📚 Collecting Like Terms and Simplifying Algebraic Expressions | 合并同类项与化简代数式
In algebra, we often work with expressions made up of numbers, letters, and operation symbols. Being able to spot and combine ‘like terms’ is a fundamental skill that makes expressions shorter and easier to use. This article will guide you step by step through identifying like terms, collecting them correctly, and simplifying a wide range of algebraic expressions. You will also learn how to avoid common pitfalls and apply these skills to real-life problems.
在代数中，我们常常与由数字、字母和运算符号组成的表达式打交道。能够识别并合并“同类项”是一项基本技能，它能让表达式变得更简短、更易用。本文将逐步引导你掌握识别同类项、正确合并以及化简各种代数表达式的方法。你还将学会如何避开常见错误，并将这些技能应用于实际生活问题。
1. What are Algebraic Terms? | 什么是代数项？
An algebraic term is a single number, a single letter, or a product of numbers and letters. For example, 7, x, 5y, -3ab, and 2x² are all terms. In an expression, terms are separated by plus or minus signs. The expression 4a + 3b – 2 consists of three terms: 4a, 3b, and -2.
代数项是指一个单独的数、一个单独的字母，或者数与字母的乘积。例如 7、x、5y、-3ab 和 2x² 都是项。在表达式中，各项由加号或减号分隔。表达式 4a + 3b – 2 由三项组成：4a、3b 和 -2。
Each term has a numerical part, called the coefficient, and a variable part. In the term 4a, the coefficient is 4 and the variable is a. For a term like x, the coefficient is understood to be 1, and for -y the coefficient is -1.
每个项都有一个数值部分，称为系数，以及一个变量部分。在项 4a 中，系数是 4，变量是 a。像 x 这样的项，系数理解为 1；对于 -y，系数是 -1。
2. Identifying Like Terms | 识别同类项
Like terms are terms that have exactly the same variable part, meaning the same letters raised to the same powers. The coefficients can be different. For instance, 2x and -5x are like terms because both contain x to the power 1. However, 3x and 3y are not like terms, and 4x and 4x² are not like terms either.
同类项是指变量部分完全相同的项，即相同的字母且指数相同。系数可以不同。例如，2x 和 -5x 是同类项，因为它们都含有 x 的一次方。然而，3x 与 3y 不是同类项，4x 与 4x² 也不是同类项。
A common way to practise this is to underline or circle the variable part. When you see 7ab and -2ab, the variable part ‘ab’ is identical, so they are like terms. Being able to recognise like terms quickly is the key to simplifying expressions accurately.
一个常用的练习方法是给变量部分画线或画圈。看到 7ab 和 -2ab 时，变量部分 ‘ab’ 完全相同，因此它们是同类项。能否快速识别同类项，是准确化简表达式的关键。
3. The Coefficient and Variable Parts | 系数与变量部分
Understanding the structure of a term helps avoid mistakes. In 6m²n, the coefficient is 6, and the variable part is m²n. The power 2 belongs only to m, not to n. Therefore, 6m²n and 2mn² are not like terms because the powers are assigned to different letters.
理解项的结构有助于避免错误。在 6m²n 中，系数是 6，变量部分是 m²n。指数 2 只属于 m，不属于 n。因此，6m²n 与 2mn² 不是同类项，因为指数分配给了不同的字母。
If a term has more than one variable, the order does not matter when checking for like terms. For example, 3pq and qp × 3 are essentially the same, as multiplication is commutative. However, it is standard practice to write variables in alphabetical order, so we would typically simplify to 3pq.
如果一个项含有多个变量，判断是否为同类项时，变量的顺序无关紧要。例如，3pq 和 qp × 3 本质上是相同的，因为乘法满足交换律。然而，通常的书写习惯是按字母顺序排列变量，因此我们一般会化简成 3pq。
4. Collecting Like Terms – Addition and Subtraction | 合并同类项 – 加法与减法
To collect like terms, we add or subtract their coefficients while keeping the variable part unchanged. This is based on the distributive property. For example, 3x + 2x = (3 + 2)x = 5x. Similarly, 7y – 4y = 3y.
合并同类项时，我们将系数相加或相减，而变量部分保持不变。这基于分配律。例如，3x + 2x = (3 + 2)x = 5x。类似地，7y – 4y = 3y。
If you have an expression like 5a + 3b + 2a – b, start by identifying like terms: 5a and 2a; 3b and -b. Then combine them: (5a + 2a) = 7a, and (3b – b) = 2b. The simplified expression is 7a + 2b.
如果遇到 5a + 3b + 2a – b 这样的表达式，首先找出同类项：5a 与 2a；3b 与 -b。然后合并：(5a + 2a) = 7a，(3b – b) = 2b。化简后的表达式为 7a + 2b。
A good tip is to reorder the expression so that like terms are grouped together. For instance, 4x – 3 + 2x + 7 becomes 4x + 2x – 3 + 7. Combine to get 6x + 4.
一个很好的技巧是重新排列表达式，将同类项放在一起。例如，4x – 3 + 2x + 7 可以写成 4x + 2x – 3 + 7。合并后得到 6x + 4。
5. Collecting Like Terms with Multiple Variables | 含有多个变量的同类项合并
When an expression contains terms with two variables, such as xy, only terms with the same combination of variables and identical powers can be combined. For example, 2xy + 5xy = 7xy, but 2xy + 3x cannot be combined because the variable sets are different.
当表达式含有带两个变量的项（例如 xy）时，只有变量组合与指数完全相同的项才能合并。例如，2xy + 5xy = 7xy，但 2xy + 3x 无法合并，因为变量集合不同。
Consider 8ab – 3a + 2ab + 4a. The like terms are 8ab and 2ab, and -3a and 4a. Combining them gives 10ab + a. Remember that 0a is simply 0 and should not be written.
考虑 8ab – 3a + 2ab + 4a。同类项有 8ab 与 2ab，以及 -3a 与 4a。合并后得到 10ab + a。请记住，0a 就是 0，不应写出。
Sometimes terms look different until you rearrange factors. For instance, p × q × 4 and 2q p are like terms because both represent 4pq and 2pq. Always rewrite terms in a consistent order before deciding if they are like terms.
有时项看起来不同，直到你重新排列因数。例如 p × q × 4 与 2q p 是同类项，因为它们分别表示 4pq 和 2pq。在判断是否为同类项之前，务必先将各项改写为统一的顺序。
6. Simplifying Expressions – Basic Examples | 化简表达式 – 基础示例
Simplifying an algebraic expression means writing it in its shortest possible form by collecting like terms and performing any arithmetic with constants. The aim is to reduce the number of terms while keeping the expression equivalent.
化简代数表达式，是指通过合并同类项并完成常数运算，将其写成尽可能简短的形式。目的在于减少项数，同时保持表达式等价。
Example 1: Simplify 9k + 3 – 4k + 6. Group like terms: (9k – 4k) + (3 + 6) = 5k + 9.
示例 1：化简 9k + 3 – 4k + 6。分组同类项：(9k – 4k) + (3 + 6) = 5k + 9。
Example 2: Simplify 2m + 5n – m + 3n – 4. Combine m terms: 2m – m = m; combine n terms: 5n + 3n = 8n; the constant is -4. Result: m + 8n – 4.
示例 2：化简 2m + 5n – m + 3n – 4。合并 m 项：2m – m = m；合并 n 项：5n + 3n = 8n；常数为 -4。结果：m + 8n – 4。
Always check that you have included all parts of the original expression. One common slip is forgetting to attach a minus sign to the term that follows it.
务必检查是否包含了原表达式的所有部分。一个常见的疏漏是忘记将减号附着在其后的项上。
7. Working with Negative Coefficients | 处理负系数
Terms can have negative coefficients, and they must be handled with care. Suppose we need to simplify 6x – 10 – 2x + 5. Identify the x terms: 6x and -2x; identify the constants: -10 and +5. Combining gives 4x – 5.
项可能带有负系数，处理时必须小心。假设我们需要化简 6x – 10 – 2x + 5。找出含 x 的项：6x 和 -2x；找出常数项：-10 和 +5。合并后得到 4x – 5。
When subtracting a negative term, it is helpful to think in terms of adding the opposite. For example, 4y – (-3y) is not a typical simplified expression, but if it occurs, it becomes 4y + 3y = 7y. In typical KS3 questions, you will mostly collect terms using straightforward addition and subtraction of coefficients.
当减去一个负项时，可以将其想象为加上其相反数。例如 4y – (-3y) 并非典型的化简表达式，但如果出现，就会变成 4y + 3y = 7y。在典型的 KS3 题目中，多数情况下只需直接加减系数来合并同类项。
Remember that the sign in front of a term belongs to that term. In the expression 12 – 7a + a – 9, the terms are 12, -7a, +a, and -9. The a term is +1a. So -7a + a = -6a; constants: 12 – 9 = 3. Final answer: -6a + 3, or 3 – 6a.
请记住，项前面的符号属于该项。在表达式 12 – 7a + a – 9 中，各项分别为 12、-7a、+a 和 -9。其中 a 的系数是 +1。因此 -7a + a = -6a；常数：12 – 9 = 3。最终答案：-6a + 3，或 3 – 6a。
8. Expressions with Constants | 含有常数项的表达式
Constants are numbers on their own, without any variable. They should always be combined together. In 15 + 3t – 8 + 7t, the like variables are 3t and 7t, giving 10t. The constants are 15 and -8, giving 7. The simplified form is 10t + 7.
常数项是指不带有任何变量的独立数字。它们应当合并在一起。在 15 + 3t – 8 + 7t 中，同类变量项为 3t 和 7t，得到 10t。常数项为 15 和 -8，得到 7。化简后的形式为 10t + 7。
Sometimes constants are written at the front or the back of an expression. Both ’10t + 7′ and ‘7 + 10t’ are correct, but it is conventional to write variable terms first, unless the question states otherwise.
常数项有时写在表达式前面，有时写在后面。“10t + 7” 与 “7 + 10t” 都是正确的，但按照惯例，通常先写变量项，除非题目另有要求。
9. Using the Commutative Property | 使用交换律
The commutative property of addition allows us to rearrange terms freely. This is extremely helpful when we want to place like terms next to each other. For example, x + 3y + 2x – y can be rewritten as x + 2x + 3y – y, making it clear that the result is 3x + 2y.
加法的交换律允许我们自由地重新排列各项。当我们希望将同类项排列在一起时，这极为有用。例如，x + 3y + 2x – y 可以改写为 x + 2x + 3y – y，这样就清楚地得出结果为 3x + 2y。
Be cautious: subtraction is not commutative, so you cannot simply move a minus sign away from its term. The expression 5 – 2x is not the same as 2x – 5. Always keep the sign attached to the term that immediately follows it.
需要注意：减法不满足交换律，因此不能随意将减号与其项分离。表达式 5 – 2x 与 2x – 5 不同。务必让符号紧跟在其后的项上。
10. Common Mistakes to Avoid | 常见错误避免
One of the most frequent errors is trying to combine unlike terms, such as adding 2x and 3 to get 5x. This is incorrect because 3 does not contain the variable x. Remind yourself: you cannot add a length to a number of apples.
最常见的错误之一是试图合并非同类项，比如将 2x 与 3 相加得到 5x。这是错误的，因为 3 并不包含变量 x。请提醒自己：不能把长度和苹果的个数直接相加。
Another mistake is mishandling the sign of the first term. If an expression begins with -4x + 9 – 3x, the first term is -4x, not 4x. Combining -4x and -3x gives -7x, so the simplified expression is -7x + 9.
另一个错误是处理不好第一项的符号。如果表达式以 -4x + 9 – 3x 开头，第一项是 -4x，而不是 4x。合并 -4x 和 -3x 得到 -7x，因此化简后的表达式为 -7x + 9。
Learners also sometimes forget that a variable on its own has a coefficient of 1. In 5y + y – 2, the y term is actually 1y, so 5y + 1y = 6y. Neglecting this leads to an answer of 5y + y, which is not fully simplified.
学习者有时会忘记，单独的变量系数为 1。在 5y + y – 2 中，y 项实际上是 1y，因此 5y + 1y = 6y。忽视这一点就会得出 5y + y 这样的答案，而这并没有彻底化简。
11. Applying to Real-world Problems | 应用于实际生活问题
Collecting like terms is not just an abstract exercise. It appears whenever we deal with perimeter, costs, or sequences. For example, the perimeter of a shape with sides 3x, 2x + 1, x, and 4 can be written as 3x + 2x + 1 + x + 4, which simplifies to 6x + 5.
合并同类项并不仅仅是抽象的练习。在处理周长、成本或数列时都会用到。例如，一个图形的边长分别为 3x、2x + 1、x 和 4，其周长可以写成 3x + 2x + 1 + x + 4，化简后为 6x + 5。
In money problems, if apples cost £a each and bananas cost £b each, buying 3 apples and 2 bananas plus an extra apple gives a total cost of 4a + 2b, after collecting the apple terms. This skill helps to keep formulas tidy and calculations quick.
在货币问题中，若每个苹果 £a，每个香蕉 £b，购买 3 个苹果和 2 个香蕉外加一个苹果，合并苹果项后总费用为 4a + 2b。这一技能有助于让公式保持整洁、计算更加快捷。
12. Practice Questions and Solutions | 练习题与解答
Test your understanding with these questions. Simplify each expression fully.
用以下题目测试你的理解。请彻底化简每个表达式。
| Expression | Your working | Simplified form |
| 7p + 3q – 2p + q | 5p + 4q | |
| 6a – 4b + a + 9b – 3 | 7a + 5b – 3 | |
| 8 – 3x + 2x + 12 – x | 20 – 2x | |
| -5mn + 2m + 7mn – 4m | 2mn – 2m | |
| 2k² + 3k + k² – k + 5 | 3k² + 2k + 5 |
Check your answers carefully. In the fourth question, the m terms and mn terms are not like terms, so they cannot be combined into a single term. In the last question, k² and k are different powers, so keep them separate. Regular practice will make collecting like terms feel automatic.
仔细核对答案。在第四题中，m 项与 mn 项不是同类项，因此不能合并成一项。在最后一题中，k² 与 k 的指数不同，应当保持分离。经常练习会使合并同类项变得得心应手。
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