📚 Simplifying Algebraic Expressions: Collecting Like Terms | 化简代数表达式：合并同类项
In IGCSE Mathematics, algebra is a core strand, and one of the very first skills you must master is simplifying algebraic expressions by collecting like terms. This single technique appears again and again — in solving equations, manipulating formulas, factorising, and even in coordinate geometry. If you can combine like terms accurately and quickly, you build a strong foundation for every later algebra topic.
在国际中等教育通用证书（IGCSE）数学中，代数是核心板块之一，而你最先需要掌握的一项技能就是通过“合并同类项”来化简代数表达式。这一技巧会反复出现——在解方程、变形公式、因式分解，甚至在坐标几何中都会用到。如果你能准确而快速地合并同类项，就为后续所有代数专题打下了坚实的基础。
1. What Is an Algebraic Expression? | 什么是代数表达式？
An algebraic expression is a combination of numbers, letters (variables), and operation signs such as +, −, × and ÷. For example, 3x + 5 and 2a − 4b + 7 are algebraic expressions. An expression has no equals sign, so it cannot be “solved” — it can only be simplified, factorised, or evaluated when the variables are given values.
代数表达式是数字、字母（变量）与运算符号（如 +、−、×、÷）组合而成的式子。例如，3x + 5 和 2a − 4b + 7 都是代数表达式。表达式没有等号，因此不能被“解出”——只能被化简、因式分解，或在变量取特定值时求值。
Each part of an expression separated by a + or − sign is called a term. In the expression 4x² − 3x + 2, there are three terms: 4x², −3x, and 2. Every term has two important parts: a numerical coefficient and a variable part. In 4x², the coefficient is 4 and the variable part is x².
表达式中用 + 或 − 号隔开的每一部分称为“项”。在表达式 4x² − 3x + 2 中，有三个项：4x²、−3x 和 2。每一个项都有两个重要部分：数字系数和变量部分。在 4x² 中，系数是 4，变量部分是 x²。
2. Like Terms and Unlike Terms | 同类项与不同类项
The word “like” means “the same type”. Like terms are terms that have exactly the same variable part — that is, the same letter or letters raised to exactly the same powers. For example, 3x and 7x are like terms because both have the variable x to the first power.
“同类”指的是“同一种类型”。同类项是指变量部分完全相同的项——也就是说，相同的字母或字母组合且各自的幂指数完全相同。例如，3x 和 7x 是同类项，因为它们的变量都是 x，且都是一次幂。
If two terms have different variables, or the same variable raised to different powers, they are unlike terms and cannot be combined by addition or subtraction. Here is a quick comparison:
如果两个项具有不同的变量，或者相同变量的幂指数不同，那么它们就是不同类项，不能通过加法和减法合并。下面做一个快速比较：
| Like terms | Unlike terms |
| 5x and −2x (same variable x) | 5x and 5y (different variables) |
| 3a² and 8a² (same a²) | 3a and 3a² (different powers) |
| 4xy and xy (same product xy) | 4xy and 4x²y (different powers) |
| 9 and −5 (constant terms) | 9 and 9x (constant vs variable) |
Note that 3xy and 2yx are like terms, because multiplication is commutative: x × y is exactly the same product as y × x. So 3xy + 2yx = 5xy.
请注意：3xy 和 2yx 是同类项，因为乘法满足交换律：x × y 与 y × x 是完全相同的乘积。因此 3xy + 2yx = 5xy。
3. The Core Rule: Combine Only the Coefficients | 核心规则：只合并系数
When you combine like terms, you add or subtract only the numerical coefficients, and you keep the variable part exactly unchanged. This is similar to counting objects: 3 apples plus 5 apples gives 8 apples, so 3a + 5a = 8a.
合并同类项时，你只需要对数字系数进行加或减，而变量部分保持完全不变。这就像数物体：3 个苹果加 5 个苹果得到 8 个苹果，所以 3a + 5a = 8a。
Study this worked example. Simplify: 6x + 3x − 2x.
请看这个完整的例子。化简：6x + 3x − 2x。
6x + 3x − 2x = (6 + 3 − 2)x = 7x
All three terms are like terms, so we combine their coefficients 6, 3 and −2. The variable x stays attached to the resulting coefficient.
这三个项都是同类项，所以我们合并它们的系数 6、3 和 −2。变量 x 始终与最终得到的系数组合在一起。
When an expression contains several groups of like terms, collect each group separately. For example:
当一个表达式包含好几组同类项时，要分别收集每一组。例如：
Simplify: 5m + 3n − 2m + 4n.
化简：5m + 3n − 2m + 4n。
Group the m terms: 5m − 2m = 3m. Group the n terms: 3n + 4n = 7n. Therefore:
先把 m 项分组：5m − 2m = 3m。再把 n 项分组：3n + 4n = 7n。因此：
5m + 3n − 2m + 4n = 3m + 7n
It is usually a good idea to rearrange the terms so that like terms are side by side before combining. Be careful: when you rearrange, you must carry the sign in front of each term with it.
通常一个好方法是先把同类项重新排列到一起，再进行合并。请注意：重新排列时，每一项前面的符号必须跟着这一项一起移动。
4. Working with Negative Coefficients | 处理负系数
Negative coefficients are extremely common in IGCSE questions, and sign errors are the most frequent cause of lost marks. The key is to treat the sign as part of the coefficient: −5x has coefficient −5, not 5.
负系数在 IGCSE 试题中非常常见，而符号错误是丢分的最主要原因。关键在于把符号视为系数的一部分：−5x 的系数是 −5，而不是 5。
Simplify: 7y − 9y.
化简：7y − 9y。
7y − 9y = (7 − 9)y = −2y
Simplify: 4a − 3b + 2a + b.
化简：4a − 3b + 2a + b。
Combine the a terms: 4a + 2a = 6a. Combine the b terms: −3b + 1b = −2b. So the answer is 6a − 2b, which can also be written as 2(3a − b) after factorising.
合并 a 项：4a + 2a = 6a。合并 b 项：−3b + 1b = −2b。因此答案为 6a − 2b，因式分解后也可以写成 2(3a − b)。
One common trap is subtracting a bracket. Remember that a minus sign in front of a bracket reverses every sign inside it: 5 − (2x − 3) = 5 − 2x + 3 = 8 − 2x.
一个常见的陷阱是减去一个括号。请记住：括号前的减号会使括号内每一项的符号都发生改变：5 − (2x − 3) = 5 − 2x + 3 = 8 − 2x。
5. Combining Terms with Powers and Letters | 含指数与字母的项的合并
Like terms must have identical powers on every variable. The term x² and the term x are not like terms, because their powers differ: x² has index 2, while x has index 1. You cannot add x² + x into a single term.
同类项要求每个变量上的指数完全相同。项 x² 与项 x 不是同类项，因为它们的指数不同：x² 的指数是 2，而 x 的指数是 1。你不能把 x² + x 合并成单独的一项。
However, x² and x² are like terms. Simplify: 2x² + 3x² = 5x². Notice that the power does not change; you do not add the exponents when you are adding terms.
但是，x² 与 x² 是同类项。化简：2x² + 3x² = 5x²。注意：当你做加法时，指数不会改变；你不需要把指数相加。
Do not confuse addition with multiplication. This is a crucial distinction:
不要把加法与乘法混淆。这是一个至关重要的区别：
-
x + x = 2x (adding like terms gives a coefficient of 2)
x + x = 2x（同类项相加得到系数 2）
-
x × x = x² (multiplying powers adds exponents)
x × x = x²（幂相乘时指数相加）
-
x² + x² = 2x² (adding like terms)
x² + x² = 2x²（同类项相加）
-
x² × x² = x⁴ (multiplying: 2 + 2 = 4)
x² × x² = x⁴（幂相乘：2 + 2 = 4）
For expressions with multiple variables and powers, collect each group separately. Simplify: 5a² + 3a − 2a² + a.
对于含多个变量和指数的表达式，要分别收集每一组。化简：5a² + 3a − 2a² + a。
Collect the a² terms: 5a² − 2a² = 3a². Collect the a terms: 3a + a = 4a. The answer is 3a² + 4a.
收集 a² 项：5a² − 2a² = 3a²。收集 a 项：3a + a = 4a。答案为 3a² + 4a。
6. Simplifying Brackets First, Then Collecting | 先去括号，再合并
Many exam questions require you to expand a bracket before combining like terms. Use the distributive law: a(b + c) = ab + ac. Multiply the term outside the bracket by every term inside the bracket.
许多考试题要求你先展开括号，再合并同类项。请使用分配律：a(b + c) = ab + ac。把括号外的项乘以括号内的每一项。
Worked example: Simplify 3(x + 2) + 4x.
完整例子：化简 3(x + 2) + 4x。
First expand: 3(x + 2) = 3x + 6. Then the expression becomes 3x + 6 + 4x. Now collect like terms: 3x + 4x = 7x, so the answer is 7x + 6.
首先展开：3(x + 2) = 3x + 6。于是原式变为 3x + 6 + 4x。现在合并同类项：3x + 4x = 7x，因此答案为 7x + 6。
Worked example: Simplify 2(3x + 1) + 3(x − 2).
完整例子：化简 2(3x + 1) + 3(x − 2)。
First expand each bracket separately: 2(3x + 1) = 6x + 2, and 3(x − 2) = 3x − 6. Combine: (6x + 2) + (3x − 6) = 6x + 3x + 2 − 6 = 9x − 4.
先分别展开每一个括号：2(3x + 1) = 6x + 2，以及 3(x − 2) = 3x − 6。合并：(6x + 2) + (3x − 6) = 6x + 3x + 2 − 6 = 9x − 4。
If there is a minus sign outside a bracket, expand with care: −2(x − 5) = −2x + 10. Every sign inside changes because −2 multiplies both x and −5.
如果括号外面有负号，展开时要格外小心：−2(x − 5) = −2x + 10。因为 −2 要同时乘以 x 和 −5，所以括号内每一项的符号都会改变。
7. Real-World Applications | 实际应用
Collecting like terms is not just a classroom exercise; it appears naturally in geometry and word problems. For instance, the perimeter of a rectangle with length p and width q is p + q + p + q. Collecting like terms gives the familiar formula:
合并同类项不仅仅是课堂练习；它会自然出现在几何和文字应用题中。例如，长为 p、宽为 q 的矩形周长为 p + q + p + q。合并同类项后就得到我们熟悉的公式：
Perimeter = p + q + p + q = 2p + 2q
Suppose a shop sells x pens in each pack, and each pack also contains 2 erasers. If you buy 3 packs of pens and 2 extra single pens, the total number of pens is 3x + 2, and the total number of erasers is 3 × 2 = 6. If the question asks for the combined total of all items, you write 3x + 2 + 6 = 3x + 8. Here, 2 and 6 are constant terms, so they can be combined.
假设每包笔中有 x 支笔，每包还含 2 块橡皮。如果你买 3 包笔并另加 2 支散装笔，那么笔的总数为 3x + 2，橡皮的总数为 3 × 2 = 6。如果题目问所有物品的总数，你可以写 3x + 2 + 6 = 3x + 8。这里，2 和 6 是常数项，因此可以合并。
In physics and coordinate geometry, formulas often contain like terms. The sum of two side lengths in a triangle, the total cost of items with the same unit price, and the average of repeated measurements can all be simplified using the same method.
在物理和坐标几何中，公式常常包含同类项。三角形的两条边长之和、单价相同的商品总价，以及重复测量的平均值，都可以用同样的方法来化简。
8. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Experienced examiners see the same errors year after year. Here are the most common traps, with corrections, so you can avoid losing easy marks.
经验丰富的考官年复一年地看到同样的错误。以下是最常见的陷阱及其修正方法，帮助你避免无谓失分。
| Mistake | Example | Correction |
| Combining unlike terms | 2x + 3y = 5xy ✗ | 2x + 3y stays as 2x + 3y ✓ |
| Adding exponents when adding terms | x² + x² = x⁴ ✗ | x² + x² = 2x² ✓ |
| Confusing x + x with x × x | x + x = x² ✗ | x + x = 2x ✓ |
| Dropping signs while rearranging | 4a − 2a + 3b written as 4a + 2a − 3b ✗ | The sign belongs to the term, so −2a stays −2a ✓ |
| Forgetting to change all signs in a bracket | 5 − (2x − 3) = 5 − 2x − 3 ✗ | 5 − (2x − 3) = 5 − 2x + 3 = 8 − 2x ✓ |
A good habit is to underline or circle like terms with the same shape before combining them. For example, underline all x-terms, circle all x²-terms, and leave constant terms unmarked. This keeps your working organised and reduces careless errors.
一个好习惯是：在合并之前，用圆圈或下划线标出同类项。例如，把所有 x 项向下划线，把所有 x² 项画圆圈，常数项留空不标。这样能让你的解题过程更加清晰，并减少粗心错误。
9. Exam-Style Practice Questions | 考试型练习题
Try these questions yourself before reading the solutions. They reflect the style of IGCSE paper questions on collecting like terms.
请先自己尝试解答以下问题，再查看解答。这些问题反映了 IGCSE 试卷中关于合并同类项的典型风格。
Question 1: Simplify 6x + 4y − 2x + 5y.
题目 1：化简 6x + 4y − 2x + 5y。
Solution: Group x terms: 6x − 2x = 4x. Group y terms: 4y + 5y = 9y. Answer: 4x + 9y.
解答：x 项分组：6x − 2x = 4x。y 项分组：4y + 5y = 9y。答案：4x + 9y。
Question 2: Simplify 5a² + 3a − 2a² + a.
题目 2：化简 5a² + 3a − 2a² + a。
Solution: a² terms: 5a² − 2a² = 3a². a terms: 3
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