📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying an expression often begins with collecting like terms. This process reduces an expression to its simplest form and is essential for solving equations, expanding brackets, and rearranging formulae. Mastering it gives you a reliable foundation for almost every algebra topic in the course.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。这一过程能把表达式化为最简形式，对解方程、展开括号和变换公式都至关重要。掌握这一技能可以为课程中几乎所有代数主题打下可靠基础。
1. What Is a Term? | 什么是项
A term is a single number, a single letter, or a product of numbers and letters. In the expression 4x + 7y − 3, the terms are 4x, 7y, and −3. Each term is separated by a plus or minus sign, and the sign belongs to the term that follows it.
项是单个数字、单个字母或数与字母的乘积。在表达式 4x + 7y − 3 中，各项分别为 4x、7y 和 −3。每一项由加号或减号分隔，符号属于它后面的那一项。
4x + 7y − 3
2. Identifying Like Terms | 识别同类项
Like terms contain exactly the same variable letters raised to exactly the same powers. The coefficients can be different. For example, 5x and −2x are like terms, but 5x and 5x² are not because the powers of x are different.
同类项含有完全相同的字母，且字母的指数完全相同；系数可以不同。例如 5x 和 −2x 是同类项，但 5x 和 5x² 不是同类项，因为 x 的指数不同。
| Terms | Like or Unlike? |
| 3x and −7x | Like terms |
| 4y² and 9y² | Like terms |
| 6xy and 2x | Unlike terms |
| 5a and 5a² | Unlike terms |
3. Why We Collect Like Terms | 为什么要合并同类项
Collecting like terms makes an expression shorter and easier to use. For example, 3a + 5a + 2b can be written as 8a + 2b. This helps you see the structure of the expression and reduces errors when substituting values or solving equations.
合并同类项能让表达式更简短、更易使用。例如 3a + 5a + 2b 可写成 8a + 2b。这样可以更容易看出表达式的结构，也能在代入数值或解方程时减少错误。
3a + 5a + 2b = 8a + 2b
4. The Basic Rule: Add the Coefficients | 基本规则：系数相加
To combine like terms, add or subtract their coefficients and keep the variable part unchanged. Think of the variable part as a label: 3 apples + 5 apples = 8 apples. In the same way, 3x + 5x = 8x.
合并同类项时，只把系数相加或相减，字母部分保持不变。可以把字母部分看作标签：3 个苹果加 5 个苹果等于 8 个苹果。同样地，3x + 5x = 8x。
3x + 5x = (3 + 5)x = 8x
5. Collecting Positive Like Terms | 合并正同类项
When all coefficients are positive, simply add them. Remember that x on its own means 1x, so 2x + 4x + x is the same as adding 2, 4, and 1 together.
当所有系数为正时，只需直接把系数相加。请记住，单独的 x 表示 1x，因此 2x + 4x + x 就相当于把 2、4 和 1 相加。
2x + 4x + x = (2 + 4 + 1)x = 7x
6. Collecting Negative Like Terms | 合并负同类项
Treat the minus sign as part of the coefficient. For example, −3y − 2y means (−3) + (−2), which gives −5. The result is therefore −5y. Never ignore the negative sign when grouping.
把减号看作系数的一部分。例如 −3y − 2y 表示 (−3) + (−2)，结果为 −5。因此答案为 −5y。分组时绝不能忽略负号。
−3y − 2y = (−3 − 2)y = −5y
7. Mixed Signs and Integer Coefficients | 混合符号与整数系数
With mixed signs, add positive and negative coefficients carefully. For example, 7p − 10p + 4p means 7 − 10 + 4, which equals 1. The simplified result is 1p, usually written as p.
符号混合时，要仔细地把正、负系数相加。例如 7p − 10p + 4p 表示 7 − 10 + 4，结果为 1。化简结果是 1p，通常写作 p。
7p − 10p + 4p = (7 − 10 + 4)p = p
8. Terms with More Than One Variable | 含多个变量的项
Terms with multiple variables must match in every variable and every power. For example, 2ab and 5ab are like terms, but 2ab and 5a are not. When simplifying, keep the variable product unchanged and combine only the coefficients.
含多个变量的项，必须在每个变量及其指数上都相同才是同类项。例如 2ab 和 5ab 是同类项，但 2ab 和 5a 不是。化简时保持变量乘积不变，只合并系数。
2ab + 5ab − ab = (2 + 5 − 1)ab = 6ab
9. Expressions with Powers | 含有幂的表达式
Powers must match exactly for terms to be like terms. x² and x are not like terms because the exponents differ. However, 3x² and −x² are like terms and can be combined.
同类项的指数必须完全一致。x² 与 x 不是同类项，因为指数不同；而 3x² 与 −x² 是同类项，可以合并。
3x² − x² + 2x² = (3 − 1 + 2)x² = 4x²
Remember that x + x² cannot be simplified further. The powers stay the same when adding like terms; the exponents are never added together.
请记住，x + x² 不能进一步化简。合并同类项时指数保持不变；指数绝不会相加。
10. Simplifying Longer Expressions | 化简较长表达式
For longer expressions, first group like terms together, keeping the sign with each term. Then combine each group separately. This organised method is especially useful when an expression contains several different variables.
对于较长表达式，先按同类项分组，注意每项保留自己的符号，然后分别合并每一组。当表达式含有多个不同变量时，这种有条理的方法尤其有用。
5x + 3y − 2x + y = (5x − 2x) + (3y + y) = 3x + 4y
11. Common Mistakes | 常见错误
Do not combine unlike terms, such as 3x + 4y. Do not change the power when adding like terms: 2x² + 3x² = 5x², not 5x⁴. Also, when reordering terms, keep the sign attached to each term.
不要合并不同类项，例如 3x + 4y 不能合并。合并同类项时不要改变指数：2x² + 3x² = 5x²，而不是 5x⁴。此外，重新排序时保持各项的符号不变。
- 3x + 4y → cannot simplify
- 2x² + 3x² = 5x² → not 5x⁴
- x + x² → cannot simplify
- 5ab − 2a → cannot simplify
12. Exam-Style Practice | 考试型练习
Simplify each expression, then check your answers. Practice like this builds the speed and accuracy needed in IGCSE non-calculator and calculator papers.
化简下列各式，然后核对答案。这类练习能提升 IGCSE 非计算器试卷和计算器试卷所需的速度与准确度。
- 6a + 9b − 2a + 4b → 4a + 13b
- 4x² + 3x − x² + 2x → 3x² + 5x
- 5mn − 2n + 3mn − n → 8mn − 3n
- 2p + 7q − 5p + q → −3p + 8q
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