📚 Collecting Like Terms | 合并同类项
In algebra, simplifying expressions by collecting like terms is one of the most essential skills. It forms the foundation for solving equations, factorising, and working with formulas.
在代数中，合并同类项是化简表达式最核心的技能之一。它是解方程、因式分解和公式计算的重要基础。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable parts, including the same powers or exponents. The numerical coefficients can be different, but the variable part must match exactly.
同类项是指变量部分完全相同的项，包括相同的字母和指数。数字系数可以不同，但变量部分必须完全一致。
3x and 5x are like terms because both have the variable x.
3x 与 5x 是同类项，因为它们都含有变量 x。
4y² and −2y² are like terms because both have y squared.
4y² 与 −2y² 是同类项，因为它们都含有 y²。
7 and 9 are like terms because both are constants.
7 和 9 是同类项，因为它们都是常数项。
Terms such as 2x and 2x² are not like terms, because the power of x is different. Similarly, xy and x²y are not like terms.
像 2x 和 2x² 这样的项不是同类项，因为 x 的指数不同。同样，xy 和 x²y 也不是同类项。
2. Why Do We Collect Like Terms? | 为什么要合并同类项？
Collecting like terms simplifies an expression, making it shorter and easier to interpret. It also reduces the chance of making mistakes when substituting values or solving equations.
合并同类项可以化简表达式，使其更简洁、更易于理解，同时减少代入数值或解方程时出错的概率。
For example, the expression 7x + 3x − 2x looks complicated, but after collecting like terms it becomes 8x. This is much easier to use.
例如，表达式 7x + 3x − 2x 看起来复杂，但合并同类项后变为 8x，使用起来就方便多了。
Another benefit is that it reveals structure. An expression like 5a + 3b − 2a + b can be reorganised into 3a + 4b, which clearly shows the relationship between the variables.
另一个好处是它能揭示结构。像 5a + 3b − 2a + b 这样的表达式可以整理为 3a + 4b，从而清晰地看出变量之间的关系。
3. The Rules for Collecting Like Terms | 合并同类项的规则
To collect like terms, we add or subtract their coefficients while keeping the variable part unchanged. Constants are also combined together.
合并同类项时，我们对系数进行加或减，而变量部分保持不变。常数项也相互合并。
a + a = 2a, 3x + 4x = 7x, 5y² − 2y² = 3y²
If a term has no coefficient written, the coefficient is understood to be 1. For example, x means 1x.
如果某项没有写出系数，其系数默认为 1。例如，x 表示 1x。
Only add coefficients of the same variable and power.
只能对相同变量和相同指数的系数进行加减。
The variable part is never changed after addition or subtraction.
相加或相减后，变量部分保持不变。
Different variables are treated separately, like x and y.
不同的变量要分别处理，例如 x 和 y 互不影响。
4. Worked Example: Simple Linear Terms | 例题：简单一次项
Simplify the expression: 6x + 4x − 3x.
化简表达式：6x + 4x − 3x。
ey are 6x, 4x and −3x. Add the coefficients: 6 + 4 − 3 = 7. Keep the variable x unchanged.
首先，找出所有含变量 x 的项，它们是 6x、4x 和 −3x。计算系数：6 + 4 − 3 = 7。变量 x 保持不变。
6x + 4x − 3x = 7x
Always double-check that every term with the same variable has been included. Missing a term is a common mistake.
务必检查是否包含了所有同变量项。漏项是常见错误。
5. Worked Example: Multiple Variable Types | 例题：多种变量
Simplify: 5a + 2b − 3a + 4b.
化简：5a + 2b − 3a + 4b。
Group the a terms: 5a − 3a = 2a. Group the b terms: 2b + 4b = 6b. Now combine the results.
将含 a 的项分组：5a − 3a = 2a。将含 b 的项分组：2b + 4b = 6b。然后合并结果。
5a + 2b − 3a + 4b = 2a + 6b
The final answer is 2a + 6b. It is conventional to write the terms in alphabetical order, but not mandatory.
最终答案为 2a + 6b。通常按字母顺序排列项，但这不是强制要求。
6. Worked Example: Squares and Products | 例题：平方项与乘积项
Simplify: 3x² + 2x + 5x² − 4x.
化简：3x² + 2x + 5x² − 4x。
Identify like terms: x² terms are 3x² and 5x²; x terms are 2x and −4x. Combine each group separately.
找出同类项：x² 项是 3x² 和 5x²；x 项是 2x 和 −4x。分别合并每一组。
3x² + 5x² = 8x², 2x − 4x = −2x
Therefore, the simplified expression is 8x² − 2x. Notice that x² and x are not combined with each other.
因此，化简后的表达式为 8x² − 2x。注意 x² 与 x 不能互相合并。
7. Worked Example: Constants and Negative Terms | 例题：常数项与负项
Simplify: 4m − 3n + 2 − 2m + 5n − 7.
化简：4m − 3n + 2 − 2m + 5n − 7。
m terms: 4m − 2m = 2m. Group the n terms: −3n + 5n = 2n. Group the constants: 2 − 7 = −5.
合并含 m 的项：4m − 2m = 2m。合并含 n 的项：−3n + 5n = 2n。合并常数项：2 − 7 = −5。
4m − 3n + 2 − 2m + 5n − 7 = 2m + 2n − 5
When subtracting a term, be careful with the sign. The minus sign belongs to the term that follows it.
进行相减时，要特别注意符号。负号属于它后面的那个项。
8. Expanding Brackets Before Collecting | 先去括号再合并
rackets before collecting like terms. The distributive law states that a(b + c) = ab + ac.
在许多表达式中，我们需要先去括号再合并同类项。分配律告诉我们 a(b + c) = ab + ac。
Example: Simplify 3(x + 2) + 2(x − 1).
例如：化简 3(x + 2) + 2(x − 1)。
First expand each bracket:
先展开每个括号：
3(x + 2) = 3x + 6, 2(x − 1) = 2x − 2
Now collect like terms: 3x + 2x = 5x and 6 − 2 = 4.
现在合并同类项：3x + 2x = 5x，6 − 2 = 4。
3(x + 2) + 2(x − 1) = 5x + 4
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Students often make the following mistakes when collecting like terms. Recognising them can help you avoid them.
学生在合并同类项时经常犯以下错误。识别这些错误能帮助你避免它们。
| Mistake 错误 | Example 示例 | Correction 正确做法 |
| Combining different powers | x² + x = x³ ❌ | x² + x cannot be simplified ✅ |
| Forgetting the coefficient 1 | x + x = x (wrong) | x + x = 2x ✅ |
| Mixing different variables | 3x + 2y = 5xy ❌ | 3x + 2y stays as 3x + 2y ✅ |
| Losing negative signs | −2a + 5a = 3 (wrong) | −2a + 5a = 3a ✅ |
To avoid mistakes, always write down the terms with their signs, then rearrange them into groups before adding.
为避免错误，务必写下带符号的项，然后将其重新分组后再进行加法。
10. Practice Questions | 练习题
Try these questions on your own. The answers are at the end of this section.
请独立完成以下练习。答案在本节末尾。
Simplify: 8x + 3x − 2x
化简：8x + 3x − 2x
Simplify: 6p + 2q − p + 3q
化简：6p + 2q − p + 3q
Simplify: 4a² + 3a − a² + 2a
化简：4a² + 3a − a² + 2a
Simplify: 2(3x + 1) + 4(x − 2)
化简：2(3x + 1) + 4(x − 2)
Answers:
答案：
1. 8x + 3x − 2x = 9x
2. 6p + 2q − p + 3q = 5p + 5q
3. 4a² + 3a − a² + 2a = 3a² + 5a
4. 2(3x + 1) + 4(x − 2) = 6x + 2 + 4x − 8 = 10x − 6
11. Real-World Applications | 实际应用
Collecting like terms appears in many real-life situations, such as calculating the total cost of items with the same price, or combining distances travelled in different directions.
合并同类项在许多实际情况中都会出现，例如计算同价格商品的总费用，或合并不同方向行驶的距离。
op, then 2 pens and 4 notebooks in another, the total cost is 3p + 2n + 2p + 4n = 5p + 6n.
如果一支钢笔价格为 p 磅，一个笔记本价格为 n 磅，你在第一店买 3 支钢笔和 2 个笔记本，在另一店买 2 支钢笔和 4 个笔记本，总费用为 3p + 2n + 2p + 4n = 5p + 6n。
+ 5N = 8N. The ability to simplify expressions makes such calculations faster and clearer.
在物理中，同方向力的合成也使用相同思路：3N + 5N = 8N。化简能力使这类计算更快、更清晰。
12. Summary | 总结
To collect like terms successfully, remember the three key steps: first, expand any brackets; second, identify terms that have the same variable part, including the same powers; third, add or subtract their coefficients.
要成功合并同类项，请记住三个关键步骤：第一，展开所有括号；第二，找出变量部分相同的项，包括相同的指数；第三，对其系数进行加减。
Always keep different variables and different powers separate. With practice, you will be able to simplify expressions quickly and correctly.
始终将不同变量和不同幂分开处理。通过练习，你将能够快速而准确地化简表达式。
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