📚 Collecting Like Terms | 合并同类项
In algebra, expressions often contain several terms. Collecting like terms is the process of simplifying an expression by combining terms that have exactly the same variable factors. This skill is essential for solving equations, rearranging formulae and working with functions in IGCSE Mathematics. A secure grasp of this single technique removes a large proportion of careless errors from later topics such as factorising, quadratic equations, inequalities and coordinate geometry.
在代数中，表达式常包含多个项。合并同类项就是通过把变量因数完全相同的项相加或相减来化简表达式。在 IGCSE 数学中，这项技能对解方程、变换公式和处理函数都至关重要。牢固掌握这一技巧，能大幅减少后面因式分解、二次方程、不等式与坐标几何等专题中的粗心错误。
1. What Are Like Terms? | 什么是同类项？
In mathematics, a term is a single number, variable, or product of numbers and variables. For example, 3x, −5y, 7x² and 4 are all terms. Terms are separated by plus or minus signs when an expression is read from left to right.
在数学中，项是指一个单独的数、变量或数与变量的乘积。例如，3x、−5y、7x² 和 4 都是项。从左到右读一个表达式时，项与项之间由加号或减号分隔。
Like terms are terms that have exactly the same variable parts, including the same powers. The coefficients can be different. For instance, 3x and 5x are like terms, while 3x and 3x² are not.
同类项是指变量部分（包括相同的幂）完全相同的项，系数可以不同。例如，3x 和 5x 是同类项，而 3x 和 3x² 不是同类项。
Constants such as 2, −7 and 0.5 are like terms with each other because they have no variable part. They can always be added together to give a single number.
常数如 2、−7 和 0.5 彼此互为同类项，因为它们都没有变量部分。它们总可以相加合并成一个数。
2. Terms, Coefficients and Indices | 项、系数与指数
Every term can be broken into two parts: the coefficient (the numerical factor) and the variable part (the letters, together with their indices). In the term 7x², the coefficient is 7 and the variable part is x².
每个项都可以拆成两部分：系数（数字因数）和变量部分（字母及其指数）。在项 7x² 中，系数是 7，变量部分是 x²。
If no number is written, the coefficient is understood to be 1, and if a minus sign stands in front, the coefficient is −1. So x means 1x and −y means −1y.
如果项前没有写数字，则默认系数为 1；如果前面是负号，则系数为 −1。因此 x 就是 1x，−y 就是 −1y。
The index (power) of a variable is part of the variable part, not the coefficient. This is why x and x² can never be collected together: they are different types of term.
变量上的指数（幂）属于变量部分，而不属于系数。这正是 x 与 x² 永远不能合并的原因：它们是不同类型的项。
3. Why Collecting Like Terms Matters | 为什么合并同类项如此重要
Simplified expressions are easier to read, easier to substitute into, and far less likely to produce errors in later steps. Almost every algebra question in IGCSE Mathematics expects a final answer in its simplest form.
化简后的表达式更易读、更易代入数值，在后续步骤中也远不容易出错。IGCSE 数学中几乎每一道代数题都要求最终答案写成最简形式。
When you solve an equation, collecting like terms on each side is usually the first move. When you rearrange a formula, you collect terms containing the required variable on one side before factorising.
解方程时，先把每一边的同类项合并通常就是第一步。变换公式时，需要先把含目标变量的项集中到一边，然后再提取公因式。
re evaluating f(6) or sketching the graph. Marks are frequently awarded for correct simplification even when the later working goes wrong.
在处理函数时，你可能需要先把 f(x) = 2x + 3x − 4 化简为 f(x) = 5x − 4，再去计算 f(6) 或画出图像。即使后面的步骤出错，正确的化简也常常能拿到分数。
4. The Basic Rule | 基本法则
To collect like terms, add or subtract the coefficients and keep the variable part unchanged. The variable part acts as a label: you are simply counting how many of that label you have.
合并同类项的方法是把系数相加或相减，而变量部分保持不变。变量部分相当于一个标签：你只是在数这个标签一共有多少个。
3x + 5x = (3 + 5)x = 8x
9y − 4y = (9 − 4)y = 5y
The same logic applies to constants, and to any shared variable part. Think of 3x + 5x as three apples plus five apples, giving eight apples; the "apple" label does not change.
同样的逻辑适用于常数，也适用于任何共同的变量部分。可以把 3x + 5x 理解为 3 个苹果加 5 个苹果，得到 8 个苹果；"苹果"这个标签本身没有变。
Terms that are not like terms must be left separate. So 4x + 3y stays as 4x + 3y, and 2x² + 5x stays as 2x² + 5x. Writing 7xy or 7x³ in these cases is a serious error.
不是同类项的项必须分开保留。因此 4x + 3y 就写成 4x + 3y，2x² + 5x 就写成 2x² + 5x。若写成 7xy 或 7x³ 就是严重错误。
5. Worked Examples with Linear Expressions | 一次式例题
5. Group the x-terms and the constants: 7x − 2x = 5x and 3 + 5 = 8. The answer is 5x + 8.
例 1：化简 7x + 3 − 2x + 5。把 x 项与常数分别归组：7x − 2x = 5x，3 + 5 = 8。答案是 5x + 8。
3b. The a-terms give 2a − a = a, and the b-terms give 5b + 3b = 8b. The answer is a + 8b.
例 2：化简 2a + 5b − a + 3b。a 项得到 2a − a = a，b 项得到 5b + 3b = 8b。答案是 a + 8b。
t the m-terms: 6m + 3m − m = 8m. Collect the numbers: −4 − 7 = −11. The answer is 8m − 11.
例 3：化简 6m − 4 + 3m − 7 − m。合并 m 项：6m + 3m − m = 8m。合并常数：−4 − 7 = −11。答案是 8m − 11。
A useful habit is to mark each family of terms with a different underline or colour before writing the answer. This prevents terms being lost or double-counted.
一个好习惯是先用不同的下划线或颜色标出每一类项，再写答案。这样可以避免漏项或重复计数。
6. Collecting Terms with Powers | 含幂的同类项
Terms are like terms only when the indices match exactly. So 4x² and 9x² are like terms and give 13x², but 4x² and 9x³ cannot be combined.
只有当指数完全相同时，项才是同类项。因此 4x² 与 9x² 是同类项，合并得 13x²；但 4x² 与 9x³ 无法合并。
2x² = 3x². The x-terms give 3x + 7x = 10x. The constant is 1. The answer is 3x² + 10x + 1.
例：化简 5x² + 3x − 2x² + 7x + 1。x² 项得到 5x² − 2x² = 3x²；x 项得到 3x + 7x = 10x；常数是 1。答案是 3x² + 10x + 1。
It is conventional to write the simplified answer in descending powers of the variable, as this matches the standard form used later in quadratic work.
通常约定把化简结果按变量的降幂排列书写，这与后面二次函数所用的标准形式一致。
7. Terms with Two or More Variables | 含两个或更多变量的项
When a term contains several letters, all of them must match for the terms to be alike. So 3ab and 5ab are like terms, but 3ab and 5a²b are not, because the power of a is different.
当一个项含有多个字母时，所有字母都必须完全相同，两项才是同类项。因此 3ab 与 5ab 是同类项，而 3ab 与 5a²b 不是，因为 a 的指数不同。
Because multiplication is commutative, ab and ba mean the same thing. So 6ab and 2ba are like terms and combine to give 8ab. Always write the letters of a term in alphabetical order to make matching easier.
由于乘法满足交换律，ab 与 ba 含义相同。因此 6ab 与 2ba 是同类项，合并得 8ab。书写时把项中字母按字母表顺序排列，会更便于比对。
y = 3xy. The x-terms give 3x − 2x = x. The y-term stays as 5y. The answer is 3xy + x + 5y.
例：化简 4xy + 3x − xy + 5y − 2x。xy 项得到 4xy − xy = 3xy；x 项得到 3x − 2x = x；y 项保留为 5y。答案是 3xy + x + 5y。
8. Signs, Brackets and Negative Coefficients | 符号、括号与负系数
Signs belong to the term that follows them. In the expression 8 − 3x + 2y, the terms are +8, −3x and +2y, not 8, 3x and 2y.
符号属于它后面的项。在表达式 8 − 3x + 2y 中，各项是 +8、−3x 和 +2y，而不是 8、3x 和 2y。
When brackets are multiplied by a number, expand first and then collect. For example, 3(x + 4) − 2(x − 5) becomes 3x + 12 − 2x + 10, which simplifies to x + 22.
当括号被一个数相乘时，要先展开再合并。例如 3(x + 4) − 2(x − 5) 展开为 3x + 12 − 2x + 10，化简得 x + 22。
le minus sign in front of a term turns an addition into a subtraction, so x + (−4x) = −3x.
注意"双重负号"：减去一个括号会使括号内每一项都变号。同时要记住，项前面的一个负号会把加法变成减法，所以 x + (−4x) = −3x。
9. Rearranging Before You Collect | 先整理再合并
Some expressions look as though they contain no like terms at first glance. Rearranging them so that similar terms sit next to each other often reveals the pattern.
有些表达式乍看之下似乎没有同类项。把它们重新排列，让相似的项相邻，往往就能看出规律。
Example: simplify 5 + 2x − 7 + 6x. Reorder to 2x + 6x + 5 − 7, then collect to get 8x − 2.
例：化简 5 + 2x − 7 + 6x。重新排列为 2x + 6x + 5 − 7，再合并得 8x − 2。
When you rearrange, move each term together with its sign. A common slip is to carry the number but drop the minus sign, which changes the whole answer.
重新排列时，要把每一项连同它的符号一起移动。常见的失误是搬走了数字却丢掉了负号，结果整个答案都变了。
10. Common Mistakes in Exams | 考试中的常见错误
e 1: combining unlike terms. Writing 3x + 4y = 7xy is wrong; the terms must stay separate.
错误 1：把非同类项合并。写成 3x + 4y = 7xy 是错的，这两项必须分开保留。
3x the answer is 5x, not 5x². Indices only change under multiplication, as in x × x = x².
错误 2：把指数相加。2x + 3x 的答案是 5x，而不是 5x²。只有在乘法中指数才变化，例如 x × x = x²。
Mistake 3: losing the sign of a term, especially after removing brackets, or forgetting that a term with no visible coefficient has coefficient 1.
错误 3：丢掉项的符号，尤其是在去括号之后；或者忘记没有写出系数的项其系数为 1。
Mistake 4: forgetting to collect the constants. Numbers such as 3 and −7 are like terms and should always be combined into a single value.
错误 4：忘记合并常数。像 3 与 −7 这样的数也是同类项，应当合并成一个数。
| Expression | 表达式 | Simplified | 化简结果 | Reason | 理由 |
| 4x + 5x | 9x | like terms, same variable | 同类项，变量相同 |
| 4x + 5x² | 4x + 5x² | different powers | 幂不同 |
| 3ab + 2ba | 5ab | order of letters is irrelevant | 字母顺序无关 |
| 6 − 11 | −5 | constants are like terms | 常数互为同类项 |
11. Exam-Style Practice Questions | 考试风格练习题
Try these questions without a calculator, showing each grouping step clearly. They mirror the style of IGCSE Paper 1 and Paper 2 simplification items.
请不用计算器完成下列题目，并把每一步的归类过程写清楚。这些题目模仿 IGCSE Paper 1 与 Paper 2 中化简题的形式。
- Simplify 7x + 3x − 2x. | 化简 7x + 3x − 2x。
- Simplify 5a − 2b + 3a + 4b. | 化简 5a − 2b + 3a + 4b。
- Simplify 4x² + 3x + 2x² − 5x. | 化简 4x² + 3x + 2x² − 5x。
- Simplify 6ab − 2a + 3ba + 7a. | 化简 6ab − 2a + 3ba + 7a。
- Expand and simplify 2(3x + 4) − 5x. | 展开并化简 2(3x + 4) − 5x。
- Expand and simplify 3(x + 2y) − 2(x − y). | 展开并化简 3(x + 2y) − 2(x − y)。
- Simplify 8p − 3q + 2p + q − 5. | 化简 8p − 3q + 2p + q − 5。
- Simplify 2x²y + 3xy² − x²y + xy². | 化简 2x²y + 3xy² − x²y + xy²。
12. Answers and Full Solutions | 答案与完整解答
Question 1: 7x + 3x − 2x = 8x, because 7 + 3 − 2 = 8 and the variable part x is unchanged.
第 1 题：7x + 3x − 2x = 8x，因为 7 + 3 − 2 = 8，变量部分 x 不变。
he a-terms give 5a + 3a = 8a and the b-terms give −2b + 4b = 2b, so the answer is 8a + 2b.
第 2 题：a 项得到 5a + 3a = 8a，b 项得到 −2b + 4b = 2b，所以答案是 8a + 2b。
-terms give 4x² + 2x² = 6x² and the x-terms give 3x − 5x = −2x, so the answer is 6x² − 2x.
第 3 题：x² 项得到 4x² + 2x² = 6x²，x 项得到 3x − 5x = −2x，所以答案是 6x² − 2x。
n 4: 3ba is the same as 3ab, so 6ab + 3ab = 9ab and −2a + 7a = 5a. The answer is 9ab + 5a.
第 4 题：3ba 与 3ab 相同，所以 6ab + 3ab = 9ab，−2a + 7a = 5a。答案是 9ab + 5a。
Question 5: 2(3x + 4) − 5x = 6x + 8 − 5x = x + 8.
第 5 题：2(3x + 4) − 5x = 6x + 8 − 5x = x + 8。
+ 2y) − 2(x − y) = 3x + 6y − 2x + 2y = x + 8y. Note the sign change on the second bracket.
第 6 题：3(x + 2y) − 2(x − y) = 3x + 6y − 2x + 2y = x + 8y。注意第二个括号内各项都变号。
Question 7: 8p + 2p = 10p and −3q + q = −2q, so the answer is 10p − 2q − 5.
第 7 题：8p + 2p = 10p，−3q + q = −2q，所以答案是 10p − 2q − 5。
Question 8: 2x²y − x²y = x²y and 3xy² + xy² = 4xy², so the answer is x²y + 4xy².
第 8 题：2x²y − x²y = x²y，3xy² + xy² = 4xy²，所以答案是 x²y + 4xy²。
If every answer here was correct, the technique is secure. If any went wrong, check whether unlike terms were combined, whether an index was added, or whether a sign was dropped.
如果以上全部正确，说明这一方法已经掌握。若有出错，请检查是否把非同类项合并了、是否把指数相加了，或者是否漏掉了某个符号。
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