📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, combining like terms is a core algebra skill. It allows you to simplify expressions such as 3x + 5x into 8x, making equations shorter and easier to solve. This guide explains how to identify like terms, combine positive and negative coefficients, remove brackets, and avoid common exam errors.
在 IGCSE 数学中，合并同类项是一项核心代数技能。它可以把 3x + 5x 这样的表达式化简为 8x，使方程更短、更容易求解。本指南将讲解如何识别同类项、合并正负系数、去括号以及避免考试中的常见错误。
1. What Are Like Terms? | 什么是同类项
A term is a number, a letter, or a product of numbers and letters. Like terms have exactly the same variable letters raised to exactly the same powers. The numerical part, called the coefficient, can be different.
项是一个数字、一个字母或数字与字母的乘积。同类项含有完全相同的字母变量，并且每个字母的指数也完全相同。数字部分称为系数，可以不同。
| Like terms | 同类项 | Unlike terms | 非同类项 |
|---|---|
| 3x and −5x | 3x and 3x² |
| 2x² and 7x² | 2x and 2y |
| 4xy and −xy | 5ab and 5a |
For example, 3x and −5x are like terms because both contain x to the power 1. However, 3x and 3x² are not like terms because the powers of x are different.
例如，3x 和 −5x 是同类项，因为两者都含有 x 的一次方。但 3x 和 3x² 不是同类项，因为 x 的指数不同。
2. Why Combine Like Terms? | 为什么要合并同类项
Combining like terms makes an expression shorter and clearer. It is often the first step in solving equations, simplifying formulae, and expanding brackets. For example, 2x + 3x + 4 can be simplified to 5x + 4, which is easier to work with.
合并同类项可以使表达式更短、更清晰。它通常是解方程、化简公式和展开括号的第一步。例如，2x + 3x + 4 可以化简为 5x + 4，更便于后续计算。
In an exam, leaving an expression such as 3x + 5x + 2 is not fully simplified. You are expected to write 8x + 2.
在考试中，把 3x + 5x + 2 这样的表达式留在答案中是不完整的。你必须写成 8x + 2。
3. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and look only at the variable part. Constants such as 5, −2, and 9 are like terms with each other because they have no variable part.
识别同类项时，忽略系数，只看变量部分。常数项例如 5、−2 和 9 彼此是同类项，因为它们没有变量部分。
Examples of like terms:
同类项示例：
- 5x and −3x
- 2x² and 6x²
- 4xy and −yx
- 7, −2, and 0.5
Notice that 4xy and −yx are like terms because multiplication is commutative: xy = yx.
注意 4xy 和 −yx 是同类项，因为乘法满足交换律：xy = yx。
4. The Basic Rule | 基本规则
You can only combine like terms by adding or subtracting their coefficients. The variable part stays exactly the same.
你只能通过加减系数来合并同类项，变量部分保持不变。
ax + bx = (a + b)x
x + 5x means (3 + 5)x = 8x. The same rule applies to subtraction: 7y − 2y = (7 − 2)y = 5y.
例如，3x + 5x 表示 (3 + 5)x = 8x。同样的规则也适用于减法：7y − 2y = (7 − 2
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