📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, simplifying algebraic expressions often starts with collecting like terms. This process combines terms that have exactly the same variable part and the same power, making an expression shorter and easier to handle. Mastering this skill is essential for solving equations, expanding brackets and working with formulae.
在 IGCSE 数学中，代数式的化简通常从合并同类项开始。这个过程把具有完全相同变量部分和相同次数的项合并起来，使表达式更短、更易处理。掌握这一技能对于解方程、去括号和使用公式都至关重要。
A student who cannot collect like terms reliably will lose marks in almost every algebra question, from simple simplification to quadratic manipulation. This guide builds the idea from the meaning of a term, through the coefficient rule, worked examples with one and several variables, the treatment of signs and powers, to exam technique and practice. Every step is explained with the reason behind it, not just the rule.
如果学生不能稳定地合并同类项，几乎每一道代数题都会失分，从最简单的化简到二次式的变形都是如此。本文从"项"的含义讲起，依次介绍系数法则、单变量与多变量的例题、符号与指数的处理，最后给出考试技巧和练习题。每一步都解释背后的道理，而不只是给出规则。
1. What Is a Term? | 什么是项
An algebraic expression is made up of terms. A term is a number, a variable, or a product of numbers and variables, separated by + or - signs. For example, in 4x + 3y - 7, the terms are 4x, 3y and -7. The number part of a term is called the coefficient, and the letter part is called the variable part.
一个代数式由项组成。项是数字、变量或数字与变量的乘积，它们之间用加号或减号分隔。例如，在 4x + 3y - 7 中，项是 4x、3y 和 -7。项中的数字部分称为系数，字母部分称为变量部分。
a term with no visible coefficient has coefficient 1, because x means 1 × x, and -x has coefficient -1.
这里有两个细节很重要。第一，项前面的符号属于这一项本身，所以 4x + 3y - 7 的第三项是 -7，系数为 -7，而不是 +7。第二，没有写出系数的项系数为 1，因为 x 就是 1 × x，而 -x 的系数是 -1。
| Term | 项 | Coefficient | 系数 | Variable part | 变量部分 |
|---|---|---|
| 4x | 4 | x |
| -7y | -7 | y |
| 5x² | 5 | x² |
| -m | -1 | m |
| 9 | 9 | none (constant) | 无（常数项） |
Understanding the structure of terms is important because collecting like terms is really just adding or subtracting coefficients. If the variable part is the same, the terms can be combined; if not, they must stay separate. A constant such as 9 has no variable part, so it can only be combined with other constants.
理解项的结构很重要，因为合并同类项实际上就是对系数进行加减。如果变量部分相同，这些项就可以合并；如果不同，它们就必须保持分开。像 9 这样的常数项没有变量部分，因此只能与其他常数项合并。
2. Like Terms and Unlike Terms | 同类项与不同类项
Like terms are terms that have exactly the same variable and the same exponent. For example, 5x and -2x are like terms because both contain x to the power of 1. However, 5x and 5x² are not like terms, because the powers of x are different. Constants such as 4 and -9 are also like terms with each other.
同类项是具有完全相同变量和相同指数的项。例如，5x 和 -2x 是同类项，因为它们都含有 x 的一次方。但是，5x 和 5x² 不是同类项，因为 x 的指数不同。常数项如 4 和 -9 彼此也是同类项。
The word "exactly" carries all the weight. 3x and 3y are not like terms because the letters differ. 2ab and 2ba are like terms, because ab and ba are the same product written in a different order. But 4a²b and 4ab² are not like terms, since in the first the a is squared and in the second the b is squared.
"完全相同"这几个字是关键。3x 和 3y 不是同类项，因为字母不同。2ab 和 2ba 是同类项，因为 ab 和 ba 只是同一乘积的不同书写顺序。但 4a²b 和 4ab² 不是同类项，因为前者 a 是平方，后者 b 是平方。
| Pair | 项对 | Like terms? | 是否同类项 | Reason | 原因 |
|---|---|---|
| 6x and 6y | No | 否 | different letters | 字母不同 |
| 6x and -2x | Yes | 是 | same letter, same power | 字母与次数相同 |
| 6x and 6x² | No | 否 | different powers | 次数不同 |
| 3ab and -7ba | Yes | 是 | same product, reordered | 乘积相同，只是重排 |
| 12 and -5 | Yes | 是 | both constants | 都是常数 |
3. Why the Rule Works: The Distributive Law | 为什么这条规则成立：分配律
ve copies of x, together there are eight copies of x. In symbols, 3x + 5x = (3 + 5)x = 8x.
合并同类项并不是为考试发明的技巧，它来源于分配律。3x 表示 3 个 x，5x 表示 5 个 x，合在一起就是 8 个 x。用符号表示就是 3x + 5x = (3 + 5)x = 8x。
Writing it formally, for any numbers a, b and any variable expression x, we have:
正式地写出来，对任意数 a、b 和任意变量表达式 x，都有：
ax + bx = (a + b)x ax - bx = (a - b)x
5x². The variable part behaves like a label or a unit: once it is identical, only the numbers in front need attention.
同样的推理适用于更复杂的变量部分。4ab + 9ab = (4 + 9)ab = 13ab，7x² - 2x² = (7 - 2)x² = 5x²。变量部分就像一个标签或单位：只要它完全相同，就只需要处理前面的数字。
A useful everyday analogy is counting fruit. Three apples plus five apples gives eight apples, but three apples plus five oranges cannot be squeezed into one number of "apples-oranges". Algebraic terms follow exactly the same logic.
一个有用的日常类比是数水果。3 个苹果加 5 个苹果等于 8 个苹果，但 3 个苹果加 5 个橙子不能硬凑成一个"苹果橙"的数量。代数项的规则与此完全相同。
4. The Coefficient Rule and Its Limits | 系数法则及其边界
The practical rule students should memorise is short: to collect like terms, add or subtract the coefficients and keep the variable part unchanged. Nothing happens to the letter or its power during addition and subtraction.
学生需要记住的实用规则很短：合并同类项时，把系数相加或相减，变量部分保持不变。在做加减法时，字母及其指数不发生任何变化。
The rule has a hard boundary. Only terms with the same variable part can be combined. This means 2x + 3y stays as 2x + 3y; it is a finished answer, not an unfinished one. Likewise 5 + 4x cannot become 9x, because 5 has no x and 4x does.
这条规则有明确的边界。只有变量部分相同的项才能合并。这意味着 2x + 3y 就保持为 2x + 3y，它是最终答案，而不是未完成的结果。同样，5 + 4x 不能变成 9x，因为 5 中没有 x，而 4x 中有。
r do: 2x + 3x = 5x. Keeping these two operations separate is the single most valuable habit in early algebra.
一个常见的误解是把加法当成乘法来处理。在乘法中指数会相加：2x × 3x = 6x²。在加法中指数永远不会相加：2x + 3x = 5x。把这两种运算清楚区分开，是初学代数时最有价值的习惯。
5. A Reliable Step-by-Step Method | 一套可靠的分步方法
Most errors in this topic come from rushing rather than from a lack of understanding. A fixed four-step routine removes almost all of them.
这个专题中的大多数错误来自急躁，而不是理解不足。一套固定的四步流程几乎可以消除所有这些错误。
Step 1: Identify every term, including its sign. | 第一步：找出每一项，包括它前面的符号。
Step 2: Group terms with the same variable part together, using underlining, circles or different colours. | 第二步：把变量部分相同的项归到一起，可以用下划线、圈画或不同颜色标记。
Step 3: Add or subtract the coefficients within each group. | 第三步：在每一组内对系数进行加减。
Step 4: Write the simplified expression with the terms in a sensible order, usually highest power first, then alphabetical. | 第四步：写出化简后的表达式，按合理顺序排列，通常先写次数最高的项，再按字母顺序排列。
For example, simplify 6a + 4 - 2a + 7b - 9. Grouping gives (6a - 2a) + (4 - 9) + 7b, which becomes 4a - 5 + 7b, normally written as 4a + 7b - 5. The letters a and b never merge, and the constants -5 stay on their own.
例如，化简 6a + 4 - 2a + 7b - 9。分组后得到 (6a - 2a) + (4 - 9) + 7b，即 4a - 5 + 7b，通常写成 4a + 7b - 5。字母 a 和 b 永远不会合并，常数 -5 也单独保留。
6. Worked Examples with One Variable | 单变量例题
Example 1: simplify 7x + 3 - 2x + 5. The x-terms are 7x and -2x, giving 5x. The constants are 3 and 5, giving 8. The answer is 5x + 8.
例 1：化简 7x + 3 - 2x + 5。含 x 的项是 7x 和 -2x，合并得 5x。常数项是 3 和 5，合并得 8。答案是 5x + 8。
4 + 9 - 1 = 4, so the a-terms give 4a. The constant -6 is unchanged. The answer is 4a - 6.
例 2：化简 -4a + 9a - 6 - a。系数是 -4、+9 和 -1，而 -4 + 9 - 1 = 4，所以含 a 的项合并为 4a。常数 -6 保持不变。答案是 4a - 6。
Example 3: simplify x² + 3x + 2x² - x. Here x² and x are different variable parts. The x² terms give 3x², and the x terms give 2x. The answer is 3x² + 2x. Many students wrongly write 5x³; that would be the result of multiplying, not adding.
例 3：化简 x² + 3x + 2x² - x。这里 x² 和 x 是不同变量部分。x² 项合并为 3x²，x 项合并为 2x。答案是 3x² + 2x。许多学生错写成 5x³，那是相乘的结果，不是相加的结果。
| Question | 题目 | Answer | 答案 |
|---|---|
| 8x + 5x | 13x |
| 12y - 4y + y | 9y |
| 5t² + 3t - 2t² - 8t | 3t² - 5t |
| -6k + 10 + 2k - 10 | -4k |
7. Worked Examples with Two or More Variables | 多变量例题
When several letters appear, the safest approach is to sort the terms by variable part before doing any arithmetic. This turns a messy expression into a set of small, separate sums.
当出现多个字母时，最稳妥的做法是先按变量部分把项分类，再做任何运算。这样就把杂乱的表达式变成几组小的、互不干扰的求和。
Example 1: simplify 3x + 2y - x + 5y. The x-terms give 2x, the y-terms give 7y. The answer is 2x + 7y.
例 1：化简 3x + 2y - x + 5y。x 项合并为 2x，y 项合并为 7y。答案是 2x + 7y。
Example 2: simplify 4ab + 2a - ab + 3a - b. The ab terms give 3ab, the a terms give 5a, and the single b term stays as -b. The answer is 3ab + 5a - b.
例 2：化简 4ab + 2a - ab + 3a - b。ab 项合并为 3ab，a 项合并为 5a，单独的 b 项保持为 -b。答案是 3ab + 5a - b。
Example 3: simplify 6p²q - 2pq² + p²q + 5pq². The p²q terms give 7p²q, and the pq² terms give 3pq². The answer is 7p²q + 3pq². Note again that p²q and pq² are different and cannot be combined.
例 3：化简 6p²q - 2pq² + p²q + 5pq²。p²q 项合并为 7p²q，pq² 项合并为 3pq²。答案是 7p²q + 3pq²。再次注意，p²q 与 pq² 不同，不能合并。
8. Signs, Negative Coefficients and Subtraction | 符号、负系数与减法
The most frequent source of lost marks is a sign error. The rule to apply without exception is that the sign immediately before a term travels with that term. In 8m - 3n - 5m - n, the terms are 8m, -3n, -5m and -n.
失分最常见的原因是符号错误。必须无条件遵守的规则是：项前面的符号跟着这一项一起移动。在 8m - 3n - 5m - n 中，项是 8m、-3n、-5m 和 -n。
3m - 4n. If you instead treat the last term as +n, you would get 3m - 2n, which is wrong. Writing the sign in front of each term explicitly, even when it is a plus, prevents this slip.
合并后得到 (8m - 5m) + (-3n - n) = 3m - 4n。如果把最后一项当成 +n，就会得到 3m - 2n，这是错的。把每项前的符号（即使是加号）都明确写出来，可以避免这种失误。
ho forget the double negative on the 4 lose the mark even though their x-terms are correct.
减去一个带括号的表达式时，括号内每一项的符号都要变号。所以 10x - (3x - 4) = 10x - 3x + 4 = 7x + 4。忘记 4 前面的双重负号的学生，即使 x 项算对了也会失分。
13x - 4. Comparing these two lines side by side is a quick way to fix the rule in memory.
加上一个括号则不改变任何符号：10x + (3x - 4) = 10x + 3x - 4 = 13x - 4。把这两行并排比较，是记住这条规则的快捷方法。
9. Powers, Roots and Mixed Variable Parts | 幂、根式与混合变量部分
Terms are like terms only when the whole variable part matches, and that includes the exponents. x, x², x³, √x and 1 ÷ x are five different variable parts, and none of them can be merged with another.
只有当整个变量部分完全一致时，项才是同类项，这其中包含指数。x、x²、x³、√x 和 1 ÷ x 是五种不同的变量部分，它们彼此都不能合并。
Consider 5√x + 3√x = 8√x. The coefficient rule works exactly as before because the variable part √x is identical in both terms. But 5√x + 3x cannot be simplified further, even though both terms contain x.
考虑 5√x + 3√x = 8√x。因为变量部分 √x 完全相同，系数法则与之前完全一样。但 5√x + 3x 不能再化简，尽管两项都含有 x。
arated by + or -, and exponents never change. To multiply, the terms are joined by × or by brackets, and exponents do change: x² × x³ = x⁵, and (x²)³ = x⁶.
同样值得把"合并"与"相乘"区分开，因为手写时两种记号很像。合并时，项之间由加号或减号分隔，指数绝不改变。相乘时，项由乘号或括号连接，指数会改变：x² × x³ = x⁵，而 (x²)³ = x⁶。
| Operation | 运算 | x-terms | 含 x 的项 | Result | 结果 |
|---|---|---|
| Add | 加 | 2x + 3x | 5x |
| Multiply | 乘 | 2x × 3x | 6x² |
| Add, different powers | 加，次数不同 | 2x² + 3x | 2x² + 3x (no change | 不变) |
10. Common Mistakes to Avoid | 常见错误
Error 1: combining unlike terms. Writing 5x + 5y as 10xy is wrong on two counts - the coefficients should not simply be added across different letters, and the letters should not be multiplied.
错误 1：合并不同类项。把 5x + 5y 写成 10xy 有两处错误：不同字母的系数不能直接相加，字母之间也不应该相乘。
. This single confusion damages later work on quadratics, so it is worth correcting early.
错误 2：把 x + x 写成 x²。两个 x 相加得到 2x；只有 x × x 才等于 x²。这一处混淆会影响后面的二次式学习，因此值得尽早纠正。
implifies to 3m, not to m³ or 3, because each m has coefficient 1. Similarly, 7n - n = 6n.
错误 3：忽略隐藏的系数。表达式 m + m + m 化简为 3m，而不是 m³ 或 3，因为每个 m 的系数都是 1。同样，7n - n = 6n。
Error 4: dropping negative signs, and Error 5: adding exponents when collecting. Write the sign with every term and keep the variable part exactly as it stands, and both problems disappear.
错误 4：漏掉负号；错误 5：合并时把指数相加。把符号写在每一项前面，并让变量部分原样保持不变，这两个问题就都消失了。
11. Collecting Like Terms in Equations and Formulae | 方程与公式中的合并同类项
get 2x + 7 = 19, then subtract 7 to get 2x = 12, and finally divide by 2 to obtain x = 6.
合并同类项是大多数一元一次方程的第一步。解 5x + 7 = 3x + 19 时，两边同时减去 3x 得到 2x + 7 = 19，再减去 7 得到 2x = 12，最后除以 2 得 x = 6。
For example, 4(x + 2) - x = 2x + 9 simplifies first to 3x + 8 = 2x + 9 and then to x = 1.
如果方程两边各含有若干个同类项，应先分别化简每一边，再移动项。例如，4(x + 2) - x = 2x + 9 先化简为 3x + 8 = 2x + 9，再得到 x = 1。
, substituting l = 2w gives P = 4w + 2w = 6w. Collecting like terms turns a two-variable formula into a single-variable one, which can then be evaluated or rearranged directly.
公式同样受益。矩形周长通常写作 P = 2l
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