📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, collecting like terms is one of the first essential algebra skills you need to master. It allows you to simplify expressions, solve equations, expand brackets and rearrange formulas with confidence. This article covers the rules, common errors and exam techniques for collecting like terms.
在 IGCSE 数学中，合并同类项是你需要掌握的首要代数技能之一。它能帮助你自信地化简表达式、解方程、去括号和变形公式。本文介绍合并同类项的规则、常见错误和考试技巧。
1. What Are Like Terms? | 什么是同类项
Like terms are terms that contain exactly the same variable letters raised to exactly the same powers. The coefficients, which are the numbers in front of the variables, can be different. For example, 4x and -7x are like terms because both contain x to the power of 1. However, 4x and 4x² are not like terms because the powers of x are different.
同类项是含有完全相同的变量字母且次数完全相同的项。变量前面的数字称为系数，系数可以不同。例如，4x 和 -7x 是同类项，因为两者都含有 x 的一次方。但是，4x 和 4x² 不是同类项，因为 x 的次数不同。
2. Why Collecting Matters | 为什么合并很重要
Collecting like terms makes an expression shorter and easier to handle. In an IGCSE exam, you often need to simplify before substituting values or solving an equation. A tidy simplified expression also helps you avoid sign errors and arithmetic mistakes under time pressure.
合并同类项使表达式更短、更容易处理。在 IGCSE 考试中，你通常需要先化简表达式，再代入数值或解方程。整洁的化简结果还能帮助你在时间紧张时避免符号错误和算术错误。
3. Recognising Like Terms | 识别同类项
To recognise like terms, ignore the coefficient and look only at the variable part. Terms such as 2xy, -xy and 5xy are like terms because they share the same variable product xy. Terms such as 3x and 3y are not like terms because the variable letters are different.
要识别同类项，可以忽略系数，只看变量部分。像 2xy、-xy 和 5xy 这样的项是同类项，因为它们都有相同的变量乘积 xy。像 3x 和 3y 这样的项不是同类项，因为变量字母不同。
4. The Golden Rule: Same Variable, Same Power | 黄金法则：同变量同次幂
The golden rule is that only terms with identical variable parts can be collected. For instance, 5x² and 3x² can be combined to give 8x², but 5x² and 3x cannot be combined. Likewise, a²b and ab² are different terms because the powers are attached to different letters.
黄金法则是：只有变量部分完全相同的项才能合并。例如，5x² 和 3x² 可以合并为 8x²，但 5x² 和 3x 不能合并。同样，a²b 和 ab² 是不同的项，因为次数对应的字母不同。
5x² + 3x² = 8x²
5x² + 3x cannot be simplified further
5. Adding and Subtracting Coefficients | 系数加减
When collecting like terms, add or subtract the coefficients while keeping the variable part unchanged. For example, 6a - 2a + a has coefficients 6 - 2 + 1, giving 5a. Always carry the sign in front of each term, because subtraction can easily be lost.
合并同类项时，将系数相加或相减，变量部分保持不变。例如，6a - 2a + a 的系数为 6 - 2 + 1，得到 5a。计算时始终带上每项前面的符号，因为减法中的负号很容易丢失。
6a - 2a + a = 5a
6. Combining with Multiple Variables | 多变量合并
Expressions with more than one variable can be simplified by collecting each type of like term separately. For example, 2x + 3y + 5x - y becomes 7x + 2y after collecting the x terms and the y terms. Different variable groups stay separate in the final answer.
含有多个变量的表达式可以通过分别合并每一类同类项来化简。例如，2x + 3y + 5x - y 合并 x 项和 y 项后得到 7x + 2y。不同的变量组在最终答案中保持分开。
2x + 3y + 5x - y = 7x + 2y
7. Constants and Mixed Expressions | 常数与混合表达式
Constant terms are plain numbers with no variable, and they can only be collected with other constants. In the expression 3n + 5 + 2n - 2, the n terms give 5n and the constants give 3, so the simplified form is 5n + 3. Never combine a constant with a variable term.
常数项是不含变量的纯数字，只能与其他常数合并。在表达式 3n + 5 + 2n - 2 中，n 项合并为 5n，常数合并为 3，所以化简结果是 5n + 3。永远不要把常数与变量项合并。
3n
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