📚 Collecting Like Terms & Simplification | 合并同类项与代数式化简
Collecting like terms is one of the most fundamental skills in algebra. It forms the foundation for solving equations, rearranging formulae, and simplifying expressions in every later topic you will encounter at IGCSE level.
合并同类项是代数中最基础的技能之一。它为你后续在IGCSE阶段遇到的解方程、变形公式以及化简表达式等所有课题奠定了基石。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable (letter) raised to exactly the same power. Only the numerical coefficient — the number in front of the variable — may differ. For example, 3x and 5x are like terms because both contain the variable x to the power 1.
同类项是指具有完全相同变量（字母）且该变量的指数完全相同的项。只有项前面的数字系数可以不同。例如，3x 和 5x 是同类项，因为它们都含有一次方的变量 x。
Consider these pairs: 4y and −2y are like terms; 7ab and 3ab are like terms. However, 2x and 2x² are not like terms, because the powers of x differ. Likewise, 5a and 5b are not like terms, because the variables themselves are different.
看这几组例子：4y 和 −2y 是同类项；7ab 和 3ab 是同类项。但是，2x 和 2x² 不是同类项，因为 x 的指数不同。同样，5a 和 5b 也不是同类项，因为变量本身不同。
Key rule: The variable part must match exactly — including all powers — for terms to be ‘like’.
关键规则：变量的部分必须完全一致——包括所有指数——这样的项才是“同类”的。
2. Identifying the Coefficient | 识别系数
The coefficient is the number that multiplies the variable. In the term 6x, the coefficient is 6. In the term −4y², the coefficient is −4. In the term x³, the coefficient is 1 (since 1 × x³ = x³). Likewise, in the term −z, the coefficient is −1.
系数是乘以变量的那个数字。在项 6x 中，系数是 6。在项 −4y² 中，系数是 −4。在项 x³ 中，系数是 1（因为 1 × x³ = x³）。同样地，在项 −z 中，系数是 −1。
Understanding coefficients is essential because when we collect like terms, we are literally adding or subtracting the coefficients while leaving the variable part untouched.
理解系数非常重要，因为当我们合并同类项时，本质上就是在对系数进行加减运算，而变量的部分保持不变。
Sometimes a term may appear without a visible coefficient, such as x or xy. In these cases, always remember the coefficient is 1 (or −1 if there is a minus sign). A common exam trap is forgetting to write the coefficient 1 when the result simplifies to a single variable.
有些项看起来没有系数，比如 x 或 xy。在这种情况下，永远记住系数是 1（如果有负号则是 −1）。一个常见的考试陷阱是：当结果化简为单个变量时，忘记写出系数 1。
3. Adding and Subtracting Like Terms | 同类项的加法与减法
To collect like terms, simply add or subtract their coefficients. The variable part does not change. For example, 3x + 5x = (3 + 5)x = 8x. Likewise, 9y − 4y = (9 − 4)y = 5y.
合并同类项时，只需对它们的系数进行加或减即可。变量部分不变。例如：3x + 5x = (3 + 5)x = 8x。同样，9y − 4y = (9 − 4)y = 5y。
3x + 5x = 8x
9y − 4y = 5y
When an expression contains multiple groups of like terms, collect each group separately. Consider the expression 2a + 3b − a + 4b. First group the a terms: 2a − a = a. Then group the b terms: 3b + 4b = 7b. The simplified result is a + 7b.
当一个表达式包含多组同类项时，分别合并每一组。考虑表达式 2a + 3b − a + 4b。先合并 a 项：2a − a = a。再合并 b 项：3b + 4b = 7b。化简结果为 a + 7b。
Be careful with negative signs. The sign belongs to the term that follows it. In the expression 4x − 6x + 2x, we compute 4 − 6 + 2 = 0, so the x terms cancel out entirely, and the expression simplifies to 0.
要特别注意负号。符号属于其后面的项。在表达式 4x − 6x + 2x 中，我们计算 4 − 6 + 2 = 0，所以 x 项完全抵消，表达式化简为 0。
4. More Than One Variable | 多个变量
Expressions often contain more than one variable. In such cases, collect terms with the same variable separately. For example, 5p + 3q − 2p + q = (5p − 2p) + (3q + q) = 3p + 4q.
表达式中经常含有多个变量。这种情况下，分别合并相同变量的项。例如：5p + 3q − 2p + q = (5p − 2p) + (3q + q) = 3p + 4q。
You should also be aware that different variables are not like terms, even if they look similar. For instance, x and y are different variables, so x + y cannot be simplified any further. It is already in its simplest form.
你还需要知道，不同的变量不是同类项，即使它们看起来很相似。例如 x 和 y 是不同的变量，所以 x + y 不能再进一步化简。它已经是最简形式。
When writing your final answer, it is conventional to arrange terms alphabetically, or to write terms with higher powers first. For example, write 4x + 3y + 2 rather than 3y + 2 + 4x, although both are technically correct.
写出最终答案时，习惯上按字母顺序排列各项，或者把指数较高的项写在前面。例如写成 4x + 3y + 2，而不写 3y + 2 + 4x，尽管两者在技术上都是正确的。
5. Powers and Indices | 幂与指数
Like terms must have the same power. The terms 2x² and 5x² are like terms because both have x raised to the power 2. However, 2x² and 5x are not like terms, because one has x² and the other has x.
同类项必须具有相同的幂。2x² 和 5x² 是同类项，因为两者都是 x 的二次方。但是 2x² 和 5x 不是同类项，因为一个是 x²，另一个是 x。
This means x² + x cannot be simplified into a single term. It must remain as x² + x. A very common student error is to write x² + x = x³, which is completely incorrect. You can only combine terms with identical variable-and-power combinations.
这意味着 x² + x 不能简化成一项。它必须保持为 x² + x。一个非常常见的学生错误是把 x² + x 写成 x³，这完全是不正确的。你只能合并具有完全相同的变量与指数组合的项。
Similarly, 3x³ + 4x³ = 7x³, but 3x³ + 4x² cannot be simplified further. The variables and powers must match perfectly before you are allowed to combine the coefficients.
同样，3x³ + 4x³ = 7x³，但 3x³ + 4x² 不能再进一步化简。只有当变量和指数完全匹配时，你才可以合并系数。
6. Products like xy | 形如 xy 的乘积项
Terms containing products of variables, such as xy, behave like single units. The term 2xy and 5xy are like terms because both contain exactly the same product xy. So 2xy + 5xy = 7xy.
含有变量乘积的项，如 xy，就像一个整体单元。2xy 和 5xy 是同类项，因为两者都含有完全相同的乘积 xy。所以 2xy + 5xy = 7xy。
However, xy and x²y are not like terms because the powers of x differ. Similarly, xy and xyz are not like terms because the product contains different variables. The variables, and their respective powers, must match exactly.
但是，xy 和 x²y 不是同类项，因为 x 的指数不同。同样地，xy 和 xyz 也不是同类项，因为乘积包含的变量不同。变量及其各自的指数必须完全匹配。
For expressions with multiple products, such as 3ab + 2bc − ab + 4bc, collect the ab terms: 3ab − ab = 2ab. Then collect the bc terms: 2bc + 4bc = 6bc. The result is 2ab + 6bc.
对于含有多个乘积项的表达式，例如 3ab + 2bc − ab + 4bc，合并 ab 项：3ab − ab = 2ab。再合并 bc 项：2bc + 4bc = 6bc。结果为 2ab + 6bc。
7. Constant Terms | 常数项
Constant terms are numbers without any variables. All constant terms are like terms with each other. For example, in the expression 4x + 7 − 2x + 3, the constants are 7 and 3. Collecting them gives 7 + 3 = 10.
常数项是不带任何变量的数字。所有常数项彼此都是同类项。例如，在表达式 4x + 7 − 2x + 3 中，常数项为 7 和 3。合并它们得到 7 + 3 = 10。
Combining everything: 4x + 7 − 2x + 3 = (4x − 2x) + (7 + 3) = 2x + 10. Note that constants are like terms with each other regardless of their values — 5 and −8 can be combined to give −3.
合并所有项：4x + 7 − 2x + 3 = (4x − 2x) + (7 + 3) = 2x + 10。注意，常数项之间无论数值如何都是同类项——5 与 −8 可以合并得到 −3。
A common mistake is combining a constant term with a variable term, such as writing 2x + 3 = 5x. This is wrong because x and 3 are not like terms. Always keep variable terms and constant terms separate in your final answer.
一个常见错误是将常数项与变量项合并，例如写成 2x + 3 = 5x。这是错误的，因为 x 和 3 不是同类项。在最终答案中，始终将变量项和常数项分开。
8. Expanding Brackets Before Collecting | 先去括号再合并
Sometimes you must expand brackets before you can collect like terms. For example, in the expression 3(x + 2) + 4x, first expand the bracket: 3(x + 2) = 3x + 6. Then collect like terms: 3x + 4x = 7x, so the result is 7x + 6.
有时你需要先展开括号才能合并同类项。例如，在表达式 3(x + 2) + 4x 中，先展开括号：3(x + 2) = 3x + 6。然后合并同类项：3x + 4x = 7x，所以结果为 7x + 6。
3(x + 2) + 4x = 3x + 6 + 4x = 7x + 6
When there is a negative sign outside the bracket, be extra careful. Consider 5x − 2(3x − 4). Expand the bracket carefully: −2(3x − 4) = −6x + 8. Now combine with the first term: 5x − 6x + 8 = −x + 8.
当括号外面有负号时，要格外小心。考虑 5x − 2(3x − 4)。仔细展开括号：−2(3x − 4) = −6x + 8。然后与第一项合并：5x − 6x + 8 = −x + 8。
The most common error here is forgetting to multiply the second term inside the bracket by the negative sign. Always double-check that every term inside the bracket has been multiplied by the factor outside, with the correct sign.
这里最常见的错误是忘记将括号内的第二项乘以负号。始终要仔细检查括号内的每一项都已与外面的因子相乘，并且符号正确。
9. Simplifying Fractions with Like Terms | 分式中的同类项化简
When expressions contain fractions, the same rules apply. For example, ½x + ¼x = (½ + ¼)x = ¾x. You may need to use a common denominator when adding the coefficients.
当表达式含有分数时，同样的规则适用。例如，½x + ¼x = (½ + ¼)x = ¾x。合并系数时，你可能需要通分。
Consider ⅔y − ½y. The common denominator of 3 and 2 is 6, so ⅔y − ½y = ⁴⁄₆y − ³⁄₆y = ⅙y. Always be precise with fractional arithmetic — the algebra itself is unchanged; it is the number work that requires care.
考虑 ⅔y − ½y。3 和 2 的最小公分母是 6，所以 ⅔y − ½y = ⁴⁄₆y − ³⁄₆y = ⅙y。分数运算要精确——代数本身没有变化，需要小心的是数字计算。
Also, pay attention to expressions like (x/2) + (x/3). Here both terms contain x to the power 1, so they are like terms. Adding gives (x/2) + (x/3) = (3x + 2x)/6 = 5x/6.
另外，注意像 (x/2) + (x/3) 这样的表达式。这里两项都含有一次方 x，所以它们是同类项。相加得到 (x/2) + (x/3) = (3x + 2x)/6 = 5x/6。
10. Collecting Like Terms in Geometry | 几何中的合并同类项
Collecting like terms appears frequently in geometry problems. For example, finding the perimeter of a rectangle with sides 2x and 3x requires adding: 2x + 3x + 2x + 3x = 10x. The perimeter simplifies neatly to a single term.
合并同类项在几何问题中经常出现。例如，求边长为 2x 和 3x 的矩形周长，需要相加：2x + 3x + 2x + 3x = 10x。周长被简洁地化简为一项。
Similarly, if a triangle has side lengths of x + 3, 2x − 1, and 4x + 2, then its perimeter is found by adding all three expressions: (x + 3) + (2x − 1) + (4x + 2) = 7x + 4.
类似地，如果三角形的三边长分别为 x + 3、2x − 1 和 4x + 2，那么求周长需要将三个表达式相加：(x + 3) + (2x − 1) + (4x + 2) = 7x + 4。
In area problems, you may also collect like terms. For instance, if a shape is split into rectangles with areas 3xy and 5xy, the total area is 3xy + 5xy = 8xy. Always look for opportunities to simplify the final answer in geometry questions.
在面积问题中，你也可以合并同类项。例如，如果一个图形被分成面积为 3xy 和 5xy 的矩形，总面积是 3xy + 5xy = 8xy。在几何题中，始终寻找化简最终答案的机会。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One of the most frequent errors is attempting to combine terms with different powers. Students often write 2x + 3x² = 5x³. This is completely wrong. You cannot add terms with different powers — they are not like terms.
最常见的错误之一是试图合并具有不同指数的项。学生常常写出 2x + 3x² = 5x³。这是完全错误的。你不能相加指数不同的项——它们不是同类项。
Another common mistake is combining numerical coefficients incorrectly, particularly with negative numbers. For example, 4x − 7x is often miscomputed as 3x instead of −3x. Always check the sign of your answer: if the larger coefficient is negative, the result will be negative.
另一个常见错误是数字系数计算不正确，特别是涉及负数时。例如，4x − 7x 经常被错误地算成 3x 而不是 −3x。始终检查答案的符号：如果较大的系数是负数，结果也是负数。
Forgetting to include a coefficient of 1 is also common. When the coefficient is not written, it is 1. For example, x + x = 2x, not x. And when you simplify 5x − 4x, the answer is x, not 0x — write the variable with coefficient 1 explicitly.
忘记写出系数 1 也很常见。当系数没有写出时，它就是 1。例如 x + x = 2x，而不是 x。当你化简 5x − 4x 时，答案是 x，而不是 0x——要明确写出系数为 1 的变量。
12. Exam Tips and Worked Examples | 考试技巧与例题解析
In IGCSE examinations, collecting like terms is usually the first step in solving problems. Always write your working clearly, grouping like terms before combining them. This helps avoid sign errors and makes it easier for the examiner to award partial credit.
在 IGCSE 考试中，合并同类项通常是解题的第一步。始终清晰地写出过程，先分组同类项再合并。这有助于避免符号错误，也让阅卷老师更容易给分。
Worked Example 1: Simplify 7p + 3q − 5p + 2q. Solution: Group p terms: 7p − 5p = 2p. Group q terms: 3q + 2q = 5q. Answer: 2p + 5q.
例题 1：化简 7p + 3q − 5p + 2q。解答：合并 p 项：7p − 5p = 2p。合并 q 项：3q + 2q = 5q。答案：2p + 5q。
Worked Example 2: Simplify 4(x + 2) − 3(x − 1). Solution: Expand brackets: 4x + 8 − 3x + 3. Collect like terms: (4x − 3x) + (8 + 3) = x + 11.
例题 2：化简 4(x + 2) − 3(x − 1)。解答：展开括号：4x + 8 − 3x + 3。合并同类项：(4x − 3x) + (8 + 3) = x + 11。
Always check whether your final answer can be simplified further. Look for any remaining like terms, and verify that all terms in your answer are indeed different from each other — that is, no two terms share the same variable and power combination.
始终检查你的最终答案是否可以进一步化简。看看是否还有剩余的同类项，并确认答案中的所有项确实彼此不同——即没有两项具有相同的变量和指数组合。
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