📚 Combining Like Terms | 合并同类项
In IGCSE Mathematics, simplifying an algebraic expression usually begins with combining like terms. This technique helps you rewrite expressions such as 3x + 2x + 5 into a shorter form, 5x + 5, before solving equations or substituting values.
在 IGCSE 数学中，化简代数式通常从合并同类项开始。这个方法可以把 3x + 2x + 5 这样的表达式改写成更简洁的 5x + 5，为后续解方程或代入求值做好准备。
1. What Are Like Terms? | 什么是同类项
Like terms are terms that contain exactly the same letters and exactly the same powers. The numbers in front, called coefficients, can be different. For example, 5x and 2x are like terms because they both contain the variable x.
同类项是指所含字母完全相同、且各字母的指数也完全相同的项。字母前面的数字称为系数，系数可以不同。例如，5x 和 2x 是同类项，因为它们的字母部分都是 x。
5x and 2x are like terms. | 5x 和 2x 是同类项。
2. Why Combining Matters | 为什么合并同类项很重要
Combining like terms is important because it reduces the number of terms in an expression. A shorter expression is easier to evaluate and helps you see the structure of a problem more clearly.
合并同类项很重要，因为它能减少表达式中的项数。表达式越简洁，求值越方便，也越容易看清题目的结构。
3. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and compare the variable part only.
判断同类项时，先忽略系数，只比较字母部分。
- 4y and −9y are like terms — both have y. | 4y 和 −9y 是同类项，因为都含 y。
- 3ab and 7ab are like terms — both have ab. | 3ab 和 7ab 是同类项，因为都含 ab。
- 6x² and 11x² are like terms — both have x². | 6x² 和 11x² 是同类项，因为都含 x²。
| Pair | 项对 | Like terms? | 同类项？ | Reason | 原因 |
|---|---|---|
| 3x and 5x | Yes | Same variable x |
| 3x and 5x² | No | Powers of x are different |
| 2ab and 3a | No | Variable parts are different |
The table shows that two terms can be combined only when their variable parts match exactly. x and x² are not like terms, and ab and a are not like terms either.
上表说明：只有字母部分完全相同时，两个项才可以合并。x 与 x² 不是同类项，ab 与 a 也不是同类项。
4. The Golden Rule: Same Letters, Same Powers | 黄金法则：字母相同且指数相同
The golden rule is that only terms with the same variable part can be combined. Coefficients do not affect this rule.
黄金法则是：只有字母部分完全相同的项才能合并。系数不影响判断。
3x² + 5x² = 8x², but 3x² + 5x cannot be simplified further.
3x² 加 5x² 等于 8x²，但 3x² 加 5x 不能继续合并。
5. Adding and Subtracting Coefficients | 系数相加减
fficients, then write the common variable part once. For example, 4x + 3x = (4 + 3)x = 7x.
合并同类项时，把系数相加或相减，然后照写公共的字母部分。例如：4x + 3x = (4 + 3)x = 7x。
9y − 4y = 5y; 2ab + 5ab = 7ab; −6p + 2p = −4p
9y − 4y = 5y；2ab + 5ab = 7ab；−6p + 2p = −4p。
6. Combining Constants | 常数项的合并
Constant terms have no variable part. All constants are like terms with each other, so combine them using ordinary arithmetic.
常数项没有字母部分。所有常数项彼此都是同类项，因此可以直接用普通算术合并。
6 + 3 − 2 = 7
2x + 3 + 5x − 2 = 7x + 1
例如：2x + 3 + 5x − 2 化简为 7x + 1。先合并字母项，再合并常数项。
7. Worked Example 1 | 例题一
Simplify 3x + 2y + 5x − y.
化简 3x + 2y + 5x − y。
- Group like terms: (3x + 5x) + (2y − y). | 分组：(3x + 5x) + (2y − y)。
- Combine coefficients: 3 + 5 = 8, 2 − 1 = 1. | 合并系数：3 + 5 = 8，2 − 1 = 1。
- Write final answer: 8x + y. | 写出最终答案：8x + y。
3x + 2y + 5x − y = 8x + y
8. Worked Example 2 | 例题二
Simplify 4x² + 3x − 2x² + 7 + 6x.
化简 4x² + 3x − 2x² + 7 + 6x。
First group the x² terms, the x terms, and the constants separately.
先把 x² 项、x 项和常数项分别分组。
4x² − 2x² = 2x², 3x + 6x = 9x, 7 stays 7
4x² − 2x² = 2x²，3x + 6x = 9x，7 保持不变。
The simplified expression is 2x² + 9x + 7.
化简结果为 2x² + 9x + 7。
9. Common Mistakes | 常见错误
- Do not combine unlike terms: 3x + 2y is not 5xy. | 不要合并不同类项：3x + 2y 不等于 5xy。
- Do not treat x and x² as the same: x + x² is not 2x². | 不要把 x 和 x² 当作同类项：x + x² 不等于 2x²。
- Do not forget constant terms: 4 + x − 1 simplifies to x + 3, not x + 4. | 不要漏掉常数项：4 + x − 1 应化简为 x + 3，而不是 x + 4。
10. Practice Questions | 练习题
Simplify each expression.
化简下列各式。
- 7a + 2a
- 5m + 3n − 2m + n
- 6p² + p − 4p² + 3p
- 8 + 3q − 5 + 2q
- 2xy + 7yx + 3x
Answers: 9a; 3m + 4n; 2p² + 4p; 3 + 5q; 9xy + 3x.
答案：9a；3m + 4n；2p² + 4p；3 + 5q；9xy + 3x。
11. Summary | 小结
- Like terms have exactly the same variable part and powers. | 同类项必须满足字母及其指数完全相同。
- Combine like terms by adding or subtracting coefficients. | 合并同类项就是系数的加减。
- Constants combine only with constants. | 常数项只能与常数项合并。
- Unlike terms must be left as separate terms. | 不同类项必须分开书写。
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