📚 Combining Like Terms | 合并同类项
In IGCSE Mathematics, simplifying algebraic expressions is one of the most important basic skills. Combining like terms turns a long expression into a shorter, clearer form, and it is essential for solving equations, rearranging formulas, and working with graphs.
在 IGCSE 数学中，化简代数式是最重要的基本技能之一。合并同类项可以把冗长的表达式化为更短、更清晰的形式，在解方程、变形公式和处理图像时都必不可少。
1. What Are Like Terms? | 什么是同类项
Like terms are terms that have exactly the same variable part, including the same variable letters and the same powers. For example, 3x and −7x are like terms, but 3x and 3x² are not.
同类项是指变量部分完全相同的项，包括相同的字母和相同的指数。例如 3x 和 −7x 是同类项，但 3x 和 3x² 不是。
Constants such as 4, −9 and 12 are also like terms because they contain no variable part at all. They can always be combined with each other.
常数项如 4、−9 和 12 也是同类项，因为它们不含变量部分。常数项之间总是可以合并。
Terms such as 2xy and −5xy are like terms, while 2xy and 2x are not because the variable parts xy and x are different.
2xy 与 −5xy 是同类项，而 2xy 与 2x 不是同类项，因为变量部分 xy 和 x 不同。
2. The Core Rule: Add or Subtract Coefficients | 核心法则：系数相加减
To combine like terms, keep the variable part unchanged and add or subtract the numerical coefficients. The variable part acts as a label of the term.
合并同类项时，保持变量部分不变，只把数字系数相加或相减。变量部分相当于项的标签。
+ 5x = (3 + 5)x = 8x, and 7y − 2y = 5y. The variable part y stays the same in the answer.
例如 3x + 5x = (3 + 5)x = 8x，7y − 2y = 5y。答案中变量部分 y 保持不变。
addition: ax + bx = (a + b)x. Once you understand this, combining terms becomes automatic.
这一法则成立的原因是乘法对加法有分配律：ax + bx = (a + b)x。一旦理解这一点，合并同类项就会变得自然。
3. Single-Variable Practice | 一元同类项练习
Start with expressions involving one variable. Combine all x terms and all constant terms separately, then write the simplified expression.
先从含一个变量的表达式开始。分别合并所有含 x 的项和所有常数项，再写出化简后的表达式。
Example: 4x + 2 + 9x − 5 = (4x + 9x) + (2 − 5) = 13x − 3.
例：4x + 2 + 9x − 5 = (4x + 9x) + (2 − 5) = 13x − 3。
You can rearrange the terms first so that like terms sit next to each other. This is safe because addition is commutative.
你可以先重新排列各项，使同类项相邻。这样做是安全的，因为加法满足交换律。
It is often helpful to underline or circle like terms with the same marking before adding them up. This reduces the chance of missing a term.
在相加之前，用相同标记给同类项画线或画圈通常会很有帮助。这样可以减少漏项的可能性。
4. Handling Negative Coefficients | 处理负系数
Negative signs belong to the coefficient of the term that follows them. Treat subtraction as adding a negative coefficient.
负号属于其后项的系数。把减法视为加上一个负数系数。
Example: 6p − 10p = −4p, because 6 + (−10) = −4. The variable part p does not change.
例：6p − 10p = −4p，因为 6 + (−10) = −4。变量部分 p 不变。
eful with an expression like 3x − 7 + 2x + 4. Combine it as (3x + 2x) + (−7 + 4) = 5x − 3.
当心类似 3x − 7 + 2x + 4 的式子。应合并为 (3x + 2x) + (−7 + 4) = 5x − 3。
x + 4 as 5x − 11, but that is wrong because the sign before 4 is positive, so −7 + 4 = −3.
一个常见错误是把 3x − 7 + 2x + 4 写成 5x − 11，但这是错误的，因为 4 前面是正号，所以 −7 + 4 = −3。
5. Combining Terms with Multiple Variables | 合并含多个变量的项
For terms with more than one variable, every variable and every power must match exactly. If one part is different, the terms are not like terms.
对于含多个变量的项，每个变量及其幂次都必须完全一致。只要有一部分不同，它们就不是同类项。
5ab = 8ab, but 3ab + 5a cannot be combined because one term has b and the other does not.
例：3ab + 5ab = 8ab，但 3ab + 5a 不能合并，因为一个项含 b，另一个不含。
Consider 2xy + 3yx. Since multiplication is commutative, xy and yx represent the same product, so these combine to 5xy.
考虑 2xy + 3yx。由于乘法满足交换律，xy 与 yx 表示同一个乘积，所以可以合并为 5xy。
You should also check the powers of each variable: 3x²y and 5xy² are not like terms because the powers attached to x and y are different.
你还要检查每个变量的幂次：3x²y 与 5xy² 不是同类项，因为 x 和 y 上的幂次不同。
6. Using Powers Correctly | 正确使用幂次
Terms with different powers are not like terms. x and x² represent different quantities and must not be combined.
幂次不同的项不是同类项。x 和 x² 表示不同的量，不能合并。
8x², but 5x² + 3x cannot be simplified further because one term is x² and the other is x.
例：5x² + 3x² = 8x²，但 5x² + 3x 不能进一步化简，因为一个是 x²，另一个是 x。
applies to higher powers: 2x³ + 7x³ = 9x³, while 2x³ + 7x² remains separate in the answer.
更高次幂同理：2x³ + 7x³ = 9x³，而 2x³ + 7x² 在答案中仍应分开。
Remember that the power never changes when you add or subtract like terms. 3x² + 4x² gives 7x², not 7x⁴.
请记住，合并同类项时幂次不会改变。3x² + 4x² 得到 7x²，而不是 7x⁴。
7. Simplifying Before Substitution | 代入前先化简
In exam questions, you often need to substitute values into an expression. Simplify first to reduce the number of calculations and the chance of error.
在考试题中，常需把数值代入表达式。先化简可减少计算量并降低出错概率。
m + 8 when m = 2, combine first to get 5m + 11, then substitute: 5(2) + 11 = 10 + 11 = 21.
例：当 m = 2 时求 7m + 3 − 2m + 8 的值，先合并同类项得 5m + 11，再代入：5(2) + 11 = 10 + 11 = 21。
This is particularly useful when the expression is long or when the same simplified expression is used several times in a question.
当表达式较长，或同一化简后的表达式在题目中需多次使用时，这一点尤其有用。
Without simplifying, you would have to calculate 7(2) + 3 − 2(2) + 8, which has more steps and more places to make an arithmetic mistake.
如果不化简，你就要计算 7(2) + 3 − 2(2) + 8，步骤更多，也更容易出现算术错误。
8. Expanding and Then Combining | 先去括号再合并
If an expression contains brackets, expand first using the distributive law, then collect like terms. Never try to combine terms through brackets.
如果表达式含括号，先用分配律去括号，再合并同类项。不要试图隔着括号合并项。
Example: 2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3.
例：2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3。
plier, distribute the negative sign as well: −4(2x − 5) = −8x + 20, because −4 × −5 = +20.
若乘数为负，负号也要一起分配：−4(2x − 5) = −8x + 20，因为 −4 × −5 = +20。
After expanding, rearrange the terms so that like terms are together, then combine their coefficients as shown earlier.
展开后，重新排列各项使同类项相邻，然后如前所述合并它们的系数。
9. Common Exam Mistakes | 常见考试错误
Mistake 1: combining unlike terms such as 3x and 5x². These must stay separate in the answer.
错误一：合并不同类项，如 3x 与 5x²。答案中它们必须分开。
Mistake 2: losing a negative sign when moving terms. Always keep the sign with its coefficient.
错误二：移动项时丢失负号。务必将符号与其系数一起保留。
Mistake 3: changing powers during addition. The power never changes when you add or subtract like terms.
错误三：合并同类项时改变幂次。合并同类项时幂次不会改变。
Mistake 4: forgetting constant terms. Constants combine with constants, and they must appear in the simplified answer.
错误四：遗漏常数项。常数项与常数项合并，且必须出现在化简结果中。
Mistake 5: treating terms with variables in a different order as different when they are actually like terms, such as xy and yx.
错误五：把变量顺序不同但实际为同类项的项当作不同项，例如 xy 和 yx。
10. Worked IGCSE-Style Questions | IGCSE 风格例题
2x + 7y. Combine x terms: 6x − 2x = 4x. Combine y terms: 3y + 7y = 10y. Answer: 4x + 10y.
问题 1：化简 6x + 3y − 2x + 7y。合并含 x 的项：6x − 2x = 4x。合并含 y 的项：3y + 7y = 10y。答案：4x + 10y。
Question 2: Simplify 5a + 3b − 2a + 4b − 3. Combine a terms, b terms, and constants: 3a + 7b − 3.
问题 2：化简 5a + 3b − 2a + 4b − 3。合并 a 项、b 项和常数项：3a + 7b − 3。
Question 3: Expand and simplify 3(2x + 1) + 2(x − 4). Expand: 6x + 3 + 2x − 8 = 8x − 5.
问题 3：展开并化简 3(2x + 1) + 2(x − 4)。展开得：6x + 3 + 2x − 8 = 8x − 5。
Question 4: Simplify 4m² + 3m − m² + 6m. Combine like powers: 3m² + 9m.
问题 4：化简 4m² + 3m − m² + 6m。合并同次幂项：3m² + 9m。
4x. Combine all x terms: 2x − x + 4x = 5x. Combine y terms: 3y + 5y = 8y. Answer: 5x + 8y.
问题 5：化简 2x + 3y − x + 5y + 4x。合并所有含 x 的项：2x − x + 4x = 5x。合并含 y 的项：3y + 5y = 8y。答案：5x + 8y。
11. Quick Check and Key Points | 速查要点
Like terms have the same variables and the same powers. Combine coefficients only.
同类项具有相同的变量和相同的幂次。只合并系数。
Unlike terms must be left separate in your final answer.
不同类项在最终答案中必须保持分开。
Expand brackets before collecting like terms, and be careful with negative multipliers.
先去括号，再合并同类项，并注意负乘数的符号。
Keep powers unchanged when adding or subtracting like terms.
合并同类项时保持幂次不变。
Always double-check negative signs and constant terms before writing the final simplified expression.
在写出最终化简式之前，务必再次检查负号和常数项。
12. Summary | 小结
Combining like terms is a foundational algebra skill. It makes expressions shorter, substitutions easier, and later work in equations and graphs more manageable.
合并同类项是代数的基础技能。它使表达式更简洁、代入更容易，也让后续的方程与图像学习更轻松。
Always look for identical variable parts, combine their coefficients, and leave everything else unchanged.
始终寻找完全相同的变量部分，合并它们的系数，其余部分保持不变。
With practice, the process becomes automatic and helps you gain marks quickly in IGCSE exams.
通过练习，这一过程会变得自动化，帮助你在 IGCSE 考试中快速得分。
Use the worked questions above to test yourself, and write each step clearly so you can spot any sign or power errors.
用上面的例题自测，并清晰地写出每一步，以便发现任何符号或幂次错误。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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