📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, collecting like terms is the essential skill of simplifying algebraic expressions by grouping terms that have identical variable parts. It appears at the very start of algebra and supports solving equations, expanding brackets, and factorising.
在 IGCSE 数学中，合并同类项是简化代数式的基本技能，方法是把变量部分完全相同的项归为一组。它是代数的起点，也为解方程、去括号和因式分解打下基础。
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 7, x, 3xy, or -5x². Terms are called ‘like terms’ when their variable parts are exactly the same, including the same powers.
项是一个单独的数、字母或数与字母的乘积，例如 7、x、3xy 或 -5x²。当两个项含有完全相同的变量部分，并且次数也相同时，它们就叫做同类项。
For example, 4x and 9x are like terms because both contain the variable x to the power 1. However, 4x and 4x² are not like terms because the powers of x are different.
例如，4x 和 9x 是同类项，因为它们都含有一次方的变量 x。但 4x 和 4x² 不是同类项，因为 x 的次数不同。
2. The Matching Test for Like Terms | 同类项的匹配判断
To decide if two terms are like terms, compare their variable parts only. Coefficients, the numbers in front, do not matter for the matching test.
判断两个项是否为同类项时，只比较它们的变量部分。变量前面的数字，也就是系数，不会影响判断。
The table below shows common cases.
下表列出常见情况。
| Terms | Like or unlike? | Reason |
| 5x and 3x | Like | Both have x¹ |
| 4x² and 6x² | Like | Both have x² |
| 2xy and 5xy | Like | Both have xy |
| 3x and 3x² | Unlike | Different powers of x |
| 4ab and 4a | Unlike | Different variable sets |
| 7 and 9 | Like | Both are constants |
3. The Coefficient Rule | 系数相加减法则
When terms are like terms, you can add or subtract their coefficients while keeping the variable part unchanged.
当项是同类项时，可以把它们的系数相加或相减，变量部分保持不变。
5x + 3x = (5 + 3)x = 8x
This works because 5x means five lots of x, and 3x means three lots of x. Together they make eight lots of x.
这个法则成立，因为 5x 表示 5 个 x，3x 表示 3 个 x，合起来就是 8 个 x。
For subtraction, subtract the coefficients carefully: 9y – 2y = (9 – 2)y = 7y.
减法时小心减系数：9y – 2y = (9 – 2)y = 7y。
4. Collecting Single-Variable Terms | 合并一元同类项
Simplify expressions with one variable by grouping all terms containing that variable and combining their coefficients. Collect constants separately.
含一个变量的代数式，可以先把含有该变量的项归为一组，把系数合并；常数项单独合并。
Example: Simplify 6x + 3 + 2x + 5.
例题：化简 6x + 3 + 2x + 5。
First collect the x terms: 6x + 2x = 8x. Then collect the constants: 3 + 5 = 8. The simplified expression is 8x + 8.
先合并 x 项：6x + 2x = 8x。再合并常数项：3 + 5 = 8。化简结果是 8x + 8。
You can also write the intermediate step as (6 + 2)x + (3 + 5) = 8x + 8.
也可以写出中间步骤：(6 + 2)x + (3 + 5) = 8x + 8。
5. Collecting Multi-Variable Terms | 合并多变量同类项
Terms with more than one variable are like terms only if they have exactly the same combination of letters and powers. The order of letters does not matter, but standard form usually writes letters alphabetically.
含有多个变量的项，只有在字母组合和次数完全相同时才是同类项。字母顺序不影响判断，但标准写法通常按字母表顺序书写。
For example, 3ab and 5ab are like terms, while 3ab and 5a are not. Also, 2xy and 2yx are like terms because xy and yx represent the same product.
例如，3ab 和 5ab 是同类项，而 3ab 和 5a 不是。2xy 和 2yx 也是同类项，因为 xy 和 yx 表示同一个乘积。
Simplify: 4ab + 3ab – ab = (4 + 3 – 1)ab = 6ab.
化简：4ab + 3ab – ab = (4 + 3 – 1)ab = 6ab。
6. Constants Are Like Terms Too | 常数项也是同类项
Numbers without any variable are called constants. All constants are like terms with each other, regardless of their size or sign.
不带变量的数叫做常数项。所有常数项彼此都是同类项，无论大小或符号如何。
Simplify: 12 + 7 – 3 = 16. This can be included in a larger expression such as 5x + 12 + 3x + 7 – 3.
化简：12 + 7 – 3 = 16。这可以用于更大的式子，如 5x + 12 + 3x + 7 – 3。
Combine x terms: 5x + 3x = 8x. Combine constants: 12 + 7 – 3 = 16. Final answer: 8x + 16.
合并 x 项：5x + 3x = 8x。合并常数项：12 + 7 – 3 = 16。最终答案：8x + 16。
7. Different Powers Must Not Be Combined | 不同次幂不能合并
A very common error is adding terms like x and x² as if they were like terms. They are not like terms because the variable has different powers.
一个常见错误是把 x 和 x² 当作同类项相加。它们不是同类项，因为变量的次数不同。
For example, 3x + 5x² cannot be simplified further. The best expression is 3x + 5x². However, 3x² + 5x² = 8x² because both terms have x².
例如，3x + 5x² 不能进一步化简，最简形式就是 3x + 5x²。但是 3x² + 5x² = 8x²，因为两项都含有 x²。
Similarly, 2xy² and 3x²y are not like terms because the powers of x and y do not match in each term.
同理，2xy² 和 3x²y 不是同类项，因为两项中 x 和 y 的次数不一致。
8. Removing Brackets Before Collecting | 先去括号再合并
When an expression contains brackets, expand them before collecting like terms. Multiply each term inside the bracket by the term outside, then collect the resulting like terms.
当式子含有括号时，要先去括号，再合并同类项。用括号外的数乘以括号内的每一项，然后合并同类项。
Example: Simplify 3(x + 2) + 2(x + 4).
例题：化简 3(x + 2) + 2(x + 4)。
Expand: 3x + 6 + 2x + 8. Collect x terms: 3x + 2x = 5x. Collect constants: 6 + 8 = 14. Final answer: 5x + 14.
去括号：3x + 6 + 2x + 8。合并 x 项：3x + 2x = 5x。合并常数项：6 + 8 = 14。最终答案：5x + 14。
9. Negative Signs and Subtraction | 负号与减法
Subtraction signs belong to the term immediately after them. When collecting, include the sign with each term. For example, in 5x – 3 + 2x – 4, the terms are 5x, -3, 2x, and -4.
减号属于它后面的那一
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