📚 Collecting Like Terms | 合并同类项
In algebra, expressions often contain several terms that can be simplified. One of the most important skills at IGCSE is collecting like terms. This process makes an expression shorter, neater and easier to use in later solving or substitution.
在代数中，表达式通常包含几个可以化简的项。IGCSE 阶段最重要的技能之一就是合并同类项。这个过程使表达式更短、更整洁，也更便于后续求解或代入。
1. What Are Like Terms? | 什么是同类项？
A term is a single number, a letter, or a product of numbers and letters. Terms are separated by + or – signs.
项是一个单独的数、一个字母，或数与字母的乘积。项之间由 + 或 – 号分隔。
Like terms are terms that have exactly the same variable part, including the same powers. For example, 3x and -5x are like terms because both contain x to the power 1.
同类项是指变量部分完全相同，包括相同的幂。例如 3x 和 -5x 是同类项，因为二者都含有 x 的一次方。
Constants such as 4, -7 and 0.5 are also like terms with each other because they have no variable part.
常数如 4、-7 和 0.5 也互为同类项，因为它们没有变量部分。
2. Recognising Like Terms | 识别同类项
To decide whether two terms are like terms, ignore the coefficient and look only at the letters and their exponents.
判断两项是否为同类项时，忽略系数，只看字母及其指数。
7xy and -2xy are like terms, but 7xy and 7x²y are not, because the exponent of x is different.
7xy 和 -2xy 是同类项，但 7xy 和 7x²y 不是，因为 x 的指数不同。
The terms 5ab and 3ba are like terms because multiplication is commutative: ab = ba. However, in IGCSE answers we usually write variables in alphabetical order.
5ab 和 3ba 是同类项，因为乘法满足交换律：ab = ba。但在 IGCSE 答案中，我们通常按字母顺序书写变量。
3. Coefficients and Constants | 系数与常数
The coefficient is the number multiplying the variable part. In -4y, the coefficient is -4.
系数是乘以变量部分的数。在 -4y 中，系数是 -4。
When no number is written, the coefficient is 1 or -1. For example, x means 1x, and -p means -1p.
当没有写出数字时，系数是 1 或 -1。例如 x 表示 1x，-p 表示 -1p。
A constant term is a number with no variable, such as 9 or -3. Like constant terms can always be collected.
常数项是不含变量的数，例如 9 或 -3。同类常数项总是可以合并。
4. The Golden Rule | 核心规则
To collect like terms, add or subtract their coefficients and keep the variable part unchanged.
合并同类项时，将系数相加或相减，变量部分保持不变。
ax + bx = (a + b)x
This works because of the distributive law. For example, 2x + 3x = (2 + 3)x = 5x.
这之所以成立，是因为分配律。例如 2x + 3x = (2 + 3)x = 5x。
Unlike terms cannot be combined into a single term. 2x + 3y stays as 2x + 3y.
不同类项不能合并为一项。2x + 3y 仍写作 2x + 3y。
5. Single Variable Examples | 单变量例题
Simplify 5m + 2m – 3m.
化简 5m + 2m – 3m。
Add the coefficients: 5 + 2 – 3 = 4, so the answer is 4m.
将系数相加：5 + 2 – 3 = 4，因此答案是 4m。
5m + 2m – 3m = 4m
Simplify 8x – 3 + 2x + 7.
化简 8x – 3 + 2x + 7。
Collect the x terms: 8x + 2x = 10x. Collect the constants: -3 + 7 = 4. The simplified expression is 10x + 4.
合并 x 项：8x + 2x = 10x。合并常数项：-3 + 7 = 4。化简后的表达式为 10x + 4。
6. Multiple Variable Examples | 多变量例题
Simplify 3a + 4b + 2a – b.
化简 3a + 4b + 2a – b。
Collect a terms: 3a + 2a = 5a. Collect b terms: 4b – b = 3b. So the result is 5a + 3b.
合并 a 项：3a + 2a = 5a。合并 b 项：4b – b = 3b。因此结果为 5a + 3b。
Simplify 6xy – 2x + 4xy + 5x.
化简 6xy – 2x + 4xy + 5x。
Collect xy terms: 6xy + 4xy = 10xy. Collect x terms: -2x + 5x = 3x. The answer is 10xy + 3x.
合并 xy 项：6xy + 4xy = 10xy。合并 x 项：-2x + 5x = 3x。答案是 10xy + 3x。
Notice that x and xy are not like terms, so they must be kept separate.
注意 x 和 xy 不是同类项，因此必须分开书写。
7. Powers and Like Terms | 幂与同类项
Terms with the same letter but different powers are not like terms.
字母相同但幂不同的项不是同类项。
x, x² and x³ are three different types of terms. 4x² + 3x² = 7x², but 4x² + 3x cannot be simplified further.
x、x² 和 x³ 是三种不同类型的项。4x² + 3x² = 7x²，但 4x² + 3x 不能进一步化简。
When terms have more than one variable, every exponent must match. 2a²b and 5ab² are not like terms.
当项含有多个变量时，每一个指数都必须相同。2a²b 和 5ab² 不是同类项。
4x² + 3x² = 7x²
Simplify 2p³ + 5p² – p³ + 3p².
化简 2p³ + 5p² – p³ + 3p²。
Collect p³ terms: 2p³ – p³ = p³. Collect p² terms: 5p² + 3p² = 8p². The result is p³ + 8p².
合并 p³ 项：2p³ – p³ = p³。合并 p² 项：5p² + 3p² = 8p²。结果为 p³ + 8p²。
8. Brackets Before Collecting | 先去括号再合并
If an expression contains brackets, simplify inside the brackets first or expand them before collecting like terms.
如果表达式含有括号，先化简括号内部，或在合并同类项前展开括号。
Simplify 3(x + 2) + 2x.
化简 3(x + 2) + 2x。
Expand: 3x + 6 + 2x. Then collect x terms: 3x + 2x = 5x. Final answer: 5x + 6.
展开：3x + 6 + 2x。然后合并 x 项：3x + 2x = 5x。最终答案：5x + 6。
Simplify 2(a + b) + 3(a – b).
化简 2(a + b) + 3(a – b)。
Expand: 2a + 2b + 3a – 3b. Collect a terms: 5a. Collect b terms: 2b – 3b = -b. Final answer: 5a – b.
展开：2a + 2b + 3a – 3b。合并 a 项：5a。合并 b 项：2b – 3b = -b。最终答案：5a – b。
9. Common Mistakes | 常见错误
Mistake 1: combining unlike terms. For example, 2x + 3y ≠ 5xy.
错误一：合并不同类项。例如 2x + 3y ≠ 5xy。
Mistake 2: changing the power when adding coefficients. 2x² + 3x² = 5x², not 5x⁴.
错误二：相加系数时改变幂。2x² + 3x² = 5x²，而不是 5x⁴。
Mistake 3: losing a negative sign. 4x – 6x = -2x, not 2x.
错误三：丢失负号。4x – 6x = -2x，而不是 2x。
Mistake 4: writing a coefficient of 1 unnecessarily. 5x – 4x = x, not 1x.
错误四：多写系数 1。5x – 4x = x，而不是 1x。
10. Exam Practice and Tips | 考试练习与技巧
Try simplifying 7n – 3 + 4n + 9 – 2n.
尝试化简 7n – 3 + 4n + 9 – 2n。
Collect n terms: 7n + 4n – 2n = 9n. Collect constants: -3 + 9 = 6. Answer: 9n + 6.
合并 n 项：7n + 4n – 2n = 9n。合并常数项：-3 + 9 = 6。答案：9n + 6。
In IGCSE exam papers, questions often ask you to simplify an expression as part of a larger problem. Always show coefficient arithmetic clearly.
在 IGCSE 考试中，题目常要求将表达式化简作为更大问题的一部分。始终清晰展示系数的运算。
Underline or circle each group of like terms with the same sign before combining them. This reduces sign errors.
在合并之前，将每组带有相同符号的同类项下划线或圈出。这可以减少符号错误。
If you see a subtraction sign before a term, include it when you collect. For example, 10y – 3y – 2y = 5y.
如果项前有减号，合并时要包含减号。例如 10y – 3y – 2y = 5y。
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