IGCSE Maths Revision: Collecting Like Terms | IGCSE 数学复习：合并同类项

📚 IGCSE Maths Revision: Collecting Like Terms | IGCSE 数学复习：合并同类项

In algebra, simplifying expressions is one of the most valuable skills you can develop. Collecting like terms is often the very first step in solving an equation or simplifying a formula. Without it, even the simplest problem can look messy and confusing.

在代数中，化简表达式是你需要掌握的最重要技能之一。合并同类项通常是解方程或化简公式的第一步。如果没有这一步，即使是最简单的问题也会显得杂乱无章、令人困惑。

1. What Are Like Terms? | 什么是同类项？

A term in algebra is a single number, a single variable, or a product of numbers and variables. For example, 3x, -5y, 7ab, and 9 are all terms. The number in front of the variable is called the coefficient.

代数中的“项”是一个单独的数字、一个单独的变量，或数字与变量的乘积。例如，3x、-5y、7ab 和 9 都是项。变量前面的数字称为“系数”。

Like terms are terms that have exactly the same variables with exactly the same exponents. The coefficient can be different, but the variable part must match perfectly.

同类项是指变量部分完全相同、变量的指数也完全相同的项。系数可以不同，但变量部分必须完全一致。

4x and -2x are like terms because both have the variable x with exponent 1.

4x 和 -2x 是同类项，因为它们都含有变量 x，且指数都是 1。
3x² and 5x² are like terms because both have x squared.

3x² 和 5x² 是同类项，因为它们都含有 x 的平方。
2xy and 7xy are like terms because both contain the product x times y.

2xy 和 7xy 是同类项，因为它们都含有 x 乘以 y 的乘积。
6x and 6y are NOT like terms because the variables are different.

6x 和 6y 不是同类项，因为变量不同。
x and x² are NOT like terms because the exponents are different.

x 和 x² 不是同类项，因为指数不同。

2. Why Do We Need to Collect Like Terms? | 为什么需要合并同类项？

Collecting like terms transforms a long, messy expression into a short, clear one. This makes further algebra easier, such as substituting values, solving equations, or rearranging formulas.

合并同类项能把一个又长又乱的多项式表达式变成一个简短清晰的形式。这会使后续的代数运算更加轻松，例如代入数值、解方程或重排公式。

It also reduces the risk of arithmetic errors. When you combine terms, you focus on the coefficients, and the variable part simply stays the same.

同时，它也能降低计算错误的风险。当你在合并同类项时，只需要关注系数，变量部分保持不变即可。

Another reason is that many exam questions explicitly ask you to “simplify” an expression. A correct final answer must be fully simplified; otherwise you may lose marks even if your working is correct.

另一个原因是，许多考试题会明确要求你“化简”一个表达式。正确的最终答案必须完全化简；否则，即使过程正确也可能被扣分。

3. The Basic Rule: Add or Subtract Coefficients Only | 基本规则：只对系数进行加减

When you collect like terms, you add or subtract the coefficients, and you keep the variable part exactly the same. For example:

合并同类项时，你只需要对系数进行加减运算，而变量部分保持完全不变。例如：

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

Notice that we only added 3 and 5. The x stayed as x. We did not write 8x² or 8x + x; the variable part is unchanged.

注意，我们只把 3 和 5 相加。x 仍然是 x。我们不会写成 8x² 或 8x + x；变量部分不变。

The same rule applies to subtraction:

同样的规则也适用于减法：

9y – 4y = (9 – 4)y = 5y

If there is no number written in front of a variable, the coefficient is 1. For example, x means 1x, and -x means -1x.

如果变量前面没有写数字，那么它的系数就是 1。例如，x 代表 1x，-x 代表 -1x。

7a + a = 7a + 1a = 8a

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

If two terms have different variables, they are not like terms and cannot be combined. For example, 2a + 3b cannot be simplified to 5ab, 5a, or 5b. It must stay as 2a + 3b.

如果两项所含的变量不同，它们就不是同类项，不能合并。例如，2a + 3b 不能化简成 5ab、5a 或 5b。它必须保持为 2a + 3b。

This is a common mistake. Students sometimes try to combine every term they see, but algebra is not addition of all numbers blindly. Only identical variable parts can be added together.

这是一个常见错误。有些学生试图把看到的每一项都合并起来，但代数不是把所有数字盲目相加。只有完全相同的变量部分才能相加。

When simplifying an expression with multiple variables, collect each variable group separately:

在化简含有多个变量的表达式时，需要把每个变量组分别合并：

4m + 2n – m + 3n = (4m – m) + (2n + 3n) = 3m + 5n

Here, we combined the m terms first, then the n terms. The final expression contains two separate parts.

在这里，我们先把含 m 的项合并，再把含 n 的项合并。最终表达式包含两个独立的部分。

5. Powers and Exponents | 幂与指数

Like terms must have the same exponent on each variable. This means x and x² are NOT like terms, and x² and x³ are NOT like terms.

同类项必须在每个变量上有相同的指数。这意味着 x 和 x² 不是同类项，x² 和 x³ 也不是同类项。

For example, 4x² + 3x² = 7x². The exponent stays 2 throughout. You never add exponents when collecting like terms; that rule belongs to multiplying powers.

例如，4x² + 3x² = 7x²。指数始终保持为 2。在合并同类项时你绝不能把指数相加；指数相加的规则属于幂的乘法。

2x² + 5x + 3x² – x = (2x² + 3x²) + (5x – x) = 5x² + 4x

Notice that 5x² and 4x cannot be combined because they have different exponents. The final answer has two terms.

注意，5x² 和 4x 不能合并，因为它们的指数不同。最终答案包含两项。

This idea also applies to algebraic fractions and expressions with indices such as y³. Always check the exponent before deciding whether terms are like terms.

这个思路同样适用于代数分式和带指数（如 y³）的表达式。在判断是否为同类项之前，一定要先检查指数。

6. Negative Coefficients and Subtraction | 负系数与减法

Negative coefficients follow the same rules. You simply combine the signed numbers. For example:

负系数遵循同样的规则。你只需要把带符号的数合并即可。例如：

-4y + 2y = (-4 + 2)y = -2y

6x – 9x = (6 – 9)x = -3x

When an expression starts with a negative sign, keep that sign with the term. For example, -5a + 2a means you have -5a and +2a combined as (-5 + 2)a = -3a.

当表达式以负号开头时，那个负号必须属于该项。例如，-5a + 2a 表示你有 -5a 和 +2a，合并为 (-5 + 2)a = -3a。

Be careful with expressions like x – 2x. Since the first term has coefficient 1, we get:

要小心像 x – 2x 这样的表达式。由于第一项的系数是 1，我们得到：

x – 2x = 1x – 2x = (1 – 2)x = -1x = -x

Writing -1x is acceptable, but the preferred final answer is -x.

写 -1x 是可以的，但更简洁的最终答案通常写成 -x。

7. Multi-Step Simplification | 多步化简

Many exam expressions contain several different variable groups and constants. The best strategy is to identify each group, then combine each group separately.

许多考试题中的表达式会包含若干个不同变量组以及常数项。最好的策略是先识别出每一组，然后分别合并每一组。

Consider this example:

来看这个例子：

5 + 3x + 2y – x + 4 – 2y

First group the x terms, the y terms, and the constants:

先把含 x 的项、含 y 的项和常数项分别归类：

(3x – x) + (2y – 2y) + (5 + 4)

Now combine each group:

然后合并每一组：

2x + 0y + 9 = 2x + 9

The y terms cancel out completely because 2y – 2y = 0. The final expression is 2x + 9.

含 y 的项完全抵消，因为 2y – 2y = 0。最终表达式是 2x + 9。

Always show your grouping steps on separate lines. This helps you avoid careless mistakes and helps the examiner follow your logic.

一定要把归组过程分步写清楚。这样既能帮助你避免粗心错误，也能让阅卷老师看清你的思路。

8. Practice Questions | 练习题目

Try the following questions before checking the answers. Use the rules we have covered: combine only like terms, keep variable parts unchanged, and simplify fully.

请先尝试以下题目，再对照答案。运用我们学过的规则：只合并同类项，保持变量部分不变，并确保完全化简。

Question \| 题目	Answer \| 答案
1. 7x + 2x	9x
2. 5a – 8a	-3a
3. 4y² + 3y – y² + 2y	3y² + 5y
4. 6 + 2x – 3 – x	x + 3
5. 3m + 4n – 2m + n	m + 5n
6. 2x² + 3x – x² – 7x	x² – 4x

For question 3, remember that y² and y are not like terms. For question 4, treat the constants 6 and -3 as their own group. For question 6, keep x² and x separate.

对于第 3 题，记住 y² 和 y 不是同类项。对于第 4 题，把常数项 6 和 -3 单独看成一组。对于第 6 题，x² 和 x 要分开处理。

9. Common Mistakes to Avoid | 常见错误

Even strong students make simple errors when collecting like terms. Here are the most common traps in IGCSE exams.

即使是成绩很好的学生在合并同类项时也会犯简单错误。以下是 IGCSE 考试中最常见的陷阱。

Mistake 1: Combining 3x and 4y as 7xy. This is wrong because x and y are different variables.

错误一：把 3x 和 4y 合并成 7xy。这是错的，因为 x 和 y 是不同变量。
Mistake 2: Adding exponents when combining x² + x² as x⁴. The correct answer is 2x².

错误二：把 x² + x² 合并成 x⁴ 时把指数相加。正确答案是 2x²。
Mistake 3: Thinking x and 5x are different because one has a coefficient. They are like terms; x + 5x = 6x.

错误三：认为 x 和 5x 不是同类项，因为它们一个带系数。它们是同类项；x + 5x = 6x。
Mistake 4: Forgetting to subtract constants. In 8 + 3x – 2, the constant part is 8 – 2 = 6, so the answer is 3x + 6.

错误四：忘记对常数项做减法。在 8 + 3x – 2 中，常数部分为 8 – 2 = 6，所以答案是 3x + 6。
Mistake 5: Dropping the negative sign on a term. For example, 4x – 2y + 3y should be 4x + y, not 4x – y.

错误五：漏掉项的负号。例如，4x – 2y + 3y 的正确结果是 4x + y，不是 4x – y。

To avoid these mistakes, always rewrite the expression with grouping symbols before combining. This simple habit will save you many marks.

要避免这些错误，在合并之前，请先用分组符号重写表达式。这个简单的习惯能帮你保住很多分数。

10. Summary | 总结

Collecting like terms is a foundational algebra skill that appears in nearly every IGCSE Maths exam. The key ideas are simple: terms must have the same variable part and the same exponents to be like terms; you only add or subtract coefficients; and the variable part stays unchanged.

合并同类项是一项基础代数技能，几乎出现在每一份 IGCSE 数学试卷中。核心思路很简单：项必须拥有相同的变量部分和相同的指数才是同类项；你只需要对系数进行加减；变量部分保持不变。

Use grouping symbols to organise your work, watch out for negative signs, and never try to combine terms that are not alike. With regular practice, this process becomes automatic and fast.

使用分组符号来整理你的解题过程，注意负号，并且永远不要尝试合并不是同类的项。通过反复练习，这个过程会变得又快又熟练。

Remember: every expression you simplify is a step toward cleaner algebra and better marks. Keep practising, and you will see steady improvement.

请记住：每一次化简表达式，都是朝着更整洁的代数表达和更高分数迈进的一步。坚持练习，你会看到稳步提升。

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IGCSE Maths Revision: Collecting Like Terms | IGCSE 数学复习：合并同类项