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Collecting Like Terms: IGCSE Algebra Essentials | IGCSE 数学：合并同类项核心技巧

2026-08-21 17:36 · 本文部分内容由AI辅助创作

📚 Collecting Like Terms: IGCSE Algebra Essentials | IGCSE 数学：合并同类项核心技巧

Collecting like terms is one of the first and most important algebra skills in IGCSE mathematics. It allows you to simplify expressions so that later work with equations, brackets and graphs becomes much easier.

合并同类项是 IGCSE 数学中最先接触也最重要的代数技能之一。它可以帮你化简表达式，让之后解方程、去括号和画图等内容都变得更简单。

In this revision guide, you will learn what terms are, how to identify like terms, how to collect them correctly, and how to avoid common exam mistakes. Every rule is illustrated with worked examples written in the style of IGCSE Core and Extended papers.

在本复习指南中，你将学习什么是项、如何识别同类项、如何正确合并它们，以及如何避免考试中的常见错误。每条规则都配有符合 IGCSE Core 与 Extended 试卷风格的例题讲解。

1. What Is a Term? | 什么是项

In algebra, an expression is made up of terms which are separated by plus or minus signs. For example, 4x + 3y − 7 has three terms: 4x, 3y and 7. Strictly speaking, the sign in front of a term belongs to that term, so we can also read the same expression as 4x, +3y and −7.

在代数中，表达式由项组成，项与项之间用加号或减号隔开。例如，4x + 3y − 7 有三个项：4x、3y 和 7。严格来说，项前面的符号属于这个项本身，所以这个表达式也可以读作 4x、+3y 和 −7。

A term can be a single number, a single letter, or a product of numbers and letters. The expression 6a is one term, while 6 + a is two separate terms.

一个项可以是一个数、一个字母，或者数与字母的乘积。6a 是一个项，而 6 + a 是两个独立的项。

Terms can also involve powers and more than one letter. Each of the following is a single term:

项还可以含指数和多个字母。下面每一个都是单独的一个项：

btraction split an expression into separate terms. So 3 × x is one term, but 3 + x is two.

关键点在于：乘法让字母留在同一个项内，而加法和减法会把表达式切成不同的项。所以 3 × x 是一个项，而 3 + x 是两个项。

2. Coefficients, Variables and Constants | 系数、变量与常数

The number part of a term is called the coefficient. In 7x, 7 is the coefficient and x is the variable. In −3ab, the coefficient is −3.

项中的数字部分叫做系数。在 7x 中，7 是系数，x 是变量。在 −3ab 中，系数是 −3。

A term with no variable, such as 5 or −8, is called a constant. Constants can always be collected together.

不含变量的项，如 5 或 −8，叫做常数项。常数项总是可以相互合并。

Watch out for hidden coefficients. When a letter stands alone, its coefficient is 1, and when it has a minus sign only, its coefficient is −1:

要特别注意“隐藏系数”。当字母单独出现时，它的系数是 1；当它只带一个负号时，系数是 −1：

x = 1x　　−x = −1x　　a² = 1a²

The table below summarises the parts of a term.

下表总结了项的各个组成部分。

Term | 项 Coefficient | 系数 Variable part | 变量部分
8x 8 x
−5y² −5 y²
ab 1 ab
−x −1 x
12 12 none (constant) | 无（常数项）

3. Like Terms and Unlike Terms | 同类项与异类项

Like terms have exactly the same variable factors raised to the same powers. For example, 3x and −2x are like terms because both contain x to the power 1.

同类项具有完全相同的变量因子，并且这些变量的指数也相同。例如，3x 和 −2x 是同类项，因为它们都含有一次方的 x。

Unlike terms have different variables or different powers. For instance, 4x and 4x² are unlike terms because x and x² are not the same variable factor.

异类项有不同的变量或不同的指数。例如，4x 和 4x² 是异类项，因为 x 和 x² 不是相同的变量因子。

The order of letters inside a term does not matter, because multiplication is commutative. This means 3ab and 7ba are like terms, and 2xy and −9yx are like terms as well.

项内字母的顺序不影响判断，因为乘法满足交换律。这意味着 3ab 和 7ba 是同类项，2xy 和 −9yx 也是同类项。

4. The Golden Rule: Same Letter, Same Power | 黄金法则：同字母、同指数

The golden rule of collecting like terms is simple: two terms can be combined only if they have the same letter or letters, each raised to the same power. Nothing else matters.

合并同类项的黄金法则很简单：只有当两个项含有的字母完全相同、并且每个字母的指数也完全相同时，才能合并。其他因素一律不影响。

Use the decision table below to train your eye before you start writing answers.

在动笔计算之前，先用下面的判断表训练你的眼力。

Pair of terms | 两个项 Like or unlike? | 同类还是异类 Reason | 理由
3x, −8x Like 同类 Same letter x, power 1 | 字母与指数相同
2y², 9y² Like 同类 Same letter, same power 2 | 字母与指数 2 相同
4x, 4x² Unlike 异类 Different powers 1 and 2 | 指数不同
5ab, −3ba Like 同类 Order does not matter | 字母顺序无关
7m, 7n Unlike 异类 Different letters | 字母不同

A useful exam habit is to circle each group of like terms in a different colour before you combine them. This makes it almost impossible to miss a term.

一个很实用的考试习惯是：在合并之前，用不同颜色把每一组同类项圈出来。这样几乎不可能漏掉任何一项。

5. How to Collect Like Terms Step by Step | 合并同类项的分步方法

Once you can identify like terms, collecting them is simply a matter of adding or subtracting their coefficients and keeping the variable part unchanged.

一旦你能识别同类项，合并它就只是把系数相加或相减，而变量部分保持不变。

Follow these four steps every time:

每次都按下面四步操作：

Here is the method applied to a typical expression.

下面用一个典型表达式演示这一方法。

5x + 3y − 2x + 7y = (5x − 2x) + (3y + 7y) = 3x + 10y

Notice that 3x and 10y cannot be combined any further, because x and y are different letters. The answer 3x + 10y is fully simplified.

注意 3x 与 10y 不能再进一步合并，因为 x 和 y 是不同的字母。答案 3x + 10y 已经是完全化简的形式。

When terms contain powers, keep the power attached to the letter and only work with the numbers:

当项中含指数时，指数要一直跟着字母，你只需处理数字：

6a² + 4a² − a² = (6 + 4 − 1)a² = 9a²

6. Dealing With Negative Signs | 正确处理负号

Most lost marks in this topic come from mishandling negative signs, not from misunderstanding the concept. Treat every minus sign as belonging to the term that follows it.

这个知识点上丢分，多数不是概念没懂，而是负号处理错了。请把每个减号都看作属于它后面紧跟的那一项。

Work through these examples carefully:

仔细看下面这些例子：

6a − 9a = −3a　　−4p − 3p = −7p　　−2x + 10x = 8x

When the negative coefficient is larger than the positive one, the answer is negative. This is exactly the same arithmetic you already use with directed numbers.

当负系数的绝对值大于正系数时，结果就是负的。这和你处理有向数时用的算术完全一样。

Be equally careful with constant terms mixed into the expression.

当常数项混在表达式中时，同样要小心。

8 − 3x − 5 + x = (8 − 5) + (−3x + x) = 3 − 2x

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ginal expression and your answer. If the two values agree, your simplification is correct.

一个快速的检验方法是：把 x = 2 这样的数值分别代入原式和你化简后的答案。如果两个结果一致，说明化简正确。

7. Collecting Like Terms With Brackets | 含括号的合并同类项

When brackets appear, expand them first, then collect like terms. The distributive law tells you to multiply the term outside the bracket by every term inside it.

当表达式含括号时，要先展开括号，再合并同类项。分配律告诉你：用括号外的项去乘以括号内的每一项。

3(x + 4) + 2x = 3x + 12 + 2x = 5x + 12

A minus sign in front of a bracket changes the sign of every term inside it. This is the single most common source of error in IGCSE algebra questions.

括号前面的负号会改变括号内每一项的符号。这是 IGCSE 代数题中最常见的错误来源。

5 − (2x − 3) = 5 − 2x + 3 = 8 − 2x

With two brackets, expand both, then collect. Keep the working on separate lines so the examiner can follow your method and award method marks.

遇到两个括号时，先把两个都展开，再合并。书写时分行写，方便阅卷老师看懂你的思路并给出方法分。

4(2x − 3) − 2(x + 5) = 8x − 12 − 2x − 10 = 6x − 22

8. Fractions, Decimals and Hidden Coefficients | 分数、小数与隐藏系数

Terms with fractional coefficients are collected in exactly the same way. Add the fractions using a common denominator, and keep the letter unchanged.

系数是分数的项，合并方式完全相同。用公分母把分数相加，字母保持不变。

x/2 + x/3 = 3x/6 + 2x/6 = 5x/6

Decimal coefficients behave like ordinary numbers. Line up the decimal points and add carefully, remembering that 0.5x is the same as x/2.

小数系数和普通数字一样处理。小数点对齐后仔细相加，并记住 0.5x 与 x/2 是同一个值。

0.5x + 0.25x = 0.75x　　1.2y − 0.4y = 0.8y

Finally, do not confuse adding terms with multiplying terms. Adding collects like terms; multiplying creates a new term with a higher power.

最后，不要把“相加项”和“相乘项”搞混。相加是合并同类项；相乘会产生一个指数更高的新项。

3a + 2a = 5a　but　3a × 2a = 6a²

9. Using Like Terms to Solve Equations | 用合并同类项解方程

Collecting like terms is the essential first move in solving most linear equations. Simplify each side, then use inverse operations to isolate the unknown.

在解大多数一次方程时，合并同类项是必不可少的第一步。先把每一边化简，再用逆运算把未知数单独留下。

Example 1: solve 3x + 5x = 40.

例 1：解 3x + 5x = 40。

8x = 40　→　x = 40 ÷ 8 = 5

Example 2: solve 2x + 3x − 4 = 11.

例 2：解 2x + 3x − 4 = 11。

5x − 4 = 11　→　5x = 15　→　x = 3

7x − 2 = 4x + 10. Here you collect the x terms on one side and the constants on the other.

例 3：解 7x − 2 = 4x + 10。这里需要把含 x 的项集中到一边，把常数项集中到另一边。

7x − 4x = 10 + 2　→　3x = 12　→　x = 4

Always substitute your answer back into the original equation to confirm both sides are equal. This costs seconds and protects marks.

务必把答案代回原方程，确认两边相等。这一步只花几秒钟，却能保住分数。

10. Common Exam Mistakes to Avoid | 考试中常见的错误

Examiners report the same errors year after year in IGCSE algebra questions. Knowing them in advance is half the battle.

在 IGCSE 代数题中，阅卷老师每年都会看到同样的错误。提前了解它们，等于赢了一半。

A final answer that is not fully simplified can still lose the last accuracy mark, even if the earlier steps were correct.

即使前面的步骤都对，如果最终答案没有完全化简，依然可能丢掉最后的准确性分数。

11. IGCSE Exam-Style Worked Examples | IGCSE 考试风格例题精讲

Work through these four examples without looking at the solutions first. They cover the styles most frequently seen in Core and Extended papers.

请先自己动手做下面四道题，再看解答。它们涵盖了 Core 与 Extended 试卷中最常见的题型。

Example 1: Simplify 7a + 4b − 3a + 2b − 5.

例 1：化简 7a + 4b − 3a + 2b − 5。

= (7a − 3a) + (4b + 2b) − 5 = 4a + 6b − 5

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

例 2：化简 3x² + 5x − x² + 2x。

= (3x² − x²) + (5x + 2x) = 2x² + 7x

Example 3: The sides of a rectangle are (3x + 1) cm and (2x + 4) cm. Find its perimeter in terms of x.

例 3：一个长方形的边长分别为 (3x + 1) 厘米和 (2x + 4) 厘米。用 x 表示它的周长。

P = 2(3x + 1) + 2(2x + 4) = 6x + 2 + 4x + 8 = 10x + 10 cm

Example 4: Show that 5(2x − 1) − 3(x − 4) simplifies to 7x + 7.

例 4：证明 5(2x − 1) − 3(x − 4) 化简后等于 7x + 7。

10x − 5 − 3x + 12 = 7x + 7

In each case, the method is identical: expand if necessary, group the like terms, then combine the coefficients.

每一题的思路都一样：必要时先展开，再把同类项分组，最后合并系数。

12. Quick Revision Checklist | 快速复习清单

Use this checklist in the last few minutes before a test to make sure every rule is at your fingertips.

在考试前的最后几分钟，用这份清单确认每条规则都已烂熟于心。