = ab + ac。以 2(3x + 4) − 3(x − 2) 为例。

First expand: 2(3x + 4) = 6x + 8, and −3(x − 2) = −3x + 6. Note carefully that the minus sign outside the second bracket changes both signs inside. Now combine like terms: 6x − 3x = 3x, and 8 + 6 = 14. The answer is 3x + 14.

先去括号：2(3x + 4) = 6x + 8，而 −3(x − 2) = −3x + 6。请特别注意，第二个括号外的负号会改变括号内每一项的符号。然后合并同类项：6x − 3x = 3x，8 + 6 = 14。答案为 3x + 14。

Another example: 5(2a − 1) + 2(3a + 4) = 10a − 5 + 6a + 8 = 16a + 3. Always double-check that every term inside the bracket has been multiplied by the factor outside.

再看一个例子：5(2a − 1) + 2(3a + 4) = 10a − 5 + 6a + 8 = 16a + 3。务必检查括号内的每一项都已被括号外的因子乘过。

8. Expressions with Fractions and Decimals | 含分数与小数的表达式

Fractions do not change the rules of collecting like terms, but you must be comfortable with fraction arithmetic. For example, (1/2)x + (1/3)x. Find a common denominator: 1/2 = 3/6 and 1/3 = 2/6. Thus, (3/6)x + (2/6)x = (5/6)x.

分数并不会改变合并同类项的规则，但你需要熟练进行分数运算。例如，(1/2)x + (1/3)x：找到公分母，1/2 = 3/6，1/3 = 2/6。因此，(3/6)x + (2/6)x = (5/6)x。

Decimals are even more straightforward: 0.4x + 1.3x = 1.7x. You may also encounter negative fractions: (3/4)y − (1/2)y = (3/4)y − (2/4)y = (1/4)y. The key is to treat the fraction as a single coefficient.

小数甚至更直接：0.4x + 1.3x = 1.7x。你也可能遇到负分数：(3/4)y − (1/2)y = (3/4)y − (2/4)y = (1/4)y。关键在于把分数当作一个整体的系数来处理。

If an expression contains a mixture of fractions and integers, convert them all to a common form before combining. For instance, (2/5)z + 0.3z = 0.4z + 0.3z = 0.7z.

如果表达式同时包含分数和整数，先将它们统一成相同形式再合并。例如，(2/5)z + 0.3z = 0.4z + 0.3z = 0.7z。

9. Common Mistakes to Avoid | 常见错误与防范

One of the most frequent errors in IGCSE exams is treating x and x² as like terms. They are not, because their exponents differ. Always check the exponent of every variable before combining.

IGCSE 考试中最常见的错误之一是把 x 与 x² 当作同类项。它们不是同类项，因为指数不同。合并前务必检查每个变量的指数。

Another common mistake is misapplying the negative sign. For example, 3x − 2(x − 5) is often incorrectly written as 3x − 2x − 5. The correct expansion is 3x − 2x + 10, which simplifies to x + 10. Remember: the minus sign applies to the whole bracket.

另一个常见错误是负号处理不当。例如，3x − 2(x − 5) 常被误写为 3x − 2x − 5，正确的去括号结果是 3x − 2x + 10，化简为 x + 10。记住：减号作用于整个括号。

A third error is combining different variables, such as adding 2x and 3y to get 5xy. This is never correct. If the variable parts differ, the terms cannot be collected together.

第三个错误是合并不同变量，例如将 2x 与 3y 相加得到 5xy。这是绝对错误的。如果变量部分不同，这些项就不能合并在一起。

10. Real-World Applications | 实际应用

Collecting like terms is not merely a classroom exercise; it appears everywhere in real-world problem solving. For example, suppose a rectangle has length (3x + 4) cm and width (2x − 1) cm. Its perimeter is 2(3x + 4) + 2(2x − 1) = 6x + 8 + 4x − 2 = 10x + 6 cm.

合并同类项不仅仅是课堂练习，它在实际问题的解决中无处不在。例如，假设一个矩形的长为 (3x + 4) cm，宽为 (2x − 1) cm。其周长为 2(3x + 4) + 2(2x − 1) = 6x + 8 + 4x − 2 = 10x + 6 cm。

In physics, when calculating total distance travelled in three stages, you might add expressions such as (2t + 3) + (5t − 1) + (t + 7), which simplifies to 8t + 9. Collecting like terms helps engineers and scientists keep their models clean and manageable.

在物理中，计算三段运动的总距离时，你可能需要将 (2t + 3) + (5t − 1) + (t + 7) 等表达式相加，化简后为 8t + 9。合并同类项帮助工程师和科学家保持模型的简洁与可控。

Even in finance, expressions like (1 + r)n + (1 + r)n−1 can be simplified using algebraic techniques. Mastering this skill allows you to handle complex formulas with confidence.

即使在金融领域，像 (1 + r)ⁿ + (1 + r)ⁿ⁻¹ 这样的表达式也可以利用代数技巧化简。熟练掌握这一技能，你就能自信地处理复杂公式。

11. Practice Problems with Solutions | 练习与解答

Now it is time to test your understanding. Simplify each of the following expressions, then check your answers below.

现在是检验你理解的时候了。请化简下列各表达式，然后对照下方答案。

1. 5x + 8x − 3x
2. 4a² + 6a − 2a² + 3a
3. 3xy − 2x + 5xy + 7x
4. 2(3b + 1) − 4(b − 5)
5. (2/3)m + (1/2)m − (1/6)m

Solutions: 1. 5x + 8x − 3x = 10x. 2. 4a² − 2a² = 2a², and 6a + 3a = 9a, so the answer is 2a² + 9a. 3. 3xy + 5xy = 8xy, and −2x + 7x = 5x, giving 8xy + 5x. 4. Expand to get 6b + 2 − 4b + 20 = 2b + 22. 5. Convert: 4/6 + 3/6 − 1/6 = 6/6 = 1, so the answer is 1m, or simply m.

解答：1. 5x + 8x − 3x = 10x。2. 4a² − 2a² = 2a²，6a + 3a = 9a，因此答案为 2a² + 9a。3. 3xy + 5xy = 8xy，−2x + 7x = 5x，结果为 8xy + 5x。4. 去括号得 6b + 2 − 4b + 20 = 2b + 22。5. 通分：4/6 + 3/6 − 1/6 = 6/6 = 1，所以答案为 1m，即 m。

12. Summary and Exam Tips | 总结与考试提示

Collecting like terms is the foundation of all algebraic manipulation. The process always follows three steps: first, identify every term and its sign; second, group together terms with identical variable parts; third, combine the coefficients in each group while keeping the variable part unchanged.

合并同类项是一切代数运算的基石。整个过程始终遵循三个步骤：首先，识别每一项及其符号；其次，将变量部分相同的项归为一组；最后，合并每组中的系数，同时保持变量部分不变。

In the IGCSE exam, you can gain easy marks by following these tips: expand brackets before collecting terms; use a pencil to circle different groups of like terms; never combine different variables; and always double-check all signs, especially after removing a bracket preceded by a minus sign.

在 IGCSE 考试中，遵循以下提示可以轻松得分：先展开括号再合并同类项；用铅笔圈出不同的同类项组；绝不合并不同变量；始终复查所有符号，尤其是去掉带有负号的括号之后。

With consistent practice, this skill will become second nature, and you will be able to simplify even the most intimidating expressions in seconds. Keep working through past paper questions, and you will see steady improvement in your algebra marks.

通过持续的练习，这项技能将成为你的本能反应，即使面对最复杂吓人的表达式，你也能在数秒内完成化简。坚持做历年真题，你的代数成绩一定会稳步提升。

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Collecting Like Terms | 合并同类项