📚 Mastering Algebraic Simplification: Collecting Like Terms | 掌握代数化简：合并同类项
Algebra is full of letters, numbers and symbols. To work confidently with expressions, you need to master the skill of combining parts that belong together. This article guides you through collecting like terms, a fundamental building block for everything from solving equations to rearranging complicated formulas.
代数中充满了字母、数字和符号。要想自如地处理代数式，你必须掌握将同类部分合并在一起的本领。本文将带你系统地学习合并同类项，这是从解方程到整理复杂公式都离不开的基本功。
1. What Are Algebraic Expressions? | 什么是代数表达式？
expression is a combination of numbers, letters (variables) and operation signs such as +, – e an equation, an expression does not contain an equals sign. Examples include 3a + 2b, 5x – ifying these expressions by collecting like terms is one of the first and most important steps towards algebraic fluency.
代数表达式是由数字、字母（变量）以及 +、–、×、÷ 等运算符号组成的式子。与方程不同，表达式不含等号。例如 3a + 2b、5x – 7 或 4p² + p。在 KS3 阶段的数学学习中，通过合并同类项来化简表达式是迈向代数流利运用的第一步，也是最为关键的一步。
2. Understanding the Term ‘Term’ | 理解“项”的概念
In algebra, a term is a single number, a single letter, or numbers and letters multiplied together. The expression 3x + 2y – 5 contains three terms: 3x, 2y and –5. Terms are separated by plus or minus signs. Being able to spot individual terms quickly stops you from adding or subtracting things that should not be combined.
在代数中，“项”是指一个单独的数、一个单独的字母，或者数和字母的乘积。表达式 3x + 2y – 5 含有三项：3x、2y 和 –5。各项之间由加号或减号隔开。能够快速认出每一个项，可以帮助你避免把不该合并的部分错误地加或减到一起。
3. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable part raised to the same power. For instance, 4a and –2a are like terms because they both contain the letter a to the power of 1. However, 3a and 3a² are not like terms, because the powers are different. Numbers without any variables, such as 6 and –11, are called constant terms and are also like each other.
同类项是指所含的字母部分完全相同，且字母的指数也相同的项。例如，4a 与 –2a 是同类项，因为它们都含有 a 的一次幂。但是 3a 和 3a² 不是同类项，因为指数不同。不含字母的数，如 6 和 –11，被称为常数项，它们彼此之间也是同类项。
4. The Golden Rule of Collecting Like Terms | 合并同类项的黄金法则
You can only add or subtract the coefficients of like terms. The variable part stays exactly the same. Think of 5a as ‘five apples’ and 3a as ‘three apples’. Adding them gives eight apples, not eight apples squared. The expression 5a + 3a is therefore 8a. This simple picture helps to avoid one of the most common early mistakes in algebra.
你只能对同类项的系数进行加减运算，字母部分始终保持不变。可以把 5a 想象成“五个苹果”，3a 是“三个苹果”。把它们加起来得到八个苹果，而不是八个苹果的平方。因此 5a + 3a 等于 8a。这个简单的图像能帮助你避开代数初学阶段最常见的错误之一。
5. Simplifying Expressions with Addition and Subtraction | 通过加减法化简表达式
To simplify an expression like 7x + 2x – x, look at the coefficients: 7 + 2 – sult is 8x. With a mixture of variables, group the like terms separately. For 3a + 2b + 5a – b, collect the a terms (3a + 5a = 8a) and the b terms (2b – ession is 8a + b. Always check that you have not changed the power or the variable letter.
化简像 7x + 2x – x 这样的式子时，只看系数：7 + 2 – 1 等于 8，所以结果是 8x。如果含有不同字母，就把同类项分别归类。对于 3a + 2b + 5a – b，合并 a 项得到 3a + 5a = 8a，合并 b 项得到 2b – b = 1b。化简后的表达式为 8a + b。务必检查你是否无意中改变了指数或字母。
6. Handling Coefficients and Negative Terms | 处理系数和负项
Pay special attention when negative signs appear. In 4y – 3y + 2, the like terms 4y and – 1y (or simply y). The constant 2 remains unchanged, so the final answer is y + 2. With 6m – 2n – 3m + n, collect m terms (6m – 3m = 3m) and n terms (–2n + n = –1n or –n). The result is 3m – n. Writing the subtraction as adding a negative often reduces sign errors.
当出现负号时要格外小心。对于 4y – 3y + 2，同类项 4y 和 –3y 合并得到 1y（即 y），常数 2 保持不变，最终答案为 y + 2。再看 6m – 2n – 3m + n，合并 m 项得 6m – 3m = 3m，合并 n 项得 –2n + n = –1n 即 –n。结果为 3m – n。把减法看作加上一个负数，常常能减少符号错误。
7. Working with Multiple Variables | 处理含有多个变量的表达式
In longer expressions such as 2x + 3y – x + 4y + 7, you need to identify three different types of term: x terms, y terms and constants. Combine 2x – x to get x, and 3y + 4y to get 7y. The constant 7 stays at the end. The simplified form is x + 7y + 7. Even when letters look similar, like a and a², they are as different as apples and apple trees: keep them separate.
对于较长的表达式，例如 2x + 3y – x + 4y + 7，你需要识别出三种不同类型的项：x 项、y 项和常数项。把 2x – x 合并为 x，3y + 4y 合并为 7y，常数 7 留在最后。化简后的形式是 x + 7y + 7。即使有些字母看起来相似，比如 a 和 a²，它们就像苹果和苹果树一样不同，一定要将它们分开处理。
8. Expressions Involving Powers | 涉及幂的表达式
Like terms must have exactly the same power. The expression 5p² + 3p – 2p² + p contains p² terms and p terms. Group the squared terms: 5p² – p. Remember that p² and p are different objects entirely, so you should never add their coefficients together in one step.
同类项必须具有完全相同的指数。表达式 5p² + 3p – 2p² + p 中含有 p² 项和 p 项。先把二次项归类：5p² – 2p² = 3p²；再把一次项归类：3p + p = 4p。化简后的答案是 3p² + 4p。请记住，p² 和 p 是完全不同的对象，绝不应该一步就把它们的系数加在一起。
9. Simplifying Before Substituting | 先化简再代入
ven values of the letters, it is almost always easier to collect like terms first. Take 4k – 2 + 3k when k = 2. Simplify 4k + 3k = 7k, so you only need to calculate 7 × 2 – 2 = 12. Without simplifying, you would do 4 × 2 – 2 + 3 × 2, which is more work and easier to miscalculate under pressure.
当题目要求你把字母取特定值代入求值时，几乎总是先合并同类项更为简便。例如在 k = 2 时计算 4k – 2 + 3k。先化简：4k + 3k = 7k，这样就只需要计算 7 × 2 – 2 = 12。如果不先化简，就要计算 4 × 2 – 2 + 3 × 2，计算量更大，且在有压力时容易出错。
10. Common Pitfalls and How to Avoid Them | 常见误区与避坑指南
Many students mistakenly add the letters themselves, treating 3a + 2a as 5a². This happens when they mix up addition and multiplication. Another trap is ignoring the sign in front of a term: –5x and +5x cancel to 0x, which simply disappears from the expression. A third pitfall is combining terms like 2xy and 2x, which are not like terms because the variable combinations differ. Constantly ask, “Are the letters and powers exactly the same?”
许多学生错误地把字母本身也相加，把 3a + 2a 算成了 5a²。这是因为他们把加法和乘法搞混了。另一个陷阱是忽略项前面的符号：–5x 和 +5x 相消为 0x，该项就直接从表达式中消失。第三个误区是把 2xy 和 2x 这类项合并，但它们并不是同类项，因为所含的字母组合不同。要不断问自己：“字母及其指数完全相同吗？”
11. Using Tables to Organise Like Terms | 用表格整理同类项
When an expression looks messy, a simple table can untangle it. List each term type in the first row and place coefficients underneath.
| Term type | x | y | Constant |
| Original expression | 3x, –2x | 4y, +y | 5 |
| Combined | x | 5y | 5 |
This visual approach helps you check that no term has been forgotten and that all signs are correct before writing the final answer x + 5y + 5.
当表达式看起来有点乱时，一张简单的表格就能理清头绪。在第一行列出每一种项的类型，然后把系数填在下方。
| 项的类型 | x | y | 常数 |
| 原表达式 | 3x, –2x | 4y, +y | 5 |
| 合并后 | x | 5y | 5 |
这种可视化方法能帮助你在写出最终答案 x + 5y + 5 之前，检查是否有遗漏的项，以及所有符号是否正确。
12. Practice Questions and Model Answers | 练习与参考答案
Try simplifying these expressions on your own before looking at the answers.
(a) 8a + 3b – 2a + b (b) 6x – 3 + 2x + 4 (c) 5m² – 2m + 3m² + m – 7
Model answers:
(a) 6a + 4b (b) 8x + 1 (c) 8m² – m – 7
Work steadily, and with each question check: Are my like terms collected? Have I kept the variable parts unchanged? Did I carry the signs correctly? Regular practice turns collecting like terms into an automatic skill that will serve you well throughout your mathematics journey.
在看答案之前，请先自己尝试化简下列表达式。
(a) 8a + 3b – 2a + b (b) 6x – 3 + 2x + 4 (c) 5m² – 2m + 3m² + m – 7
参考答案：
(a) 6a + 4b (b) 8x + 1 (c) 8m² – m – 7
一步一步稳扎稳打，每做一题都问问自己：同类项是否已经合并？字母部分是否保持不变？符号是否处理正确？坚持练习，合并同类项就会变成一种自动化的技能，让你的整个数学学习之路都走得更加顺利。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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