📚 Collecting Like Terms | 合并同类项
In IGCSE mathematics, algebraic expressions often contain several similar parts. 'Collecting like terms' is the skill of simplifying an expression by combining those parts into a single term. It is essential for solving equations, rearranging formulae and working with functions, and it appears in almost every algebra question on the paper.
在 IGCSE 数学中，代数表达式常包含多个相似部分。"合并同类项"就是把这些相似部分合并为一项，从而化简表达式的技能。它对解方程、变换公式和处理函数都至关重要，几乎出现在试卷的每一道代数题中。
1. What Are Like Terms? | 什么是同类项
A term in algebra is a single number, a single letter, or a product of numbers and letters. An expression is a sum of such terms, joined by + or − signs. Before you can simplify anything, you must be able to split an expression into its separate terms.
在代数中，项是指单个数字、单个字母或数字与字母的乘积。表达式就是这些项的和，各项之间用 + 或 − 连接。要化简任何表达式，首先必须能够把它拆分成若干个独立的项。
Like terms are terms that have exactly the same variable part, including the same powers. For example, 3x and −5x are like terms because both contain the variable x raised to the power 1. The numbers in front, called coefficients, are allowed to be different.
同类项是指变量部分完全相同、包括幂次也相同的项。例如，3x 和 −5x 是同类项，因为它们都含有 x 的一次方。前面的数称为系数，系数可以不同。
Terms such as x and x² are not like terms because the powers of x are different. Likewise, 4xy and 4x are not like terms because the first one contains the extra variable y. A constant such as 7 has no variable part at all, so it is only like another constant.
x 和 x² 不是同类项，因为 x 的幂次不同。同样，4xy 和 4x 也不是同类项，因为前者多出一个变量 y。而像 7 这样的常数项没有字母部分，因此只能与另一个常数项构成同类项。
2. Coefficients, Signs and the Variable Part | 系数、符号与字母部分
Every term has three features: a sign (+ or −), a coefficient (the numerical factor) and a variable part (the letters together with their powers). In the term −7ab² the sign is −, the coefficient is 7 and the variable part is ab².
每一项都有三个特征：符号（+ 或 −）、系数（数字因数）以及字母部分（字母及其幂）。在项 −7ab² 中，符号是 −，系数是 7，字母部分是 ab²。
When a term is written with no visible number in front, such as x or ab, its coefficient is 1. A term such as −y has coefficient −1. It helps to write that invisible 1 in your head before you start counting.
若某一项前面没有可见数字，例如 x 或 ab，它的系数为 1；像 −y 这样的项系数为 −1。开始计算前先在脑中补上这个看不见的 1，会很有帮助。
| Term 项 | Coefficient 系数 | Variable part 字母部分 |
| 5x | 5 | x |
| −3y² | −3 | y² |
| ab | 1 | ab |
| −k | −1 | k |
| 7 | 7 | none 无 |
3. Why We Collect Like Terms | 为什么合并同类项
Collecting like terms makes an expression shorter and easier to work with. In IGCSE questions, you are often asked to simplify an expression before substituting values or solving an equation. A simplified expression also reduces the chance of arithmetic mistakes later on.
合并同类项能使表达式更短、更易于计算。在 IGCSE 考试中，题目通常要求你在代入数值或解方程之前先化简表达式。化简后的表达式还能减少后续的算术错误。
For example, instead of working with 4a + 3b − 2a + 5b, you can write 2a + 8b. This is clearer and much faster to evaluate when a and b are known.
例如，与其处理 4a + 3b − 2a + 5b，不如写成 2a + 8b。当 a 和 b 已知时，这样更清晰、计算更快。
Marks are awarded for the simplification itself, not only for the final value, so a fully collected answer is part of good exam technique. It is also the first step in solving linear equations, where you must tidy each side before balancing.
考试评分不仅看最终数值，也看化简过程，因此把答案合并到最简形式是应试技巧的一部分。它也是解一元一次方程的第一步：在两边平衡之前，必须先把每一边整理清楚。
4. Identifying Like Terms | 识别同类项
To decide whether two terms are like terms, ignore the coefficients and compare only the variable parts. The terms 7p, −2p and ½p are all like terms, because each of them is a multiple of p.
判断两项是否为同类项时，先忽略系数，只比较字母部分。7p、−2p 和 ½p 都是同类项，因为它们都是 p 的倍数。
in which letters are multiplied never matters, so qp is simply another way of writing pq.
6pq 和 3qp 也是同类项，因为乘法满足交换律：pq = qp。字母相乘的顺序不影响结果，所以 qp 只是 pq 的另一种写法。
However, watch the powers carefully. The terms 5x²y and 3xy² are not like terms, because in the first term x is squared while in the second term y is squared. In the same way, 5x²y and −2x²y are like terms, but 5x²y and 5xy are not.
但要特别留意幂次。5x²y 与 3xy² 不是同类项，因为前者 x 是平方，后者 y 是平方。同理，5x²y 与 −2x²y 是同类项，而 5x²y 与 5xy 不是同类项。
A quick test is to read each term aloud as "how many of what". Three of x, minus five of x, and one half of x are all counts of the same thing, so they can be added; three of x and four of y are counts of different things, so they cannot.
一个简便的方法是逐项读成"多少个什么"：3 个 x、−5 个 x、½ 个 x 都是在数同一种东西，可以相加；3 个 x 和 4 个 y 数的是不同的东西，不能相加。
5. How the Rule Works | 合并法则的原理
Collecting like terms works because of the distributive law. If two terms share the same variable part, that part can be taken out as a common factor and the coefficients added together.
合并同类项的依据是乘法分配律。如果两项的字母部分相同，就可以把这个字母部分作为公因式提出，再把系数相加。
a x + b x = (a + b) x
In practice you never need to write the factor out every time. Simply add the coefficients and keep the variable part unchanged: 3x +
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