Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, combining like terms is one of the most essential skills you will ever learn. It allows you to simplify long expressions, solve equations with confidence, and build a strong foundation for everything from coordinate geometry to calculus. In this revision guide, we will break down exactly what like terms are, how to combine them correctly, and how to avoid the common traps that examiners love to set.

在代数中，合并同类项是你将学到的最重要技能之一。它能帮助你化简冗长的表达式、自信地解方程，并为从坐标几何到微积分的一切内容打下坚实的基础。在本复习指南中，我们将详细讲解什么是同类项、如何正确合并它们，以及如何避开出题人最爱设置的常见陷阱。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part, meaning the same letter(s) raised to the same power(s). The only thing that may differ between like terms is the numerical coefficient (the number in front of the variable).

同类项是指变量部分完全相同的项，即相同的字母且幂次完全相同。同类项之间唯一可以不同的是数值系数（变量前面的数字）。

For example, 3x and 5x are like terms because both have the variable x to the power 1. Similarly, 2x² and 7x² are like terms because both have x². However, 3x and 3x² are not like terms, because the powers of x are different.

例如，3x 和 5x 是同类项，因为两者都有变量 x 且幂次都是 1。类似地，2x² 和 7x² 是同类项，因为两者都有 x²。然而，3x 和 3x² 不是同类项，因为 x 的幂次不同。

Expression \| 表达式	Terms \| 各项	Can they combine? \| 能合并吗？
3x + 5x	3x , 5x	Yes, both are x¹ ✅
4x + 4x²	4x , 4x²	No, powers differ ❌
2xy + 7yx	2xy , 7yx	Yes, xy = yx ✅
6a + 2a + 3b	6a , 2a , 3b	6a & 2a yes; 3b cannot join ❌

2. Identifying the Variable and Power | 识别变量与幂次

To decide whether two terms are like terms, you must compare two things: the variable letter and the exponent. Both must match exactly.

要判断两个项是否为同类项，必须比较两样东西：变量字母和指数。两者都必须完全一致。

For example, x means x¹ (the power of 1 is usually not written). Therefore x and x² are not like terms because 1 ≠ 2. Similarly, 3ab and 3a are not like terms, because one has the factor b and the other does not.

例如，x 表示 x¹（通常不写出1次幂）。因此 x 和 x² 不是同类项，因为 1 ≠ 2。同样，3ab 和 3a 不是同类项，因为其中一个含有因式 b，另一个不含。

One helpful fact: the order of multiplied letters does not matter. By the commutative property, xy = yx, so 5xy and −2yx are like terms and can be combined to give 3xy.

一个有用的结论：字母相乘的顺序无关紧要。根据乘法交换律，xy = yx，所以 5xy 和 −2yx 是同类项，合并后得到 3xy。

3. Adding Like Terms | 加法合并同类项

When adding like terms, you add the coefficients and keep the variable part exactly the same. The variable does not change; only the number in front changes.

合并加法同类项时，将系数相加，变量部分保持完全不变。变量不改变，只有前面的数字改变。

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

Here the x part stayed as x; only 3 + 5 was computed. Now try with three terms:

这里 x 部分仍然是 x；只计算了 3 + 5。再来看看三项的情况：

4xy + 9xy + 2xy = (4 + 9 + 2)xy = 15xy

7m + 2m + 3m = 12m

Notice that even if coefficients are large or awkward, the rule never changes: combine only the numbers, keep the letter(s) untouched.

注意，即使系数很大或很复杂，规则也永远不会改变：只合并数字，字母保持不变。

4. Subtracting Like Terms | 减法合并同类项

When subtracting like terms, you subtract the coefficients and keep the variable part unchanged. Be especially careful with negative answers.

做减法合并时，将系数相减，变量部分保持不变。要特别注意结果为负数的情况。

6y − 2y = 4y

9x − 5x − 3x = (9 − 5 − 3)x = 1x = x

5a − 7a = −2a

Remember that the sign in front of a term belongs to that term. In the expression 9x − 5x − 3x, the 5x and 3x are both being subtracted. A common error is to subtract 5x but add 3x by mistake.

请记住，项前面的符号属于该项。在表达式 9x − 5x − 3x 中，5x 和 3x 都在被减去。一个常见错误是减去了 5x 却误加了 3x。

5. Combining Terms with Different Variables | 合并不同变量的项

Terms with different variables cannot be combined by addition or subtraction. The expression 2x + 3y cannot be simplified to 5xy or to 5(x + y); it simply remains 2x + 3y.

不同变量的项不能通过加法或减法合并。表达式 2x + 3y 不能化简为 5xy 或 5(x + y)；它只能保持为 2x + 3y。

However, you can reorder an expression so that like terms sit next to each other, then combine them group by group:

不过，你可以重新排列表达式，让同类项相邻，然后逐组合并：

3x + 4y + 5x − 2y = (3x + 5x) + (4y − 2y) = 8x + 2y

6p − 2q + 3q − p = (6p − p) + (−2q + 3q) = 5p + q

This grouping strategy is the key to handling long, messy expressions in your IGCSE exam.

这种分组策略是你在 IGCSE 考试中处理冗长复杂表达式的关键。

6. Constant Terms | 常数项

Numbers without any variable are called constant terms. All constants are like terms with each other, because they all have no variable part. So you can always combine them.

不带任何变量的数字称为常数项。所有常数项彼此都是同类项，因为它们都没有变量部分。因此它们之间总能合并。

7 + 3x − 2 + 5x = (7 − 2) + (3x + 5x) = 5 + 8x

Here the constants 7 and −2 combined to give 5, while the x-terms 3x and 5x combined to give 8x. The final answer is written with the variable term first: 8x + 5.

这里常数项 7 和 −2 合并得到 5，而 x 项 3x 和 5x 合并得到 8x。最终答案通常把变量项写在前面：8x + 5。

4 + 9y − 1 + 2y = 11y + 3

7. Simplifying Expressions with Brackets | 含括号表达式的化简

When brackets appear, you must first expand them using the distributive law, and only then combine like terms. Many exam questions test this two-step process.

当出现括号时，必须先利用乘法分配律去括号，然后才能合并同类项。许多考试题目考查的就是这一两步过程。

3(2x + 4) + 5x = 6x + 12 + 5x = 11x + 12

Watch out for a negative sign in front of a bracket. Every term inside the bracket changes sign:

特别小心括号前的负号。括号内每一项都要变号：

2(3a − 1) − 3(a + 2) = 6a − 2 − 3a − 6 = 3a − 8

−2(x − 3) = −2x + 6

In the last example, −2 multiplies both x and −3, giving −2x and +6. Students often forget the +6 and wrongly write −2x − 3.

在最后一个例子中，−2 分别乘以 x 和 −3，得到 −2x 和 +6。学生常常忘记 +6，错误地写成 −2x − 3。

8. Combining Fractions and Decimals | 分数与小数的合并

When coefficients are fractions, you need a common denominator before adding or subtracting them. The variable part is ignored until the arithmetic is done.

当系数是分数时，需要先通分再进行加减。变量部分在计算完成前不用管。

½x + ¼x = ²⁄₄x + ¼x = ¾x

−⅓y + ½y = −²⁄₆y + ³⁄₆y = ⅙y

For decimal coefficients, simply perform normal decimal arithmetic:

对于小数系数，直接进行普通的小数运算：

0.5x + 0.25x = 0.75x

1.3a − 0.8a = 0.5a

Fraction and decimal questions often appear in the non-calculator paper, so practise these by hand.

分数和小数的题目经常出现在非计算器试卷中，所以一定要动手练习。

9. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Below is a summary of the most frequent errors IGCSE students make when combining like terms, together with the correct approach.

下面总结 IGCSE 学生在合并同类项时最常见的错误，以及正确的做法。

Wrong \| 错误	Why it is wrong \| 错误原因	Correct \| 正确
3x + 2x² = 5x³	x and x² are not like terms; never multiply by combining	3x + 2x² stays as it is
2x + 3y = 5xy	Different variables cannot be added	2x + 3y stays as it is
4a − 2a = 2	The variable a was dropped	4a − 2a = 2a
−3(x + 2) = −3x + 2	Only x was multiplied by −3, not +2	−3(x + 2) = −3x − 6
5x − 3 − 2x = 3x + 5	Sign of 3 changed wrongly	5x − 2x − 3 = 3x − 3

Notice the pattern behind all these errors: students either forget that variable parts must match exactly, or they lose signs when moving terms. Always bring the sign along with the term.

注意所有这些错误背后的共同规律：学生要么

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Combining Like Terms | 合并同类项