📚 Collecting Like Terms | 合并同类项
In algebra, collecting like terms is the process of simplifying an expression by adding or subtracting terms that have exactly the same variable part. This skill is essential for solving equations, expanding brackets, and rearranging formulas in IGCSE Mathematics.
在代数中，合并同类项是通过相加或相减具有完全相同变量部分的项来化简表达式的过程。这项技能对于 IGCSE 数学中解方程、展开括号和变形公式至关重要。
1. What Are Like Terms? | 什么是同类项
A term is a single number, a variable, or a product of numbers and variables. Like terms have exactly the same variable part, including the same powers. For example, 3x and 5x are like terms, while 3x and 3x² are not.
项是单个数字、变量或数字与变量的乘积。同类项具有完全相同的变量部分，包括相同的幂次。例如，3x 和 5x 是同类项，而 3x 和 3x² 不是。
2. Why Collect Like Terms? | 为什么要合并同类项
Collecting like terms makes an expression shorter and easier to use. It helps you see the structure of an expression before substituting values or solving an equation. In an exam, simplified answers are usually required for full marks.
合并同类项使表达式更短、更易使用。它能帮助你在代入数值或解方程之前看清表达式的结构。在考试中，通常需要化简后的答案才能获得满分。
3. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and look only at the variable part. For example, 7ab, −2ab and ab are like terms because they all contain ab. Constants such as 4, −9 and 12 are also like terms with each other.
要识别同类项，请忽略系数，只看变量部分。例如，7ab、−2ab 和 ab 是同类项，因为它们都含有 ab。常数项如 4、−9 和 12 也彼此是同类项。
| Expression | Like terms? | Reason |
|---|---|---|
| 3x and 5x | Yes | Same variable x |
| 3x and 3x² | No | Different powers of x |
| 7ab and −2ab | Yes | Same variable part ab |
| 4 and −9 | Yes | Both are constants |
| 5x and 3y | No | Different variables |
4. The Basic Rule: Add or Subtract Coefficients | 基本法则：系数相加或相减
e variable part unchanged. This works because of the distributive law: ax + bx = (a + b)x.
要合并同类项，将系数相加或相减，变量部分保持不变。这是分配律的运用：ax + bx = (a + b)x。
ax + bx = (a + b)x
5. Single-Variable Examples | 单变量示例
Start with simple expressions containing one variable. For 4x + 7x, the coefficients 4 and 7 add to 11, so the result is 11x. For 10y − 3y, subtract 3 from 10 to get 7y.
从含有一个变量的简单表达式开始。对于 4x + 7x，系数 4 和 7 相加得 11，因此结果为 11x。对于 10y − 3y，用 10 减去 3 得到 7y。
4x + 7x = 11x
10y − 3y = 7y
6. Multiple Variables and Powers | 多变量与幂
When an expression has several variables or powers, only combine terms whose variable parts match exactly. For instance, 3xy and 5xy combine to 8xy, but 3xy and 3x²y do not.
当表达式含有多个变量或幂时，只能合并变量部分完全一致的项。例如，3xy 和 5xy 合并为 8xy，但 3xy 和 3x²y 不能合并。
3xy + 5xy = 8xy
2x² + 5x² = 7x²
The expression 5x + 3y cannot be simplified further because the variables are different. Likewise, 2x and 3x² are not like terms because the powers are different.
表达式 5x + 3y 不能进一步化简，因为变量不同。同样，2x 和 3x² 不是同类项，因为幂次不同。
7. Dealing with Negative Coefficients | 处理负系数
−3: 8 + (−3) = 5, so the result is 5x. For −4a + 6a, the result is 2a because −4 + 6 = 2.
负系数遵循相同的规则。对于 8x − 3x，将第二个系数视为 −3：8 + (−3) = 5，因此结果为 5x。对于 −4a + 6a，结果为 2a，因为 −4 + 6 = 2。
8x − 3x = 5x
−4a + 6a = 2a
5p − 9p = −4p
8. Simplifying Longer Expressions | 化简较长表达式
When simplifying a longer expression, first identify the different groups of like terms. You can underline or circle them mentally. Then combine each group separately, and write the final expression in a tidy order, usually with higher powers first.
化简较长表达式时，首先识别不同的同类项组。可以在心里为它们画线或圈出。然后分别合并每一组，最后按整洁的顺序写出结果，通常高次项在前。
7x + 3y − 2x + 5y = (7x − 2x) + (3y + 5y) = 5x + 8y
For mixed powers, group x² terms with x² terms, x terms with x terms, and constants with constants.
对于混合幂次，将 x² 项与 x² 项合并，x 项与 x 项合并，常数项与常数项合并。
9. Common Mistakes | 常见错误
A common error is adding coefficients while also changing the variable part. For example, 2x + 3x is not 5x²; it is 5x. Another mistake is combining unlike terms such as 2x and 3y. Always check that both the letters and their powers are identical before combining.
常见错误是在相加系数的同时改变了变量部分。例如，2x + 3x 不是 5x²，而是 5x。另一个错误是合并不同类项，例如 2x 和 3y。合并前一定要检查字母及其幂是否完全相同。
Also, do not forget the sign in front of each term. In 6x − 2x, the second coefficient is −2, so the answer is 4x, not 8x.
另外，不要忘记每项前面的符号。在 6x − 2x 中，第二个系数是 −2，因此答案是 4x，而不是 8x。
10. Exam-Style Practice and Answers | 考试风格练习与答案
Try these exam-style questions before checking the answers below. Simplify each expression by collecting all like terms.
在查看下列答案之前，请先尝试这些考试风格的问题。通过合并所有同类项来化简每个表达式。
- a) 7x + 2x − 3x
- b) 4y² + 3y − 2y² + 6y
- c) 5ab + 2a − 3ab + 4a
- d) 6m + 3n − 4m + 7n − 2n
Answers: a) 6x, b) 2y² + 9y, c) 2ab + 6a, d) 2m + 8n.
答案：a) 6x，b) 2y² + 9y，c) 2ab + 6a，d) 2m + 8n。
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