📚 Collecting Like Terms | 合并同类项
In algebra, expressions are built from terms that contain variables and constants. Simplifying an expression by combining terms that share the same variable part is one of the most essential skills in IGCSE Mathematics. This process of 合并同类项 (collecting like terms) makes expressions cleaner, easier to evaluate, and is essential for solving equations.
在代数中，表达式由含有变量和常数的项构成。把具有相同变量部分的项合并在一起，是IGCSE数学中最核心的技能之一。合并同类项能让表达式更简洁、便于求值，也是解方程的重要基础。
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 3x, 5y², or −2ab. Like terms have exactly the same variable part, including the same powers of each variable. For example, 4x and −7x are like terms because both contain the variable x to the power 1. Similarly, 3x² and 8x² are like terms, but 3x and 3x² are not like terms because the exponents differ.
项是一个数、一个变量，或数与变量的乘积，例如3x、5y²或−2ab。同类项具有完全相同的变量部分，每个变量的幂也必须相同。例如4x与−7x是同类项，因为两者都只含x的一次方；3x²与8x²也是同类项；但3x与3x²不是同类项，因为指数不同。
2. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficients (the numbers in front) and look only at the variable part. Terms with the same letters raised to the same powers are like terms. Constant terms (numbers without variables) are also like terms with each other.
识别同类项时，先忽略系数（前面的数字），只观察变量部分。字母相同且对应幂也相同的项为同类项；没有变量的常数项之间也互为同类项。
| Expression | Like Terms |
| 5a, −2a, 7b, 3a | 5a, −2a and 3a are like terms; 7b is alone |
| 2xy, 3x²y, 5xy, −x²y | 2xy and 5xy; 3x²y and −x²y |
| 4, −9, 6p | 4 and −9 are like constants; 6p has no like term |
Remember that x²y and xy² are not like terms, because the same variables are raised to different powers.
注意：x²y 与 xy² 不是同类项，因为相同变量的指数不同。
3. Adding and Subtracting Like Terms | 同类项的加与减
When you collect like terms, add or subtract their coefficients while keeping the variable part unchanged. For example:
合并同类项时，只对系数进行加减，变量部分保持不变。例如：
3x + 5x = (3 + 5)x = 8x
9y − 4y = (9 − 4)y = 5y
The variable y does not change: you are only combining how many y’s you have. This is the central idea of collecting like terms.
变量y始终不变，你只是在合并”有几个y”。这正是合并同类项的核心思想。
4. Coefficients and Variables | 系数与变量
A coefficient is the number multiplied by the variable. If a term is just x, its coefficient is 1; if it is −x, the coefficient is −1. Remember to include the sign of the term when collecting.
系数是乘在变量前的数字。单独一个x，系数为1；−x的系数为−1。合并时务必带上每一项自带的正负号。
x + 2x − 3x = (1 + 2 − 3)x = 0x = 0
When the result is 0x, the whole term disappears, so the simplified answer is simply 0.
当结果为0x时，整个项消失，所以化简后的答案就是0。
5. Simplifying Expressions with Multiple Terms | 多组同类项的化简
A typical expression may contain several groups of like terms. Deal with each group separately, then write the simplified expression. For example:
一个表达式中常包含多组不同的同类项。逐组处理，最后写出化简结果。例如：
4a + 3b − 2a + 7b = (4a − 2a) + (3b + 7b) = 2a + 10b
It is often helpful to rearrange the terms so that each group is next to each other. Always keep every sign attached to its term when moving it.
先把同一组的项移到相邻位置会很有帮助。移动任何项时，都要让该项的符号始终跟着它。
6. Dealing with Negative Coefficients | 负系数的处理
Subtraction can be tricky because the minus sign belongs to the term that follows it. For instance:
减法容易出错，因为减号属于它后面的那一项。例如：
5m − 2m − 3n + n = (5 − 2)m + (−3 + 1)n = 3m − 2n
Think of each term as carrying its own sign: +5m, −2m, −3n and +n. Only combine the coefficients inside each group.
把每一项都看作自带符号：+5m、−2m、−3n和+n。只在各组内部合并系数即可。
7. Multiplying and Dividing Terms | 项的乘除
Collecting like terms works only for addition and subtraction. When multiplying terms, multiply coefficients and variables separately:
合并同类项只适用于加减运算。做乘法时，应分别相乘系数与变量：
2x × 3x = 6x²
When dividing, divide the coefficients and cancel common variable factors:
做除法时，先除系数，再约去相同的变量因子：
6x² ÷ 2x = 3x
These multiplication and division rules are often used together with collecting when simplifying more complex expressions.
这些乘除法则常与合并同类项配合，用于化简更复杂的代数式。
8. Expanding Brackets Then Collecting | 先展开括号再合并
In IGCSE exams, you are often asked to expand brackets first and then collect like terms. For example:
IGCSE考试常要求先展开括号，再合并同类项。例如：
2(x + 3) + 4(x + 1) = 2x + 6 + 4x + 4 = 6x + 10
The key is to expand carefully, then group the like terms before simplifying. Do not try to collect like terms across the brackets before expansion.
关键在于细心展开，然后再分组合并、化简。不要在展开括号之前就尝试合并”括号外与括号内”的同类项。
9. Common Mistakes | 常见错误
The most common errors include combining unlike terms, forgetting the coefficient of 1, and losing negative signs. Watch out for these traps:
最常见的错误包括：合并非同类的项、忘记系数1、丢失负号。请特别注意以下陷阱：
- Writing x² + x = x³: this is wrong because x² and x are not like terms. 写成x²+x=x³是错误的，因为x²与x不是同类项。
- Forgetting that x means 1x, so x + 3x = 4x. 忘记x就是1x，所以x+3x=4x。
- Dropping a minus sign, for example writing 4a − 2a + 3 as 2a + 3 correctly, but writing 4a − (2a + 3) as 2a + 3 incorrectly. 丢掉减号：4a−(2a+3)应化为2a−3，而不是2a+3。
Always double-check that the variables and exponents match exactly before adding or subtracting.
在进行加减之前，始终核对各变量的幂是否完全一致。
10. Exam Tips | 考试技巧
Show your working in clear steps. Underline or circle each group of like terms before combining them. Check your final answer by substituting a simple value, such as x = 1.
分步骤工整书写；合并前先用下划线或圆圈标出各组同类项；最后代入x=1之类的简单数值检验答案。
- Write down each intermediate line, especially when negative numbers are involved. 涉及负数时，务必写出每一步中间过程。
- Check signs twice: the most common lost mark is a misplaced minus sign. 把符号检查两遍：最常见的失分点是减号位置写错。
- Leave the final answer in alphabetical order, e.g. 2a + 10b, not 10b + 2a. 最后答案按字母顺序书写，例如写成2a+10b而不是10b+2a。
11. Practice Questions | 练习与答案
Try these exam-style questions. Simplify each expression fully.
请完成下列考试风格练习，将每个表达式彻底化简。
| Question | Answer |
| 1. 7x + 3y − 2x + 5y | 5x + 8y |
| 2. 4a² − a + 3a + 2a² | 6a² + 2a |
| 3. 3(2p + 1) + 2(4p − 3) | 14p − 3 |
If you got all three correct, you have mastered the basics. If not, review the sections above and try again.
如果三题全对，说明你已掌握基本要领；若有错误，请回顾对应小节后再试一次。
12. Conclusion | 小结
Collecting like terms is the foundation of algebra. Once you master identifying, adding, subtracting and combining terms correctly, solving equations, factorising and working with expressions becomes much easier. Practice daily to build both speed and accuracy.
合并同类项是代数学习的基石。一旦熟练掌握识别、加减和合并同类项，解方程、因式分解和处理表达式都会变得更加轻松。每天坚持练习，才能同时提升速度与准确率。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导