📚 Collecting Like Terms: Rules, Examples and Exam Tips | 合并同类项：规则、例题与考试技巧
In IGCSE algebra, simplifying expressions is one of the first and most important skills you will master. The key step in almost every simplification is to collect like terms - that is, to group terms that have exactly the same variable part. This article explains the rules, the notation behind coefficients, common pitfalls and plenty of exam-style examples so that you can collect like terms quickly and accurately, whether the question involves linear expressions, powers, several variables, brackets, fractions or negative signs.
在 IGCSE 代数中，化简表达式是你首先要掌握也是最重要的技能之一。几乎每一次化简的关键步骤都是合并同类项——也就是说，把变量部分完全相同的项归为一组。本文讲解相关规则、系数背后的记号含义、常见错误，以及大量考试型例题，帮助你快速、准确地合并同类项；无论题目涉及一次式、幂、多个变量、括号、分数还是负号，你都能从容应对。
1. What Are Terms and Like Terms? | 什么是项与同类项
An algebraic expression is made up of terms. A term is a number, a variable, or a product of numbers and variables. For example, 3x, −5, 2x² and 4xy are all terms. The numerical part of a term is called its coefficient, and the variable part is sometimes called the variable part or the letter part. In the term 3x, the coefficient is 3 and the variable part is x; in −5, the coefficient is −5 and there is no variable part, so it is called a constant term.
代数表达式由项组成。项是一个数、一个变量，或数与变量的乘积。例如，3x、−5、2x² 和 4xy 都是项。项中的数字部分称为系数，字母部分称为变量部分。在 3x 中，系数是 3，变量部分是 x；在 −5 中，系数是 −5，没有变量部分，因此称为常数项。
Like terms are terms that contain exactly the same variable or variables, each raised to the same power. The coefficients do not need to be the same. For instance, 3x and 5x are like terms, and −2x² and 7x² are like terms. The order in which variables are written inside a term does not matter, so 4xy and −9yx are also like terms, because multiplication is commutative.
同类项是指含有完全相同的变量，且每个变量的指数也相同的项。系数不必相同。例如，3x 和 5x 是同类项，−2x² 和 7x² 也是同类项。项内变量书写的顺序并不影响判断，因为乘法满足交换律，所以 4xy 与 −9yx 也是同类项。
Terms such as 3x and 3x² are not like terms, because the powers of x are different. Similarly, 4xy and 4x are not like terms because one contains y and the other does not. The table below summarises the most common cases.
像 3x 和 3x² 这样的项不是同类项，因为 x 的指数不同。同样，4xy 和 4x 不是同类项，因为前者含 y 而后者不含。下表总结了最常见的几种情况。
| Pair of terms | 项的组合 | Like terms? | 是否同类项 | Reason | 原因 |
| 3x and 5x | Yes | 是 | Same variable x, same power 1 | 变量 x 相同，指数同为 1 |
| −2x² and 7x² | Yes | 是 | Same variable x, same power 2 | 变量 x 相同，指数同为 2 |
| 3x and 3x² | No | 否 | Different powers of x | x 的指数不同 |
| 4xy and 4x | No | 否 | One has y, the other does not | 一项含 y，另一项不含 |
| 4xy and −9yx | Yes | 是 | Same variables, order is irrelevant | 变量相同，顺序无关 |
| 5 and −2 | Yes | 是 | Both are constants | 两者都是常数项 |
2. The Core Rule: Add or Subtract the Coefficients | 核心法则：系数相加减
Once you have identified like terms, the rule for combining them is simple: keep the variable part exactly the same and add or subtract the coefficients. Think of 3x + 5x as "three x's plus five x's", which gives eight x's, written as 8x. The letter does not change and the power does not change.
一旦识别出同类项，合并规则就很简单：变量部分原封不动，只把系数相加或相减。可以把 3x + 5x 理解为"三个 x 加五个 x"，得到八个 x，写作 8x。字母不变，指数也不变。
3x + 5x = 8x −2x² + 7x² = 5x² 5x − 8x = −3x
as a sum of terms together with their signs: 5x − 8x is really (+5x) + (−8x), and 5 + (−8) = −3, so the answer is −3x.
只要把符号看作紧跟在后面的项自带的符号，减法与加法完全一样。一个好习惯是把表达式读成"带符号的项之和"：5x − 8x 实际上是 (+5x) + (−8x)，而 5 + (−8) = −3，所以答案是 −3x。
Terms that are not like terms cannot be merged into a single term. You may still rewrite an expression by reordering it, but the number of separate terms stays the same. For example, 2x + 3y has two terms and cannot be simplified further, because x and y are different variables.
不是同类项的项不能合并成一个项。你可以重新排列表达式的顺序，但独立项的个数不会改变。例如 2x + 3y 有两项，无法再化简，因为 x 与 y 是不同的变量。
3. Coefficients, Signs and Hidden 1s | 系数、符号与隐藏的 1
ewise, −x = −1x, and y² = 1y². Writing the hidden 1 in your working, especially in the first few practice questions, makes it much harder to lose a term.
本主题中的许多错误都源于漏看隐藏的系数或符号。写成 x 的项其实有一个看不见的系数 1，所以 x = 1x。同样，−x = −1x，y² = 1y²。在解题过程（尤其是最初几道练习）中把隐藏的 1 写出来，会大大降低漏项的概率。
6a + a = 7a. These two results differ by 2a, which is exactly the kind of small difference examiners like to test.
紧贴在括号或项前面的负号属于整个项。因此在 6a − a 中，第二项是 −1a，于是 6a − a = 6a − 1a = 5a。相反，6a + a = 7a。这两个结果相差 2a，正是出题人喜欢考查的那种细小差别。
x = 1x −x = −1x 6a − a = 5a 6a + a = 7a
b. All of them are the same term, so they are all like terms with each other. The convention in IGCSE is to write variables in alphabetical order, so ab rather than ba, and x²y rather than yx².
还值得记住的是，像 2ab 这样的项可以写成多种等价形式：2ba、a × 2b 或 2 × a × b，它们都是同一个项，因此彼此都是同类项。IGCSE 的书写习惯是把变量按字母顺序排列，因此写成 ab 而不是 ba，写成 x²y 而不是 yx²。
4. Worked Example 1: Linear Expressions | 例题一：一次式
Let us simplify 7a + 3b − 4a + 5b − 2. Step 1: identify groups of like terms - the a-terms are 7a and −4a, the b-terms are 3b and 5b, and the constants are −2 alone. Step 2: combine each group. Step 3: write the answer with the terms in a tidy order.
我们来化简 7a + 3b − 4a + 5b − 2。第一步：找出同类项分组——含 a 的项是 7a 与 −4a，含 b 的项是 3b 与 5b，常数项只有 −2。第二步：分别合并。第三步：按整齐的顺序写出答案。
7a − 4a = 3a 3b + 5b = 8b 7a + 3b − 4a + 5b − 2 = 3a + 8b − 2
Notice that 3a, 8b and −2 are three different types of term, so the expression stops there. A common wrong answer is 11ab − 2 or 11ab, both of which come from incorrectly adding coefficients of unlike terms.
注意 3a、8b 和 −2 是三种不同的项，所以化简到此结束。常见错误答案包括 11ab − 2 或 11ab，两者都是错误地把非同类项的系数相加造成的。
the constants: 8 + 2 = 10. The simplified expression is −8m + 10, or equivalently 10 − 8m.
第二个有用的例子以减法开头：−5m + 8 − 3m + 2。合并含 m 的项：−5m − 3m = −8m。合并常数项：8 + 2 = 10。化简结果为 −8m + 10，也可以写成 10 − 8m。
5. Worked Example 2: Expressions with Powers | 例题二：含幂的表达式
Powers must match exactly before terms can be collected. In an expression such as 4x² + 3x − 2x² + 7x − 5x², there are two families: the x² terms and the x terms. Treat them as separate queues and simplify each one.
幂必须完全相同才能合并。在 4x² + 3x − 2x² + 7x − 5x² 这样的表达式中，有两类项：x² 项与 x 项。把它们当作两条独立的队列，分别化简。
(4 − 2 − 5)x² = −3x² (3 + 7)x = 10x Answer: −3x² + 10x
You may also write the answer as 10x − 3x². Either order is acceptable unless the question asks for descending powers, which is the more conventional presentation in IGCSE mark schemes.
答案也可以写成 10x − 3x²。除非题目要求按降幂排列，否则两种顺序都可以；而在 IGCSE 评分标准中，降幂排列更为常见。
ou get x⁴, because the indices add under multiplication. The rule for collecting like terms only ever involves adding or subtracting coefficients, never multiplying or dividing them.
不要把"加幂"与"乘幂"混为一谈。x² + x² = 2x²，因为你有两个相同的 x² 项；而 x² × x² = x⁴，因为乘法中指数相加。合并同类项的规则永远只是系数的加减，绝不涉及系数的乘除。
x² + x² = 2x² x² × x² = x⁴ x² + x³ cannot be simplified | x² + x³ 无法合并
6. Worked Example 3: Several Variables, Including xy Terms | 例题三：多变量，含 xy 项
ing 3xy. The y terms are 3y and −y, that is 3y − 1y = 2y. The y² term is 4y² on its own, and the constant is 6.
含两个变量的表达式常常包含四五种不同的项。看 5xy + 3y − 2xy + 4y² − y + 6。xy 项是 5xy 与 −2xy，得 3xy。y 项是 3y 与 −y，即 3y − 1y = 2y。y² 项只有 4y²，常数项是 6。
5xy + 3y − 2xy + 4y² − y + 6 = 3xy + 2y + 4y² + 6
Each of the four resulting terms is different from the others, so the expression is fully simplified. A handy check is to count the number of distinct variable parts before and after: the original had five distinct parts (xy, y, y², constant) and so does the answer, though the coefficients have changed.
得到的四个项彼此都不同，因此表达式已完全化简。一个实用的检查方法是比较化简前后"不同变量部分"的个数：原式与答案都包含 xy、y、y² 与常数这几类，只是系数发生了变化。
Remember that x²y and xy² are not like terms, even though they contain the same letters. In x²y the power of x is 2, while in xy² the power of y is 2, so the variable parts are different and the terms stay separate.
请记住，x²y 与 xy² 不是同类项，尽管它们含有相同的字母。x²y 中 x 的指数是 2，而 xy² 中 y 的指数是 2，变量部分不同，因此两项必须分开保留。
7. Expanding Brackets Before Collecting | 先展开括号再合并
get 3x + 6 + 2x − 10, then collect: 3x + 2x = 5x and 6 − 10 = −4, so the answer is 5x − 4.
当表达式含括号时，应先展开，再合并。对于 3(x + 2) + 2(x − 5)，展开得 3x + 6 + 2x − 10，然后合并：3x + 2x = 5x，6 − 10 = −4，所以答案是 5x − 4。
3(x + 2) + 2(x − 5) = 3x + 6 + 2x − 10 = 5x − 4
e −3 × (−1) = +3. Collecting gives 8x − 3x = 5x and −12 + 3 = −9, so the answer is 5x − 9.
括号前面的负号会改变括号内每一项的符号。对于 4(2x − 3) − 3(x − 1)，第二个括号展开后变为 −3x + 3，因为 −3 × (−1) = +3。合并得 8x − 3x = 5x，−12 + 3 = −9，所以答案是 5x − 9。
4(2x − 3) − 3(x − 1) = 8x − 12 − 3x + 3 = 5x − 9
x + 3. Forgetting to flip the sign of the second term is one of the most frequently penalised errors in IGCSE algebra questions.
同样的道理适用于括号前只有一个负号的情形：−(x − 3) = −x + 3。忘记改变第二项的符号，是 IGCSE 代数题中最常被扣分的错误之一。
8. Fractions, Decimals and Negative Coefficients | 分数、小数与负系数
The collecting rule does not care whether the coefficients are integers, fractions or decimals. For ½x + ⅓x, the coefficients ½ and ⅓ add to 5/6, so the answer is (5/6)x. Using a common denominator of 6 is the safest approach, and the final fraction should normally be left in its simplest form.
合并规则并不在意系数是整数、分数还是小数。对于 ½x + ⅓x，系数 ½ 与 ⅓ 相加得 5/6，因此答案是 (5/6)x。用公分母 6 是最稳妥的做法，最终分数通常要化为最简形式。
½x + ⅓x = (3/6)x + (2/6)x = (5/6)x
1.6y. A quick estimate helps: 0.7 − 2.3 is roughly −1.6, which confirms the negative sign.
处理小数时，要仔细对齐数位：1.5x + 2.4x = 3.9x，0.7y − 2.3y = −1.6y。快速估算有助于检查：0.7 − 2.3 大约是 −1.6，可确认结果应为负数。
3, giving 3p. Doing this in two small steps rather than one long jump reduces sign errors.
负系数遵循有理数运算的常规规则。对于 −4p + 9p − 2p，可以分步合并：−4 + 9 = 5，再 5 − 2 = 3，得 3p。分成两小步计算，比一步跳过去更能减少符号错误。
− 3) = 2x + 3 + 3x − 1 = 5x + 2. Expanding before collecting keeps all the signs visible.
如果分数写在括号前面作为系数，应先展开。例如 ½(4x + 6) + ⅓(9x − 3) = 2x + 3 + 3x − 1 = 5x + 2。先展开再合并，可以让所有符号都清晰可见。
9. Common Pitfalls and Exam Traps | 常见错误与考试陷阱
y is wrong, because xy means x multiplied by y, which is not what addition produces. Similarly, 3x + 3x² is not 6x³; the correct answer is simply 3x + 3x², two separate terms.
最常见的错误是非法合并不同类项。把 2x + 3y 写成 5xy 是错误的，因为 xy 表示 x 乘 y，而加法不会产生这种结果。同样，3x + 3x² 不等于 6x³；正确答案就是 3x + 3x²，两项各自保留。
A second trap is losing a minus sign when rearranging the expression. Always carry the sign in front of a term with that term, and rewrite subtraction as adding a negative before you rearrange. A third trap is dropping a term entirely, which often happens when an expression contains a hidden coefficient of 1, such as the −a in 5b − a.
第二个陷阱是重新排列项时丢失负号。始终把项前面的符号与该项绑在一起，并在整理前把减法改写为加上负数。第三个陷阱是整项漏写，这在表达式含有隐藏系数 1 时尤其常见，例如 5b − a 中的 −a。
uently place an addition and a multiplication side by side in the same question to test whether you can tell them apart.
最后，注意不要混淆运算。x + x = 2x，但 x × x = x²；2x + 3x = 5x，但 2x × 3x = 6x²。出题人常常在同一道题里同时安排加法和乘法，以检验你能否区分它们。
x + x = 2x ≠ x² 2x × 3x = 6x² 2x + 3y ≠ 5xy
10. Checking Your Answer by Substitution | 用代入法检验答案
Every simplification can be checked numerically by substituting a convenient value for the variable into the original expression and into your answer. If the two values agree for a well-chosen number, the simplification is very likely correct. Values such as 2, 3 or −1 are quick to compute.
任何化简都可以用代入数值的方法检验：把一个方便的数值分别代入原式与你的答案。如果两个值相等，化简就极可能正确。像 2、3 或 −1 这样的数值计算起来很快。
2, and the right side is 6 + 8 − 2 = 12. The two sides agree, so the answer is consistent.
以之前的例子 7a + 3b − 4a + 5b − 2 = 3a + 8b − 2 为例。取 a = 2、b = 1。左边为 14 + 3 − 8 + 5 − 2 = 12，右边为 6 + 8 − 2 = 12。两边相等，说明答案自洽。
tead. This habit is especially valuable in examinations, where a single lost sign can cost several marks.
对于含幂的表达式，避免代入 1 或 0，因为这些值可能掩盖错误，改用 x = 2 或 x = 3 更可靠。这个习惯在考试中尤其有价值，因为一个丢失的符号可能让你失掉好几分。
11. Exam-Style Questions with Worked Solutions | 考试型练习与解答
− 3a + 6b + 4b, then collect: 10a − 3a = 7a and −5b + 6b + 4b = 5b. The answer is 7a + 5b.
题目 1：化简 5(2a − b) − 3(a − 2b) + 4b。先展开得 10a − 5b − 3a + 6b + 4b，再合并：10a − 3a = 7a，−5b + 6b + 4b = 5b。答案为 7a + 5b。
the x terms: −3x + 7x = 4x. Collect the constants: 2 − 9 = −7. The answer is 3x² + 4x − 7.
题目 2：化简 8x² − 3x + 2 − 5x² + 7x − 9。合并 x² 项：8x² − 5x² = 3x²；合并 x 项：−3x + 7x = 4x；合并常数项：2 − 9 = −7。答案为 3x² + 4x − 7。
ree expressions: (3x + 5x + 4x) + (2 − 1 + 3) = 12x + 4, so the perimeter is (12x + 4) cm.
题目 3：三角形的三边长分别为 (3x + 2) cm、(5x − 1) cm 和 (4x + 3) cm。用 x 表示周长的化简结果。把三个表达式相加：(3x + 5x + 4x) + (2 − 1 + 3) = 12x + 4，所以周长是 (12x
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