📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, expressions such as 3x + 5x – 2x are much easier to handle when we simplify them. The process of adding or subtracting terms that have exactly the same variable and the same power is called collecting like terms. This topic is a core skill for solving equations, expanding brackets and rearranging formulae.
在 IGCSE 代数中，像 3x + 5x – 2x 这样的代数式在化简后会更容易处理。将字母相同且指数也相同的项相加或相减，这个过程叫做合并同类项。这是解方程、去括号和公式变形的基础技能。
1. What Are Like Terms? | 什么是同类项？
A term is a single number, letter or product of numbers and letters, such as 4, x, 7y or 3x². Two terms are like terms if their variable parts are exactly the same. The coefficients, or number parts, can be different.
项是一个单独的数、字母或数与字母的乘积，例如 4、x、7y 或 3x²。如果两个项的字母部分完全相同，它们就是同类项；系数，也就是数字部分，可以不同。
For example, 5x, -2x and ½x are like terms because they all contain x to the power 1. The numbers 5, -2 and ½ are different coefficients, but the letter part x is identical.
例如，5x、-2x 和 ½x 是同类项，因为它们都含有 x 的一次方。数字 5、-2 和 ½ 是不同的系数，但字母部分 x 完全相同。
| Like terms: 3y, -8y, ½y, 12y | Unlike terms: 3x, 3x², 3xy, 3 |
The table shows that changing the letter, the power or the number of letters stops terms from being like terms. Only the coefficient may change while the variable part stays the same.
上表说明，只要改变了字母、指数或字母的个数，它们就不再是同类项。只有系数可以改变，字母部分必须保持不变。
2. The Basic Rule: Same Letter, Same Power | 基本规则：字母相同且指数相同
To decide if terms are like, check two things: the variable letters must match, and each variable must have the same exponent. For example, 3x and 3x² are not like terms because the powers of x are different. Similarly, 2xy and 2x are not like terms because one has y and the other does not.
判断是否为同类项要检查两点：字母必须相同，并且每个字母的指数必须相同。例如，3x 和 3x² 不是同类项，因为 x 的指数不同；同样，2xy 和 2x 也不是同类项，因为一个含有 y，另一个没有。
2x + 3x² cannot be simplified to 5x or 5x²
This rule is the foundation of collecting like terms. If you remember that powers must match, you will avoid most common mistakes.
这条规则是合并同类项的基础。只要记住指数必须相同，你就能避开大多数常见错误。
3. Why We Collect Like Terms | 为什么要合并同类项？
Collecting like terms makes expressions shorter and reveals their structure. Instead of writing x + x + x + x, we write 4x. This also helps when substituting values because fewer terms mean fewer arithmetic steps.
合并同类项可以把代数式化短，展示其结构。比如 x + x + x + x 可以写成 4x。代入数值时，项数越少，计算步骤也就越少。
In later topics, you will need to simplify before solving equations, factorising quadratics or using graphs. A clean simplified expression reduces errors and makes patterns easier to spot.
在后续学习中，解方程、二次因式分解或画图像前都需要先化简。一个整理清楚的代数式能减少错误，也更容易发现规律。
4. Adding and Subtracting Like Terms | 同类项的加减
When terms are like, you add or subtract their coefficients and keep the variable part unchanged. For example, 7a + 3a means 7 lots of a plus 3 lots of a, giving 10 lots of a.
当项是同类项时，只需把系数相加或相减，字母部分保持不变。例如，7a + 3a 表示 7 个 a 加 3 个 a，得到 10 个 a。
7a + 3a = (7 + 3)a = 10a
For subtraction, the same rule applies. 9y – 4y means 9 lots of y take away 4 lots of y, leaving 5 lots of y.
减法也遵循同样的规则。9y – 4y 表示 9 个 y 减去 4 个 y，剩下 5 个 y。
9y – 4y = (9 – 4)y = 5y
5. Combining Terms in Longer Expressions | 在较长的代数式中合并
In longer expressions, find and group all like terms before simplifying. A useful method is to underline the x terms, circle the y terms and box the constants.
在较长的代数式中，先找出所有同类项并分组，再化简。一个有用的方法是：在 x 项下划线，圈出 y 项，再用方框标出常数项。
For example, simplify 4x + 2y – x + 5y. Group the x terms: 4x – x = 3x. Group the y terms: 2y + 5y = 7y. The simplified expression is therefore 3x + 7y.
例如，化简 4x + 2y – x + 5y。合并 x 项：4x – x = 3x；合并 y 项：2y + 5y = 7y。因此化简结果为 3x + 7y。
4x + 2y – x + 5y = 3x + 7y
6. Working with Constants | 处理常数项
Constant terms are numbers without variables. They are like terms with each other, so 7 + 4 = 11. In an expression such as 2x + 5 – x + 8, combine the x terms to get x and the constants to get 13.
常数项是没有字母的数。它们彼此是同类项，因此 7 + 4 = 11。在式子 2x + 5 – x + 8 中，合并 x 项得到 x，常数项合并得到 13。
2x + 5 – x + 8 = x + 13
A constant and a variable term are not like terms. For example, 3x + 4 cannot be simplified further because the 4 has no letter part matching x.
常数项和含字母的项不是同类项。例如，3x + 4 不能进一步化简，因为 4 没有与 x 匹配的字母部分。
7. Common Mistakes: Unlike Terms Cannot Be Combined | 常见错误：不同类项不能合并
A very common error is writing 3x + 2x² = 5x² or 3x + 2x² = 5x. These are both incorrect because x and x² are not like terms. The expression 3x + 2x² must be left as it is.
一个非常常见的错误是把 3x + 2x² 写成 5x² 或 5x。这两种写法都是错误的，因为 x 和 x² 不是同类项。3x + 2x² 必须保留原样。
Another mistake is treating 5xy and 5x as like terms. They are different because xy has an extra y, so 5xy + 5x cannot be combined into 10xy or 10x.
另一个错误是把 5xy 和 5x 当作同类项。它们不同，因为 xy 多了一个 y，所以 5xy + 5x 不能合并成 10xy 或 10x。
Always match both the letter and the power. If either is different, the terms must stay separate.
一定要同时匹配字母和指数。只要有一个不同，这两项就必须分开。
8. Removing Brackets Before Collecting | 先去括号再合并
Often you must expand brackets before collecting like terms. For 2(x + 3) + 3(x – 1), first expand to 2x + 6 + 3x – 3. Then collect x terms and constants: 2x + 3x = 5x and 6 – 3 = 3, giving 5x + 3.
通常要先展开括号再合并同类项。对于 2(x + 3) + 3(x – 1)，先展开为 2x + 6 + 3x – 3。然后合并 x 项和常数项：2x + 3x = 5x，6 – 3 = 3，得到 5x + 3。
2(x + 3) + 3(x – 1) = 2x + 6 + 3x – 3 = 5x + 3
Be careful with negative signs when expanding. If the bracket is multiplied by a negative number, every term inside changes sign.
展开时要注意负号。如果括号前乘的是负数，括号内每一项都要变号。
9. Collecting Terms with Multiple Variables | 多个变量的合并
Expressions can contain several different variable parts. Only combine parts that match exactly. For 3ab + 2a – ab + 5a, collect the ab terms: 3ab – ab = 2ab. Then collect the a terms: 2a + 5a = 7a.
代数式中可以有多个不同的字母部分。只有完全相同的部分才能合并。对于 3ab + 2a – ab + 5a，先合并 ab 项：3ab – ab = 2ab；再合并 a 项：2a + 5a = 7a。
3ab + 2a – ab + 5a = 2ab + 7a
Remember that ab and a are not like terms, and ab and b are not like terms either. The order of letters in a term does not change the variable part: 3ab and 5ba are like terms because multiplication is commutative.
记住，ab 和 a 不是同类项，ab 和 b 也不是同类项。项中字母的顺序不会改变字母部分：3ab 和 5ba 是同类项，因为乘法满足交换律。
10. Word Problems and Perimeter | 文字题与周长
Collecting like terms appears in perimeter problems. A rectangle with length 2x + 3 and width x – 1 has perimeter 2(2x + 3) + 2(x – 1). Expanding gives 4x + 6 + 2x – 2, which simplifies to 6x + 4.
合并同类项常用于周长问题。长为 2x + 3、宽为 x – 1 的长方形，周长为 2(2x + 3) + 2(x – 1)。展开得到 4x + 6 + 2x – 2，化简后为 6x + 4。
Perimeter = 2(2x + 3) + 2(x – 1) = 6x + 4
You may also see problems asking you to write an expression for total cost. If a pen costs x pounds and a notebook costs y pounds, then 3 pens and 2 notebooks cost 3x + 2y pounds. This expression cannot be simplified further because x and y are unlike terms.
你也可能遇到要求写出总费用表达式的问题。如果一支笔 x 英镑，一个笔记本 y 英镑，那么 3 支笔和 2 个笔记本的总价为 3x + 2y 英镑。这个式子不能进一步化简，因为 x 和 y 不是同类项。
11. Practice Questions with Solutions | 练习题与解答
Try simplifying these expressions before reading the solutions. (a) 6m + 3n – 2m + n (b) 5x² + 3x – 2x² + 7x (c) 4(p + 2) – 3(p – 1)
在看答案之前，请先尝试化简下列式子：(a) 6m + 3n – 2m + n (b) 5x² + 3x – 2x² + 7x (c) 4(p + 2) – 3(p – 1)
For (a), group m terms: 6m – 2m = 4m. Group n terms: 3n + n = 4n. The answer is 4m + 4n.
对于 (a)，合并 m 项：6m – 2m = 4m；合并 n 项：3n + n = 4n。答案是 4m + 4n。
For (b), group x² terms: 5x² – 2x² = 3x². Group x terms: 3x + 7x = 10x. The answer is 3x² + 10x.
对于 (b)，合并 x² 项：5x² – 2x² = 3x²；合并 x 项：3x + 7x = 10x。答案是 3x² + 10x。
For (c), expand first: 4p + 8 – 3p + 3. Collect p terms to get p and constants to get 11, so the answer is p + 11.
对于 (c)，先展开：4p + 8 – 3p + 3。合并 p 项得到 p，常数项合并得到 11，所以答案是 p + 11。
12. Exam Tips | 考试技巧
In exams, underline like terms with matching symbols before simplifying. Write one step at a time, and always check that the letter part has not changed. If you are asked to simplify, leave unlike terms separated by + or -.
考试时，可以用相同的记号标出同类项再化简。一步一步写，并检查字母部分是否保持不变。如果题目要求化简，不同类项之间用 + 或 – 隔开。
Show your working clearly, especially when expanding brackets first. A method mark may be awarded even if the final answer has a small arithmetic error. Double-check signs, as sign errors are the most common cause of lost marks in this topic.
清晰展示解答过程，尤其是在需要先展开括号的题目中。即使最终答案有小的计算错误，过程正确也可能得到步骤分。最后检查正负
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