Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In IGCSE algebra, one of the first essential skills is simplifying expressions by collecting like terms. This topic appears in almost every algebra question, from solving equations to expanding brackets.

在 IGCSE 代数中，一项基础技能是通过合并同类项化简表达式。该知识点几乎出现在每一道代数题中，从解方程到去括号都会用到。

1. What Are Like Terms? | 什么是同类项

A term is a number, a variable, or a product of numbers and variables such as 5x, −3a², or 7.

项是一个数、一个变量或数与变量的乘积，例如 5x、−3a² 或 7。

Like terms are terms whose variable parts are exactly the same, including the same powers. 4x and 9x are like terms, but 4x and 4x² are not.

同类项是指变量部分完全相同的项，包括相同的幂。4x 和 9x 是同类项，但 4x 和 4x² 不是。

The coefficient can be different, but the variable factor must match exactly.

系数可以不同，但变量部分必须完全一致。

2. Why Collect Like Terms? | 为什么合并同类项

Expressions often contain repeated variable terms. Collecting like terms makes them shorter and easier to use in equations, graphs, and problem solving.

表达式经常包含重复的变量项。合并同类项能让表达式更简短，便于在方程、图像和解题中使用。

When we collect like terms, we add or subtract only the coefficients while keeping the variable part unchanged.

合并同类项时，我们只对系数进行加减，变量部分保持不变。

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

For example, 4a + 3b + 2a simplifies to 6a + 3b, which is shorter but still equivalent.

例如，4a + 3b + 2a 可化简为 6a + 3b，表达式更短但值不变。

3. Identifying Like Terms | 识别同类项

Check both the variable letters and their exponents. For example, 2xy and −7xy are like terms, but 2xy and 3x²y are not because the exponent of x is different.

检查变量字母及其指数。例如 2xy 和 −7xy 是同类项，但 2xy 和 3x²y 不是，因为 x 的指数不同。

Constants such as 4, −9, and 12 are all like terms because they have no variable part.

常数项如 4、−9 和 12 都是同类项，因为它们没有变量部分。

Useful grouping: underline or circle terms with the same variable part before simplifying.

实用技巧：化简前先把变量部分相同的项划线或圈出分组。

Expression	Like terms?	Reason
2x and −7x	Yes	Same variable x
2xy and 3x²y	No	Different power of x
5 and −9	Yes	Both constants

4. Single-Variable Examples | 单变量例题

Simplify 7x + 2x − 3x. All three terms are like terms, so combine coefficients: 7 + 2 − 3 = 6.

化简 7x + 2x − 3x。三项都是同类项，因此合并系数：7 + 2 − 3 = 6。

7x + 2x − 3x = 6x

Example: Simplify 12a − 5a + a. Remember that a means 1a.

例题：化简 12a − 5a + a。记住 a 表示 1a。

12a − 5a + 1a = 8a

Always attach the sign in front of each term to its coefficient.

始终把每项前面的符号归入它的系数。

5. Multiple Variables and Powers | 多变量与幂

For expressions with several variables, collect each type separately: x terms with x terms, y terms with y terms, and so on.

对于含多个变量的表达式，要分别合并各类型：x 项与 x 项合并，y 项与 y 项合并，依此类推。

Example: 3x + 4y + 2x − y. Combine 3x + 2x = 5x and 4y − y = 3y.

例如：3x + 4y + 2x − y。合并 3x + 2x = 5x，4y − y = 3y。

3x + 4y + 2x − y = 5x + 3y

If an expression contains x² and x terms, collect powers separately: 2x² + 5x + 3x² − 2x gives 5x² + 3x.

如果表达式同时含 x² 和 x 项，要分别合并幂：2x² + 5x + 3x² − 2x 得到 5x² + 3x。

Powers must match exactly. x² and x are different types and cannot be combined.

幂必须完全一致。x² 和 x 属于不同类型，不能合并。

6. Collecting with Negative Coefficients | 含负系数的合并

Treat the sign in front of a term as part of its coefficient. For 5x − 8x + 2x, the coefficients are 5, −8, and 2.

把项前面的符号看作系数的一部分。对于 5x − 8x + 2x，系数分别是 5、−8 和 2。

5x − 8x + 2x = (5 − 8 + 2)x = −1x = −x

If a final coefficient is 1 or −1, write just x or −x, not 1x or −1x, unless a question asks for it.

如果最终系数是 1 或 −1，只写 x 或 −x，不写 1x 或 −1x，除非题目另有要求。

7. Constants and Mixed Terms | 常数项与混合项

Simplify 6 + 3x − 2 + 5x. First collect the x terms: 3x + 5x = 8x. Then collect the constants: 6 − 2 = 4.

化简 6 + 3x − 2 + 5x。先合并 x 项：3x + 5x = 8x。再合并常数项：6 − 2 = 4。

6 + 3x − 2 + 5x = 8x + 4

Present the answer in a tidy order: variable terms first, then the constant, unless a particular order is requested.

答案按整齐顺序书写：一般先写变量项，再写常数项，除非题目有特定顺序要求。

8. Brackets Before Collecting | 先去括号再合并

If brackets appear, expand them first. For 3(x + 2) + 4x, multiply: 3x + 6 + 4x, then collect: 7x + 6.

如果出现括号，要先展开。例如 3(x + 2) + 4x，先乘开：3x + 6 + 4x，再合并：7x + 6。

With a negative sign before a bracket, change signs of all terms inside: 5x − (2x − 3) = 5x − 2x + 3 = 3x + 3.

括号前是负号时，括号内所有项的符号都要变号：5x − (2x − 3) = 5x − 2x + 3 = 3x + 3。

For two brackets, expand each: 2(x + 1) + 3(x − 2) = 2x + 2 + 3x − 6 = 5x − 4.

如果有两个括号，分别展开：2(x + 1) + 3(x − 2) = 2x + 2 + 3x − 6 = 5x − 4。

9. Common Mistakes | 常见错误

Adding unlike terms: 3x + 2y is not 5xy. It stays as 3x + 2y.

把不同类项相加：3x + 2y 不等于 5xy，应保持 3x + 2y。
Forgetting the sign: 4x − 2x + 5x = 7x, not 11x.

忘记符号：4x − 2x + 5x = 7x，而不是 11x。
Changing exponents: x + x = 2x, not x².

错误改变指数：x + x = 2x，不是 x²。

10. Exam-Style Practice | 考试风格练习

Try these IGCSE-style questions before checking the answers.

在查看答案前，先尝试以下 IGCSE 风格练习。

Simplify 9p + 3q − 4p + 2q.
Simplify 7x − 3 + 2x + 8.
Simplify 4(x + 3) − 2(x − 1).

Answers: 5p + 5q; 9x + 5; 2x + 14.

答案：5p + 5q；9x + 5；2x + 14。

11. Extension: Fractions and Decimals | 拓展：分数与小数系数

The same rule applies with fractional or decimal coefficients. For ½x + ¼x, write 2/4 + 1/4 = 3/4, so the result is ¾x.

分数或小数系数遵循同样规则。对于 ½x + ¼x，先通分：2/4 + 1/4 = 3/4，结果是 ¾x。

0.2x + 0.7x = 0.9x

Keep coefficients as fractions or decimals according to the question’s instructions.

根据题目要求，系数可以保留为分数或小数。

12. Summary Checklist | 总结清单

Identify terms with exactly the same variable part and power.

识别变量部分和幂完全相同的项。
Add or subtract coefficients only; do not change the variable part.

只加减系数，不变变量部分。
Handle negative signs carefully and expand brackets first.

小心处理负号，并先去括号。
Write the final expression in a clear, simplified order.

最终表达式按清晰、简化的顺序书写。

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Collecting Like Terms | 合并同类项