📚 Collecting Like Terms in Algebra | 代数中的合并同类项
In IGCSE algebra, expressions often look messy because several terms are written together. Collecting like terms means spotting the terms that have exactly the same variable part and combining their coefficients. This skill is essential before solving equations, expanding brackets, factorising, or working with graphs.
在 IGCSE 代数中，代数式常常因为多个项写在一起而显得复杂。合并同类项就是找出变量部分完全相同的项，并把它们的系数相加。无论是解方程、去括号、因式分解还是处理图像，这项技能都是基础。
It is one of the first algebraic techniques you learn, yet it appears in almost every later topic, from linear equations and inequalities to quadratic expressions and coordinate geometry. If you cannot collect terms confidently, later work becomes slow and error-prone.
这是你最先学会的代数技巧之一，但它几乎出现在之后所有主题中，从一次方程、不等式到二次式和解析几何。如果合并同类项不够熟练，后面的学习就会又慢又容易出错。
1. What Are Like Terms? | 什么是同类项？
A term is a single number, a single variable, or a product of numbers and variables, together with the sign written in front of it. An algebraic expression is simply a collection of such terms joined by plus or minus signs.
项是指单个数字、单个变量，或者数字与变量的乘积，再加上它前面的符号。代数式就是由这样的项用加号或减号连接起来的整体。
Like terms have exactly the same variables raised to exactly the same powers. Only the coefficient, the number in front, may be different. In other words, the variable part must match perfectly; the numerical part is allowed to change.
同类项是指所含的变量完全相同，并且每个变量的指数也完全相同。只有系数，也就是变量前面的数，可以不同。换句话说，变量部分必须完全一致，数字部分可以不同。
For example, 4x and -7x are like terms because both contain x to the power 1. Similarly, 2x² and 5x² are like terms because both contain x². Constants such as 3 and -8 are also like terms because they have no variable part at all.
例如，4x 和 -7x 是同类项，因为它们都含有 x 的一次方。同样，2x² 和 5x² 是同类项，因为它们都含有 x²。像 3 和 -8 这样的常数项也是同类项，因为它们没有变量部分。
However, 3x and 3x² are not like terms, and 3x and 3y are not like terms. The variable part must match exactly, and it does not matter that the coefficients happen to be equal.
但是，3x 和 3x² 不是同类项，3x 和 3y 也不是同类项。变量部分必须完全相同，系数偶然相等并不能让它们成为同类项。
Like terms: 4x, -7x, 0.5x | Unlike terms: 3x, 3x², 3y
同类项：4x, -7x, 0.5x | 非同类项：3x, 3x², 3y
Collecting like terms therefore means grouping the matching terms and replacing each group with a single simpler term, so that the whole expression becomes as short as possible.
因此，合并同类项就是把相匹配的项分组，并用一个更简单的项替换每一组，使整个代数式尽可能简短。
2. The Coefficient and the Variable Part | 系数与变量部分
Every term can be split into two parts: a coefficient and a variable part. In the term 7x, the coefficient is 7 and the variable part is x. In the term -3ab, the coefficient is -3 and the variable part is ab.
每个项都可以拆成两部分：系数和变量部分。在项 7x 中，系数是 7，变量部分是 x。在项 -3ab 中，系数是 -3，变量部分是 ab。
The sign in front of a term always belongs to the term itself. This matters when you collect terms, because subtracting a term is the same as adding a negative term: 8x - 3x is really 8x + (-3x), so the coefficients 8 and -3 are added.
项前面的符号永远属于这个项本身。这在合并同类项时很重要，因为减去一个项等于加上一个负项：8x - 3x 实际上就是 8x + (-3x)，所以是把系数 8 和 -3 相加。
orgetting this invisible 1 is one of the most frequent sources of error in examination questions.
如果变量前面没有写数字，系数就默认为 1。所以 x = 1x，y = 1y，-m = -1m。忘记这个"隐形的 1"是考试中最常见的错误来源之一。
Constants such as 12, -4 and 0.5 have no variable part, so they are all like terms with each other, no matter what their values are.
像 12、-4、0.5 这样的常数项没有变量部分，因此无论数值是多少，它们彼此之间都是同类项。
Term = coefficient × variable part → 7x = 7 × x
项 = 系数 × 变量部分 → 7x = 7 × x
3. Which Terms Are Not Like Terms | 哪些项不是同类项
Most mistakes in this topic come from treating unlike terms as if they matched. The following rules summarise the cases you must watch out for in IGCSE papers.
这一主题中的大多数错误，都来自把非同类项当成了同类项。下面这些规则总结了 IGCSE 试卷中你必须警惕的情况。
- Different letters are never like terms: 3x and 3y cannot be combined; 2a + 5b stays as it is.
字母不同的一定不是同类项：3x 与 3y 不能合并；2a + 5b 只能保持原样。 - Different powers of the same letter are not like terms: x and x² do not match, so x + x² cannot be simplified.
同一个字母但指数不同也不是同类项：x 与 x² 不匹配，所以 x + x² 无法化简。 - Different combinations of letters are not like terms: 2xy and 2x²y are different because the power of x differs; 4ab and 4a²b² are also different.
字母组合不同也不是同类项：2xy 与 2x²y 因为 x 的指数不同而不同；4ab 与 4a²b² 同样不同。 - Powers and roots do not mix: √x, x and x² are three different variable parts.
根式与幂不能混用：√x、x 和 x² 是三种不同的变量部分。 - tter: xy and yx are like terms, because multiplication is commutative, so 3xy + 4yx = 7xy.
字母顺序不影响：xy 与 yx 是同类项，因为乘法满足交换律，所以 3xy + 4yx = 7xy。 - Fractions and decimals are just coefficients: ½x and 0.5x are like terms, since ½ = 0.5.
分数和小数只是系数：½x 与 0.5x 是同类项，因为 ½ = 0.5。
The key question to ask yourself every time is simple: are the variable parts exactly identical, including the indices? If yes, you may add the coefficients. If no, leave the terms alone.
每次你只需问自己一个简单的问题：变量部分是否完全一致，包括指数？如果一致，就可以把系数相加；如果不一致，就让这些项保持原样。
4. The Rule for Collecting Like Terms | 合并同类项的法则
is: a·x + b·x = (a + b)x, and a·x - b·x = (a - b)x, where x stands for any variable part.
合并同类项时，把系数相加，变量部分保持不变。用符号表示就是：a·x + b·x = (a + b)x，a·x - b·x = (a - b)x，其中 x 代表任意变量部分。
5a + 3a = (5 + 3)a = 8a
9m - 4m = (9 - 4)m = 5m
a common factor. This is exactly why only the coefficients are added and the variable part is untouched.
这条法则的依据是分配律的逆用。因为 5a + 3a = a(5 + 3)，变量部分相当于公因式。这正是为什么只加系数、而变量部分保持不变的原因。
When an expression contains several groups of like terms, work through them systematically. Many students find it helpful to underline or circle the different groups first, then add each group separately.
当一个代数式中含有好几组同类项时，要有系统地逐组处理。很多学生发现，先用下划线或圆圈把不同的组标出来，然后分别相加，效果很好。
An expression is usually regarded as fully simplified when no further like terms remain and the terms are written with the highest power first, with letters in alphabetical order, for example 4x² + 3xy - 5y + 2.
当式中不再有同类项，并且各项按次数从高到低排列、字母按字母表顺序书写时，例如 4x² + 3xy - 5y + 2，通常就认为已经化简到最简形式。
5. Worked Example: Simple Linear Expressions | 例题：简单一次式
Consider the expression 7x + 3 - 2x + 5. First identify the like terms: 7x and -2x share the variable part x, while 3 and 5 are both constants. Then combine each group separately.
考虑代数式 7x + 3 - 2x + 5。首先找出同类项：7x 与 -2x 的变量部分都是 x，而 3 和 5 都是常数项。然后把每一组分别合并。
ts together gives 5x + 8, which cannot be simplified any further because 5x and 8 are not like terms.
关于 x 的项：7x - 2x = 5x。关于常数项：3 + 5 = 8。把两个结果合起来得到 5x + 8，由于 5x 与 8 不是同类项，所以无法再化简。
The table below shows several one-step examples that you should be able to do mentally before moving on to harder questions.
下面的表格列出了几个一步到位的例子，在进入更难的题目之前，你应该能够直接口算出来。
| Expression 原式 | Collected form 合并结果 | Reason 理由 |
| 4y + 6y - 9 | 10y - 9 | y terms combine; -9 stays alone |
| 12 - 5t + 3t - 4 | 8 - 2t | -5t + 3t = -2t and 12 - 4 = 8 |
| 3x + 2x + x | 6x | x means 1x, so 3 + 2 + 1 = 6 |
| 8p - p | 7p | 8 - 1 = 7 |
Notice that the order of the final answer is flexible: 8 - 2t and -2t + 8 mean exactly the same thing. However, examiners generally prefer the term with the variable first, or the arrangement that matches the original question.
注意，最后答案的顺序是灵活的：8 - 2t 与 -2t + 8 含义完全相同。不过，考官通常更偏好把含变量的项写在前面，或者与题目原有的排列保持一致。
6. Collecting Terms with More Than One Variable | 含多个变量的合并
When an expression has several different letters, treat each letter as its own family. For example, in 3a + 2b + 5a - b, the a terms are 3a and 5a, and the b terms are 2b and -b.
当一个式子里有多个不同字母时，把每个字母看成各自的一"族"。例如在 3a + 2b + 5a - b 中，含 a 的项是 3a 和 5a，含 b 的项是 2b 和 -b。
e b terms gives 2b - b = b, because 2b - b = 2b - 1b = 1b = b. The final answer is 8a + b.
合并 a 的项得到 3a + 5a = 8a。合并 b 的项得到 2b - b = b，因为 2b - b = 2b - 1b = 1b = b。最终答案是 8a + b。
wrong: the correct product 11pq would only appear if the terms were multiplied, not added.
字母不同的项永远不能合并成一项，所以 4p + 7q 保持为 4p + 7q。一个常见错误是写成 4p + 7q = 11pq，这是错的：只有当这些项相乘时才会出现乘积 11pq，而不是在相加时。
y = 5xy, and group the x terms to get 5x - 3x = 2x. The simplified expression is 5xy + 2x.
同样的思路适用于混合乘积。在 3xy + 5x + 2xy - 3x 中，把 xy 项合起来得 3xy + 2xy = 5xy，把 x 项合起来得 5x - 3x = 2x。化简后的式子是 5xy + 2x。
A useful habit is to rewrite the expression with all the x terms first, then the xy terms, then the y terms, and finally the constants, before you start collecting.
一个很有用的习惯是：在开始合并之前，先把所有 x 项写在前面，接着是 xy 项、y 项，最后是常数项。
7. Powers, Indices and Higher Order Terms | 幂、指数与高次项
including the index must be identical. So 3x² and 5x² are like terms, and 3x² + 5x² = 8x².
幂与简单变量的规则完全相同：包括指数在内的变量部分必须一致。因此 3x² 与 5x² 是同类项，3x² + 5x² = 8x²。
However x and x² are not like terms, because the powers differ. The expression x + x² is already as simple as it can be, and writing it as x³ would be completely wrong.
但 x 与 x² 不是同类项，因为指数不同。式子 x + x² 已经是最简形式，把它写成 x³ 是完全错误的。
x² + x² = 2x² (NOT x⁴) | x² × x³ = x⁵
x² + x² = 2x² （不是 x⁴） | x² × x³ = x⁵
Remember the key distinction: when you add or subtract like terms, you add the coefficients and leave the indices alone; when you multiply powers of the same letter, you add the indices. Mixing these two rules up is a very common exam error.
记住这个关键区别：加减同类项时，你加的是系数，指数保持不变；而同底数幂相乘时，
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