📚 Collecting Like Terms in IGCSE Maths | IGCSE 数学：合并同类项
In IGCSE Mathematics, collecting like terms is one of the first algebraic skills you need to master. It allows you to simplify expressions before solving equations, substituting values, or factorising. This article explains the rules step by step and helps you avoid the most common errors.
在 IGCSE 数学中，合并同类项是你需要掌握的首批代数技能之一。它帮助你在解方程、代入数值或因式分解之前先化简表达式。本文逐步解释规则，并帮助你避免最常见的错误。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that contain exactly the same variables raised to exactly the same powers. The numbers in front, called coefficients, may be different. For example, 3x and −5x are like terms because both contain x to the power 1.
同类项是指含有完全相同的变量、且变量的指数也完全相同的项。项前面的数字称为系数，系数可以不同。例如，3x 和 −5x 是同类项，因为两者都含有指数为 1 的 x。
Constant terms such as 6, −2, and 0.75 are also like terms with each other, because they have no variable part at all.
常数项（如 6、−2 和 0.75）彼此之间也是同类项，因为它们完全没有变量部分。
The table below shows some pairs of like terms and pairs that are not like terms.
下表给出一些同类项和非同类项的例子。
| Like terms | Not like terms |
|---|---|
| 4x and −7x | 4x and 4x² |
| 2xy and 5xy | 2xy and 2y |
| 9 and −3 | 9 and 9x |
2. Why Do We Collect Like Terms? | 为什么要合并同类项？
Algebraic expressions often contain several repeated terms. Collecting like terms makes an expression shorter, neater, and easier to use in later steps. For example, solving or substituting into a simplified expression reduces arithmetic mistakes.
代数表达式经常包含多个重复的项。合并同类项可以使表达式更短、更清晰，也更便于后续步骤使用。例如，在求解或代入化简后的表达式时，可以减少算术错误。
In IGCSE exam questions, you may be asked to "simplify" an expression. Unless another instruction is given, this usually means expand brackets if needed and then collect like terms.
在 IGCSE 考试题目中，你可能会被要求“化简”一个表达式。除非题目另有说明，这通常意味着：如果需要就先去括号，然后合并同类项。
3. The Golden Rule: Only Add or Subtract Coefficients | 黄金法则：只加减系数
When you collect like terms, keep the variable part exactly the same. Add or subtract only the coefficients. For example, 3x + 5x means 3 lots of x plus 5 lots of x, which gives 8 lots of x.
合并同类项时，变量部分必须保持不变。只对系数进行加法或减法运算。例如，3x + 5x 表示 3 个 x 加 5 个 x，得到 8 个 x。
3x + 5x = (3 + 5)x = 8x
The same rule applies to subtraction: 10a − 3a = (10 − 3)a = 7a.
同样的规则也适用于减法：10a − 3a = (10 − 3)a = 7a。
4. Handling Positive and Negative Coefficients | 处理正负系数
The sign in front of a term belongs to that term. Think of subtraction as adding a negative coefficient. For example, 5x − 9x + 2x can be read as +5x − 9x + 2x.
项前面的符号属于该项本身。可以把减法看作是加上一个负数系数。例如，5x − 9x + 2x 可以读作 +5x − 9x + 2x。
5x − 9x + 2x = (5 − 9 + 2)x = −2x
Be especially careful when a negative sign comes in front of a bracket. You may need to change the signs of all terms inside before collecting.
当括号前面出现负号时要特别小心。在合并之前，你可能需要改变括号内所有项的符号。
5. Collecting Terms with Multiple Variables | 合并含多个变量的项
Terms with more than one variable are like terms only if every variable and every power match exactly. For instance, 2xy and 7xy are like terms, but 2xy and 2x are not, because the first contains y while the second does not.
含多个变量的项，只有当每个变量和每个指数都完全相同时才是同类项。例如，2xy 和 7xy 是同类项，但 2xy 和 2x 不是，因为前者含有 y，而后者没有。
3ab + 4a − 2ab + 6a = (3ab − 2ab) + (4a + 6a) = ab + 10a
Grouping the like terms first is helpful so you do not accidentally combine terms that look similar but are not the same.
先对同类项进行分组会很有帮助，这样你就不会意外地合并那些看起来相似但实际上不同的项。
6. Collecting After Expanding Brackets | 先去括号再合并
ting like terms. Use the distributive law such as a(b + c) = ab + ac, and watch the signs.
许多 IGCSE 化简题都含有括号。你必须先去括号，再合并同类项。使用分配律，如 a(b + c) = ab + ac，并注意符号。
