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Collecting Like Terms | 合并同类项

2026-08-21 18:18 · 本文部分内容由AI辅助创作

📚 Collecting Like Terms | 合并同类项

In IGCSE and A-Level mathematics, algebraic expressions often look more complicated than they really are. The single most useful skill for taming them is collecting like terms - adding or subtracting the terms that share exactly the same variable part. This technique underpins solving equations, rearranging formulae, factorising, working with sequences and sketching graphs.

在 IGCSE 与 A-Level 数学中,代数表达式往往看起来比实际更复杂。驯服它们最有用的一个技能就是合并同类项——把字母部分完全相同的项相加或相减。这一技巧是解方程、变形公式、因式分解、数列计算和绘制图像的基础。

1. What is a Term? | 什么是项?

An algebraic expression is built from terms. Terms are separated by plus or minus signs, and each term can be a number, a variable, or a product of numbers and variables. In the expression 3x + 5y − 7 there are three terms: 3x, 5y and −7.

代数表达式由项组成。项与项之间用加号或减号分隔,每一项可以是数字、字母,或数字与字母的乘积。在表达式 3x + 5y − 7 中共有三项:3x、5y 和 −7。

Every term has two parts. The coefficient is the number in front, and the variable part is the letter or letters together with their powers. So in 6ab² the coefficient is 6 and the variable part is ab². A term with no letter, such as −7, is called a constant term.

每一项都有两部分。系数是前面的数字,字母部分是字母及其指数的组合。因此在 6ab² 中,系数是 6,字母部分是 ab²。没有字母的项(如 −7)叫做常数项。

When a term is written as x or as −y, the coefficient is 1 or −1 respectively; the 1 is simply hidden. Recognising this helps when you later add and subtract such terms.

当项写成 x 或 −y 时,系数分别是 1 或 −1,只是这个 1 被省略了。识别这一点有助于你后面加减这类项。

2. What Are Like Terms? | 什么是同类项?

Like terms are terms whose variable parts are exactly the same. Their coefficients may differ, but the letters and the powers of those letters must match perfectly. For example, 4x and −11x are like terms because both have the variable part x.

同类项是指字母部分完全相同的项。它们的系数可以不同,但字母以及字母的指数必须完全一致。例如 4x 和 −11x 是同类项,因为两者的字母部分都是 x。

Terms that do not share the same variable part are called unlike terms and can never be combined into a single term. So 3x and 3x² are not like terms, because x and x² have different powers; 2ab and 2a are unlike because their letters differ.

字母部分不相同的项称为非同类项,永远不能合并成一项。因此 3x 与 3x² 不是同类项,因为 x 与 x² 的指数不同;2ab 与 2a 也不是同类项,因为它们的字母不同。

Terms | 项Like terms? | 是否同类项Reason | 原因
3x and 5xYes | 是Same variable part x | 字母部分同为 x
3x and 3x²No | 否Powers of x differ (1 vs 2) | x 的指数不同(1 与 2)
2ab and 2aNo | 否Different letters | 字母不同
−7y and 4yYes | 是Same variable part y | 字母部分同为 y
5 and −9Yes | 是Both are constants | 两者都是常数项
2xy and −6yxYes | 是xy and yx are the same product | xy 与 yx 是同一个乘积

The last row is worth remembering: multiplication is commutative, so xy and yx represent the same variable part, which means 2xy and −6yx are like terms. The order of letters never changes whether terms are like or unlike.

最后一行值得记住:乘法满足交换律,所以 xy 与 yx 表示同一个字母部分,因此 2xy 与 −6yx 是同类项。字母的书写顺序不会改变两项是否同类。

3. The Sign Belongs to the Term | 符号属于项

Each term carries the sign written in front of it. The expression 8 − 2x has the terms 8 and −2x, not 8 and 2x. This sign rule is the most common source of lost marks when collecting like terms.

每一项都带着它前面的符号。表达式 8 − 2x 的项是 8 和 −2x,而不是 8 和 2x。这个符号规则是合并同类项时最常见的丢分点。

A reliable habit is to rewrite every subtraction as adding a negative term before you start. For instance, 5a − 3b + a can be read as 5a + (−3b) + a.

一个可靠的习惯是:在动笔之前,先把所有减法改写成"加上一个负项"。例如 5a − 3b + a 可以读作 5a + (−3b) + a。

5a − 3b + a = 5a + (−3b) + a = 6a − 3b

e a-family gives 5 + 1 = 6, and the b-family stays alone because there is only one b term.

一旦符号正确地归属到相应的项,合并系数就变成了机械操作:a 家族给出 5 + 1 = 6,而 b 家族因为只有一项而保持不动。

4. The Four-Step Method | 四步法

Step 1: rewrite the expression so that every term keeps its own sign, turning subtractions into additions of negative terms. Step 2: identify the like terms by comparing variable parts. Step 3: add or subtract the coefficients within each family, keeping the variable part unchanged. Step 4: write the answer in a conventional order, usually with the highest power first and the constant last.

第一步:重写表达式,让每一项都保留自己的符号,把减法变成加上负项。第二步:通过比较字母部分找出同类项。第三步:在每一类内部把系数相加或相减,字母部分保持不变。第四步:按常规顺序写出答案,通常指数高的项在前,常数项在最后。

the y-family gives 3 − 1 = 2, so the whole expression collapses into a much shorter form.

把方法用于 5x + 3y + 2x − y:x 家族给出 5 + 2 = 7,y 家族给出 3 − 1 = 2,于是整个表达式收缩成一个短得多的形式。

5x + 3y + 2x − y = 7x + 2y

Note that the variable parts never change during collection. You are only counting how many x's and how many y's you have, in exactly the same way that 5 apples plus 2 apples equals 7 apples.

注意,在合并过程中字母部分从不改变。你只是在数自己有几个 x、几个 y,就像 5 个苹果加 2 个苹果等于 7 个苹果一样。

5. Worked Examples with One Variable | 单变量例题

7x + 4x. Both terms have variable part x, so add the coefficients: 7 + 4 = 11, giving 11x.

例 1:7x + 4x。两项的字母部分都是 x,因此把系数相加:7 + 4 = 11,得到 11x。

. Treat this as 9a + (−4a) + 2a. The coefficients give 9 − 4 + 2 = 7, so the answer is 7a.

例 2:9a − 4a + 2a。把它看作 9a + (−4a) + 2a。系数给出 9 − 4 + 2 = 7,所以答案是 7a。

nt 5 stands alone because there is no other constant. The simplified expression is 9m + 5.

例 3:12m + 5 − 3m。m 家族是 12m − 3m = 9m,常数 5 单独保留,因为没有其他常数项。化简结果为 9m + 5。

t the p terms: −6p + 4p = −2p. The answer is 7 − 2p, which can also be written as −2p + 7.

例 4:15 − 6p + 4p − 8。先合并数字:15 − 8 = 7。再合并含 p 的项:−6p + 4p = −2p。答案是 7 − 2p,也可以写成 −2p + 7。

15 − 6p + 4p − 8 = 7 − 2p

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Both orderings are accepted in exams, although writing the letter term first is more conventional in IGCSE mark schemes.

两种顺序在考试中都被接受,不过把含字母的项写在前面更符合 IGCSE 评分标准的习惯。

6. Powers and Products | 指数与乘积

nd 7y³ − 2y³ = 5y³. The exponent is part of the variable part and is never added together.

当出现指数时,只要指数相同,合并同类项的方法完全一样。所以 3x² + 5x² = 8x²,7y³ − 2y³ = 5y³。指数属于字母部分,绝不参与相加。

term; it stays as x² + x. In the same way 4ab + 2ab = 6ab, but 3ab + 4a stays as 3ab + 4a.

然而 x² 与 x 不是同类项,所以像 x² + x 这样的表达式无法合并成一项,只能保持为 x² + x。同理,4ab + 2ab = 6ab,但 3ab + 4a 只能保持为 3ab + 4a。

to a new term: 2x × 3x = 6x². These two rules are completely different and mixing them up is a very common error.

注意不要把"合并"和"相乘"混淆。合并处理的是加减:2x + 3x = 5x。相乘则是把项变成一个全新的项:2x × 3x = 6x²。这两条法则完全不同,把它们搞混是非常常见的错误。

2x + 3x = 5x   but   2x × 3x = 6x²

7. Several Variables at Once | 同时含多个变量

Real exam questions usually mix several families of like terms in one expression. The strategy is unchanged: sort the terms into families, then collect each family separately.

真正的考题通常会把好几类同类项混在一个表达式里。策略不变:先把项按家族分类,再分别合并每一类。

2a + 7b, the a-family is 4a − 2a = 2a and the b-family is 3b + 7b = 10b, giving 2a + 10b.

对于 4a + 3b − 2a + 7b,a 家族是 4a − 2a = 2a,b 家族是 3b + 7b = 10b,得到 2a + 10b。

4a + 3b − 2a + 7b = 2a + 10b

ult is 6xy. For 3x²y + 4x²y, the result is 7x²y, but x²y and xy² remain unlike terms and cannot be collected.

同样的思路可以处理 xy、x²y 这类乘积项。对于 2xy + 5xy − xy,系数是 2 + 5 − 1 = 6,结果是 6xy。对于 3x²y + 4x²y,结果是 7x²y;但 x²y 与 xy² 仍然是非同类项,不能合并。

When an expression contains many families, underlining each family in a different way - or writing them in separate columns - is a quick way to avoid leaving a term out.

当表达式中家族很多时,用不同方式给每一类做标记,或把它们分列写开,是避免漏掉某一项的快捷方法。

8. Negative Coefficients and Subtraction | 负系数与减法

the answer is −3x. For −2a − 6a, the coefficients give −2 − 6 = −8, so the answer is −8a.

负系数遵循普通的有理数运算。对于 5x − 8x,系数给出 5 − 8 = −3,所以答案是 −3x。对于 −2a − 6a,系数给出 −2 − 6 = −8,所以答案是 −8a。

oefficient does not make the term disappear; it simply means the total is being reduced, or in the last case, that it ends up positive overall.

对于 −3p + 9p,系数给出 −3 + 9 = 6,所以答案是 6p。负系数并不会让项消失,它只是说明总量在减少,或者在最后这个例子里说明总体结果仍然是正的。

adding. So 4q − (−2q) becomes 4q + 2q = 6q, and 7r − (−3r) − 5r becomes 7r + 3r − 5r = 5r.

减去一个负项等于加上它。因此 4q − (−2q) 变成 4q + 2q = 6q,而 7r − (−3r) − 5r 变成 7r + 3r − 5r = 5r。

4q − (−2q) = 4q + 2q = 6q

Writing brackets around negative terms at the start of a calculation is a simple way to keep track of every sign and to reduce careless mistakes under exam pressure.

在计算开始时把负项用括号写出来,是跟踪每一个符号、减少考试压力下粗心错误的简单方法。

9. Collecting Terms in Equations | 在方程中合并同类项

the left-hand side simplifies to 5x, so the equation becomes 5x = 25 and therefore x = 5.

解一元一次方程时,合并同类项通常是第一步。在 3x + 2x = 25 中,左边化简为 5x,因此方程变成 5x = 25,于是 x = 5。

1, subtract x from both sides to get 2x + 5 = 11, then subtract 5 to get 2x = 6, so x = 3.

当两边都出现字母时,把含字母的项归到一边,把数字归到另一边。从 3x + 5 = x + 11 出发,两边同减 x 得 2x + 5 = 11,再减 5 得 2x = 6,所以 x = 3。

3x + 5 = x + 11 → 2x = 6 → x = 3

ion: with x = 3, the left side is 14 and the right side is 14, so the solution is correct.

一个有用的检验方法是把答案代回原方程:当 x = 3 时,左边等于 14,右边也等于 14,说明解是正确的。

10. Common Mistakes to Avoid | 需要避免的常见错误

Most lost marks in this topic come from a small number of repeated errors. Checking your work against the list below takes only a few seconds.

这个知识点上丢掉的分数,大多来自少数几个反复出现的错误。对照下面的清单检查一遍,只需要几秒钟。