📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, simplifying algebraic expressions is often the first step in solving equations, rearranging formulae, and sketching graphs. The idea of collecting like terms helps you turn a long expression such as 3x + 5y + 2x − y into the cleaner form 5x + 4y.
在IGCSE数学中，化简代数式通常是解方程、变换公式和画图的第一步。合并同类项这一思路可以帮你把 3x + 5y + 2x − y 这样的长表达式整理成更简洁的 5x + 4y。
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 5, x, −3y or 2ab. Like terms are terms whose variable parts are exactly the same, including their powers.
项是一个单独的数、字母或数与字母的乘积，例如 5、x、−3y 或 2ab。同类项是指变量的部分完全相同，包括次数也相同的项。
For example, 6x and −2x are like terms because both contain x to the power 1. However, 6x and 6x² are not like terms, because x and x² have different powers.
例如，6x 和 −2x 是同类项，因为两者都含有一次方的 x。但是，6x 和 6x² 不是同类项，因为 x 和 x² 的次数不同。
The coefficient can be any number, positive or negative, but the variable part must match exactly.
系数可以是正数或负数，但变量部分必须完全一致。
2. Why Collecting Like Terms Matters | 为什么要合并同类项？
Collecting like terms simplifies an expression so that it is shorter and easier to use. It also helps you see the structure of an expression before solving equations or substituting values.
合并同类项可以化简表达式，使其更短、更容易使用。它还能帮助你在解方程或代入数值之前看清表达式的结构。
In IGCSE exams, many algebra questions require you to simplify first. Even if the final answer is correct, clear simplification often earns method marks.
在IGCSE考试中，许多代数题都要求先化简。即使最终答案正确，清晰的化简过程通常也能获得方法分。
Collecting like terms is also used when adding algebraic fractions, expanding brackets, and solving linear equations.
合并同类项还用于代数分式相加、展开括号以及解一元一次方程。
3. Key Vocabulary: Term, Coefficient, Variable | 关键术语：项、系数、变量
A coefficient is the number part of a term. For example, in 7xy the coefficient is 7, while the variable part is xy. A constant is a term with no variable, such as 4 or −9.
系数是项中的数字部分。例如，在 7xy 中系数是 7，变量部分是 xy。常数是不含变量的项，例如 4 或 −9。
You should recognise these words in exam instructions: 'simplify', 'collect like terms' and 'write in its simplest form' all mean the same process.
你应当认识考试指令中的这些词：'化简'、'合并同类项'和'写成最简形式'都表示同一种过程。
If a term is written as just x, its coefficient is 1. If it is written as −x, its coefficient is −1.
如果一项只写作 x，它的系数是 1。如果写作 −x，它的系数是 −1。
4. The Basic Rule: Add Coefficients | 基本规则：系数相加
To combine like terms, keep the variable part unchanged and add or subtract the coefficients only.
合并同类项时，变量部分保持不变，只对系数进行加法或减法运算。
8n − 3n = (8 − 3)n = 5n
This rule works because multiplication is distributive over addition: each n is counted eight times, then three are removed, leaving five n's.
这个规则成立是因为乘法对加法有分配律：先数了 8 个 n，再去掉 3 个，剩下 5 个 n。
Never add or subtract the variable parts. The letter or letters and their powers stay exactly the same.
永远不要对变量部分做加法或减法。字母及其次数保持完全不变。
5. Single-Variable Examples | 单一变量示例
e some basic examples with one variable: x + 2x + 3x = 6x; 5p − 2p = 3p; 7m + m − 4m = 4m.
下面是一些含一个变量的基本示例：x + 2x + 3x = 6x；5p − 2p = 3p；7m + m − 4m = 4m。
Remember that a variable written alone has coefficient 1, so m means 1m and −m means −1m.
请记住，单独写的变量系数为 1，因此 m 表示 1m，−m 表示 −1m。
x² + 3x² = 4x² but x² + x cannot be combined
Only terms with identical powers are like terms. x² and x look similar but are not the same variable part.
只有次数完全相同的项才是同类项。x² 和 x 看起来相似，但变量部分并不相同。
For example, 4y − 9y = −5y, and 10t + 3t − 6t = 7t.
例如，4y − 9y = −5y，10t + 3t − 6t = 7t。
6. Multi-Variable Terms | 多变量项
For terms with more than one variable, the variables and their powers must match exactly. The order of the variables does not matter: 3ab and −5ba are like terms because multiplication is commutative.
对于含有多个变量的项，变量及其次数必须完全一致。变量的顺序无关紧要：3ab 和 −5ba 是同类项，因为乘法满足交换律。
However, 2x²y and 3xy² are not like terms, because the powers of x and y are not the same in both terms.
但是，2x²y 和 3xy² 不是同类项，因为两项中 x 和 y 的次数不同。
| Expression | 表达式 | Like terms? | 是否同类项？ | Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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