Collecting Like Terms & Simplifying Expressions | 合并同类项与化简代数式

📚 Collecting Like Terms & Simplifying Expressions | 合并同类项与化简代数式

Algebra is the language of patterns, and one of the first skills you must master is collecting like terms — the process of simplifying an expression by grouping terms that share the same variable part. This skill forms the foundation for solving equations, factorising, and working with functions in IGCSE Mathematics.

代数是描述规律的语言，而你需要掌握的第一项核心技能就是合并同类项——将含有相同字母部分的项归类合并，从而化简代数式。这项技能是解方程、因式分解以及处理函数的基础，在 IGCSE 数学中至关重要。

1. What Is an Algebraic Expression? | 什么是代数式？

An algebraic expression is a combination of numbers, letters (variables), and operation signs such as +, −, × and ÷. For example, 3x + 5y − 2 is an expression. Each part separated by a + or − sign is called a term.

代数式是由数字、字母（变量）以及运算符号（如 +、−、×、÷）组合而成的。例如，3x + 5y − 2 就是一个代数式。每个用 + 或 − 分隔开的部分称为一个项。

A variable is a letter that represents an unknown number, such as x, y, or n.

变量是代表未知数的字母，例如 x、y 或 n。
A coefficient is the number multiplying the variable, e.g. in 4x, the coefficient is 4.

系数是与变量相乘的数字，例如在 4x 中，系数是 4。
A constant is a term without a variable, e.g. 7 or −3.

常数项是不含变量的项，例如 7 或 −3。

Consider the expression 6x + 3y − 2x + 5. It has four terms: 6x, 3y, −2x and 5. The coefficient of x in the first term is 6, and the constant term is 5.

观察表达式 6x + 3y − 2x + 5，它包含四个项：6x、3y、−2x 和 5。第一项中 x 的系数是 6，常数项是 5。

2. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part — same letter(s) raised to the same power. For example, 3x and −5x are like terms because both contain x to the power 1. Similarly, 4xy and 2xy are like terms.

同类项是指变量部分完全相同的项——即字母相同且字母的指数也相同。例如，3x 和 −5x 是同类项，因为它们都含有 x 的一次幂。同样地，4xy 和 2xy 也是同类项。

Like terms: same letters, same powers ⇔ 同类项：字母相同、指数相同

2x and 7x → like terms (both have x¹)

2x 和 7x → 同类项（都含有 x¹）
5a² and −3a² → like terms

5a² 和 −3a² → 同类项
4x and 4x² → not like terms (different powers)

4x 和 4x² → 不是同类项（指数不同）
3ab and 3a → not like terms (different variable parts)

3ab 和 3a → 不是同类项（变量部分不同）

Remember: 2x means x + x, so adding another x gives 3x. This is exactly why we can combine like terms — they represent the same quantity.

请记住：2x 表示 x + x，再添加一个 x 就得到 3x。这正是我们可以合并同类项的原因——它们表示相同的数量。

3. The Golden Rules of Collecting Like Terms | 合并同类项的基本法则

When collecting like terms, you only add or subtract the coefficients. The variable part stays exactly the same. For instance, 3x + 5x = (3 + 5)x = 8x. Note that 3x + 5x ≠ 8x² — the power of x does not change.

合并同类项时，你只需要对系数进行加减运算，变量部分保持不变。例如，3x + 5x = (3 + 5)x = 8x。请注意，3x + 5x ≠ 8x²——x 的指数不会改变。

ax + bx = (a + b)x where x is the same variable | ax + bx = (a + b)x，其中 x 为同一变量

Expression 表达式	Simplified 化简后
2x + 3x	5x
7y − 2y	5y
4a + a	5a (because 1a is just a \| 因为 1a 就是 a)
−3x − 5x	−8x

Be careful with the invisible coefficient 1: the term x actually means 1x, and −y means −1y.

要注意隐含系数 1：项 x 实际上表示 1x，而 −y 表示 −1y。

4. Simplifying Expressions with Multiple Variable Types | 化简含多种变量的表达式

When an expression contains different variables, you must group each type separately. For example, simplify 5x + 3y − 2x + 4y. First collect the x-terms: 5x − 2x = 3x. Then collect the y-terms: 3y + 4y = 7y. The simplified expression is 3x + 7y.

当表达式中含有不同变量时，你必须分别对每一类进行合并。例如，化简 5x + 3y − 2x + 4y。先合并 x 项：5x − 2x = 3x；再合并 y 项：3y + 4y = 7y。化简后的表达式为 3x + 7y。

Notice that 3x and 7y cannot be added together because they are not like terms. The plus sign between them must be kept in the final answer.

请注意，3x 和 7y 不能相加，因为它们不是同类项。最终答案中必须保留它们之间的加号。

5x + 3y − 2x + 4y = 3x + 7y | 先合并 x，再合并 y

Here is a useful strategy: underline or circle each family of like terms with a different symbol before combining. This reduces careless mistakes in longer expressions.

这里有一个实用的策略：在合并之前，用不同的符号为每一组同类项画下划线或圈出。这可以减少长表达式中的粗心错误。

5. Working with Negative Terms | 处理含负号的项

When subtracting terms, treat the sign before a term as belonging to that term. For example, in 4x − 6x, the second term is −6x, so 4x − 6x = −2x. Always keep the sign with its term when rearranging.

在减法运算中，要把项前面的符号视为该项的一部分。例如，在 4x − 6x 中，第二项是 −6x，所以 4x − 6x = −2x。在重新排列项时，务必让符号跟随它所属的项。

e.g. 9a − 4a = 5a; −2b − 3b = −5b; c − 5c = −4c

Consider the more complex example: 6x − 3y − 4x + 2y. Grouping x-terms: 6x − 4x = 2x. Grouping y-terms: −3y + 2y = −1y = −y. The answer is 2x − y.

看一个更复杂的例子：6x − 3y − 4x + 2y。合并 x 项：6x − 4x = 2x；合并 y 项：−3y + 2y = −1y = −y。答案为 2x − y。

A common mistake is to write 2x + y by forgetting the negative sign. Always check the sign of each collected coefficient carefully.

常见的错误是忘记负号而写成 2x + y。合并后一定要仔细检查每一项系数的符号。

6. Combining Terms with Powers (x², x³) | 合并含幂的项（x²、x³）

Terms with different powers of the same letter are not like terms. For instance, x² and x cannot be combined. However, 3x² and 5x² are like terms and simplify to 8x².

同一字母的不同幂次项不是同类项。例如，x² 和 x 不能合并。但 3x² 和 5x² 是同类项，可化简为 8x²。

x + x = 2x, but x × x = x²; x² + x² = 2x², not 2x⁴

Wrong: 4x + 2x = 6x² ✗

错误：4x + 2x = 6x² ✗
Correct: 4x + 2x = 6x ✓

正确：4x + 2x = 6x ✓
Correct: 4x² + 2x² = 6x² ✓

正确：4x² + 2x² = 6x² ✓

Simplify the full example: 7x² + 3x − 2x² + x. Combine x²-terms: 7x² − 2x² = 5x². Combine x-terms: 3x + x = 4x. Final answer: 5x² + 4x.

化简完整示例：7x² + 3x − 2x² + x。合并 x² 项：7x² − 2x² = 5x²；合并 x 项：3x + x = 4x。最终答案：5x² + 4x。

7. Products Like xy and ab | 形如 xy、ab 的乘积项

Terms such as xy form their own family. Only terms with exactly the same variable letters and powers can be combined. For example, 3xy and −xy are like terms because xy is xy. But 3xy and 3x are not like terms.

xy 这样的项自成一组。只有变量字母及指数完全相同的项才可以合并。例如，3xy 和 −xy 是同类项，因为 xy 就是 xy。但 3xy 和 3x 不是同类项。

5xy + 2xy = 7xy; 4ab − ab = 3ab; xy² + 3xy² = 4xy²

Note the ordering of letters: 3ab and 5ba are actually like terms because multiplication is commutative (ab = ba). In IGCSE exams, it is conventional to write variables in alphabetical order.

注意字母的顺序：3ab 和 5ba 实际上是同类项，因为乘法满足交换律（ab = ba）。在 IGCSE 考试中，通常按字母顺序书写变量。

Simplify: 2xy + 3x − xy + 4x. The xy-terms: 2xy − xy = 1xy = xy. The x-terms: 3x + 4x = 7x. Answer: xy + 7x.

化简：2xy + 3x − xy + 4x。xy 项：2xy − xy = 1xy = xy；x 项：3x + 4x = 7x。答案为：xy + 7x。

8. Simplifying Expressions with Brackets | 化简含括号的表达式

Sometimes you must first expand brackets before collecting like terms. Use the distributive law: a(b + c) = ab + ac. For example, 2(x + 3) = 2x + 6. Then combine any like terms that emerge.

有时你需要先去括号再合并同类项。运用分配律：a(b + c) = ab + ac。例如，2(x + 3) = 2x + 6，然后再合并产生的同类项。

3(x + 2) + 4x = 3x + 6 + 4x = 7x + 6

Watch out for negative signs outside brackets: −(x − 5) means −1 × (x − 5) = −x + 5. Every sign inside the brackets flips.

要特别注意括号外的负号：−(x − 5) 表示 −1 × (x − 5) = −x + 5，括号内每一项的符号都要变号。

Example 1: 5(x + 2) − 2x = 5x + 10 − 2x = 3x + 10

例 1：5(x + 2) − 2x = 5x + 10 − 2x = 3x + 10
Example 2: 4 − (2x + 1) = 4 − 2x − 1 = 3 − 2x

例 2：4 − (2x + 1) = 4 − 2x − 1 = 3 − 2x

9. Real-World Applications | 实际应用

Collecting like terms appears in geometry when finding perimeters. Suppose a rectangle has sides of length 2x + 1 and x + 3. Its perimeter P = 2(length + width) = 2[(2x + 1) + (x + 3)]. Simplifying inside: (2x + x) + (1 + 3) = 3x + 4. Then P = 2(3x + 4) = 6x + 8.

合并同类项在几何求周长时经常出现。假设一个矩形的边长为 2x + 1 和 x + 3。其周长 P = 2(长 + 宽) = 2[(2x + 1) + (x + 3)]。化简括号内：(2x + x) + (1 + 3) = 3x + 4，所以 P = 2(3x + 4) = 6x + 8。

Another application: combining two algebraic expressions. The total of 3a + 2b and a − b is (3a + a) + (2b − b) = 4a + b.

另一个应用是合并两个代数式。3a + 2b 与 a − b 的和为 (3a + a) + (2b − b) = 4a + b。

10. Common Mistakes to Avoid | 常见错误避坑指南

Mistake 错误	Correction 纠正
3x + 4y = 7xy ✗	3x + 4y stays as 3x + 4y (not like terms) 3x 与 4y 不是同类项，保持原样
x + x = x² ✗	x + x = 2x (addition, not multiplication) 加法不是乘法
3x + 2x = 5x² ✗	3x + 2x = 5x (power stays 1) 指数保持不变
6 − 2x + 3 = 6 − 5x ✗	Combine constants only: 6 + 3 − 2x = 9 − 2x 只合并常数项

Always ask yourself: “Do these terms have exactly the same variable part?” If the answer is yes, add or subtract the coefficients. If no, leave them alone.

始终问自己：“这些项的变量部分完全相同吗？” 如果答案是肯定的，就对系数进行加减；如果答案是否定的，就保持原样。

11. Practice Questions | 练习巩固

Q1. Simplify: 7x + 2x

练习 1：化简：7x + 2x

Answer: 9x | 答案：9x

Q2. Simplify: 5a + 3b − 2a + b

练习 2：化简：5a + 3b − 2a + b

Answer: 3a + 4b | 答案：3a + 4b

Q3. Simplify: 4x² + 3x − x² − 7x

练习 3：化简：4x² + 3x − x² − 7x

Answer: 3x² − 4x | 答案：3x² − 4x

Q4. Simplify: 3(2x + 1) + 5x

练习 4：化简：3(2x + 1) + 5x

Answer: 6x + 3 + 5x = 11x + 3 | 答案：6x + 3 + 5x = 11x + 3

Q5. Simplify: 8ab − 3a + 2ab − a

练习 5：化简：8ab − 3a + 2ab − a

Answer: 10ab − 4a | 答案：10ab − 4a

12. Summary | 总结

Collecting like terms is a systematic skill: identify families of identical variable parts, add or subtract their coefficients, and keep the variable part unchanged. Always respect the signs, remember that x means 1x, and never try to combine terms with different powers or different variables.

合并同类项是一项系统性的技能：识别变量部分相同的项族，对其系数进行加减，并保持变量部分不变。始终注意符号，牢记 x 就是 1x，并且绝不尝试合并指数不同或变量不同的项。

With consistent practice, this process becomes second nature — and it will serve you well across all areas of IGCSE Mathematics, from solving equations to calculus in later years.

通过持续练习，这一过程将成为你的本能反应——它将在 IGCSE 数学的各个领域为你服务，从解方程到更高年级的微积分。

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Collecting Like Terms & Simplifying Expressions | 合并同类项与化简代数式