📚 Collecting Like Terms: A Complete IGCSE Algebra Guide | 合并同类项：IGCSE 代数完整指南
In IGCSE algebra, many questions ask you to simplify an expression such as 6x + 3y - 2x + 5y. Collecting like terms is the first essential skill: it lets you replace a long expression with a shorter equivalent one. This guide explains what like terms are, how to combine them correctly, how to deal with brackets, negative signs, fractions and powers, and how to avoid the common exam errors that quietly cost marks.
在 IGCSE 代数中，许多题目要求你化简如 6x + 3y - 2x + 5y 这样的表达式。合并同类项是第一项核心技能：它能把一个较长的表达式写成等价的简短形式。本文说明什么是同类项、如何正确合并，以及如何处理括号、负号、分数和指数，并指出那些悄悄扣分的常见考试错误。
1. What Are Like Terms? | 什么是同类项
A term is a single number, a single letter, or a product of numbers and letters, such as 4x, -3y, 7x² or 9. Like terms have exactly the same variable part and the same powers. The coefficient, which is the number in front of the letter part, can be different.
项是单个数字、单个字母或数字与字母的乘积，例如 4x、-3y、7x² 或 9。同类项具有完全相同的字母部分和相同的指数。系数，也就是字母前面的那个数字，可以不同。
For example, 5x and -3x are like terms because both have x to the power 1. However, 4x and 4x² are not like terms, because x and x² are different powers. Also, 2xy and -7xy are like terms, while 5 and -9 are like constant terms.
例如，5x 和 -3x 是同类项，因为两者都含有 x 的一次方。然而，4x 与 4x² 不是同类项，因为 x 和 x² 是不同的幂。另外，2xy 与 -7xy 是同类项，而 5 和 -9 属于同类常数项。
A useful way to remember the rule is the phrase "same letter, same power". If either the letter or the power differs, the terms cannot be merged into one term.
一个实用的记忆口诀是"字母相同、指数相同"。只要字母或指数有一项不同，这两个项就不能合并成一个项。
2. Why Collecting Like Terms Matters in IGCSE | 为什么合并同类项在 IGCSE 中很重要
Many IGCSE questions do not only ask you to simplify an expression; they also expect simplified answers in algebra, equation solving and function work. If you do not collect like terms, your final answer may be mathematically equivalent but not in the required simplified form, so marks can be lost.
许多 IGCSE 题目不仅要求你化简表达式，在代数、解方程和函数等内容中也要求答案必须化简。如果不合并同类项，最终答案可能在数学上等价，但不是题目要求的化简形式，因此可能丢分。
This matters because examiners mark against a simplified answer in the mark scheme. Writing 6x + 3y - 2x + 5y when the answer is 4x + 8y shows you have not finished the question, even though the two expressions are equal for every value of x and y.
这一点很重要，因为评分标准中的答案是化简后的形式。当正确答案是 4x + 8y 时，你写成 6x + 3y - 2x + 5y，说明你还没有完成这道题，尽管这两个表达式对任意 x、y 取值都相等。
The skill also supports later topics: expanding brackets, factorising, solving linear equations, rearranging formulae, and simplifying algebraic fractions all depend on it.
这项技能还支撑后续内容：去括号、因式分解、解一元一次方程、公式变形以及代数分式化简，都以它为基础。
3. Identifying the Terms and Their Signs | 识别各项及其符号
Before you can combine anything, you must split the expression into separate terms. The sign in front of a term belongs to that term. In 6x + 3y - 2x + 5y, the four terms are +6x, +3y, -2x and +5y.
在合并之前，必须先把这个表达式拆成一个个独立的项。项前面的符号属于该项。在 6x + 3y - 2x + 5y 中，四项分别是 +6x、+3y、-2x 和 +5y。
Treating a subtraction as "adding a negative" removes most sign errors. So 6x + 3y - 2x + 5y can be read as 6x + 3y + (-2x) + 5y, and now every term is being added.
把减法看成"加上一个负数"，可以消除大部分符号错误。因此 6x + 3y - 2x + 5y 可以读作 6x + 3y + (-2x) + 5y，此时每一项都是在做加法。
The table below sorts a small list of terms into groups that can and cannot be combined.
下表把一组项按"可以合并"和"不能合并"分成两类。
| Terms 各项 | Like or not? 是否同类项 | Combined form 合并结果 |
| 5x and -3x | Like terms 同类项 | 2x |
| 2xy and -7xy | Like terms 同类项 | -5xy |
| 5 and -9 | Like constants 同类常数项 | -4 |
| 4x and 4x² | Not like 不是同类项 | 4x + 4x² cannot be shortened 无法再合并 |
| 3ab and 3a | Not like 不是同类项 | 3ab + 3a cannot be shortened 无法再合并 |
4. The Golden Rule: Combine Coefficients Only | 黄金法则：只合并系数
When terms are alike, you add or subtract the coefficients and keep the variable part completely unchanged. The letter part is a label; it does not change during the addition.
当两项是同类项时，你只把系数相加或相减，字母部分完全保持不变。字母部分只是一个标签，在加法过程中不发生变化。
5x + 3x = (5 + 3)x = 8x
7a - 4a = (7 - 4)a = 3a
c = (a + b)c. Replacing c with x gives the rule used thousands of times in algebra papers.
这其实是分配律的逆向使用：ac + bc = (a + b)c。把 c 换成 x，就得到了在代数试卷中要用上千次的规则。
multiply or to add the powers, writing 5x + 3x = 8x². That is wrong: 5x + 3x means five x's plus three more x's, which is eight x's, not eight x-squared.
最常见的错误是把指数也相加或相乘，写成 5x + 3x = 8x²。这是错的：5x + 3x 表示五个 x 再加三个 x，等于八个 x，而不是八个 x 的平方。
5. Step-by-Step Method | 合并同类项的分步方法
A reliable routine makes the process almost automatic, even in long expressions with many different letters. Follow the same five steps every time.
一套固定的流程可以让这个过程几乎变成自动化，即使面对含有多种字母的长表达式也适用。每次都按同样的五步来做。
Step 1: Rewrite the expression as a sum, attaching the sign to each term. 第一步：把表达式改写成求和形式，把符号归给每一项。
Step 2: Identify groups of like terms by circling or underlining each group. 第二步：找出同类项分组，可以圈出或划线标记每一组。
Step 3: Add the coefficients within each group, keeping the sign in front of each coefficient. 第三步：在每一组内把系数相加，注意保留每个系数前面的符号。
Step 4: Write the combined groups in one line, usually in alphabetical order such as x before y. 第四步：把各组结果写在一行，通常按字母顺序排列，例如 x 在 y 之前。
Step 5: Check the answer by substituting a simple number for each letter. 第五步：用简单数字代入每个字母，检验答案是否等价。
Keeping the working tidy is not decoration. In a two-mark question the method mark is usually awarded for showing the grouped coefficients.
书写的整洁不是为了好看。在满分两分的题目中，方法分通常是靠展示分组后的系数来获得的。
6. Worked Example: 6x + 3y - 2x + 5y | 例题详解：6x + 3y - 2x + 5y
Let us apply the five steps to the expression used throughout this guide.
我们把五步法用在本篇文章反复使用的表达式上。
Step 1 - rewrite as a sum: 6x + 3y + (-2x) + 5y.
第一步——改写成求和形式：6x + 3y + (-2x) + 5y。
Step 2 - group the like terms. The x-terms are 6x and -2x. The y-terms are 3y and 5y.
第二步——把同类项分组。含 x 的项是 6x 和 -2x。含 y 的项是 3y 和 5y。
(6x - 2x) + (3y + 5y)
Step 3 - add the coefficients in each group: 6 - 2 = 4 and 3 + 5 = 8.
第三步——各组系数相加：6 - 2 = 4，3 + 5 = 8。
Step 4 - write the simplified expression.
第四步——写出化简后的表达式。
6x + 3y - 2x + 5y = 4x + 8y
The expression has gone from four terms to two, and no further simplification is possible because 4x and 8y are not like terms.
这个表达式从四项变成了两项，而且无法继续化简，因为 4x 与 8y 不是同类项。
7. Negative Signs and Subtraction | 负号与减法
Negative coefficients cause more lost marks than any other part of this topic. The rule is simple: the minus sign in front of a term travels with it into the calculation.
负系数在这个知识点中造成的失分最多。规则很简单：项前面的负号要跟着这一项一起参与运算。
-4a + 9a = 5a, because -4 + 9 = 5. -4a + 9a = 5a，因为 -4 + 9 = 5。
3b - 8b = -5b, because 3 - 8 = -5. 3b - 8b = -5b，因为 3 - 8 = -5。
-2m - 6m = -8m, because -2 - 6 = -8. -2m - 6m = -8m，因为 -2 - 6 = -8。
11n - 11n = 0, and the term disappears completely. 11n - 11n = 0，这一项会完全消失。
When a term sums to zero, do not write 0n. Simply leave it out of the final answer.
当某一项相加结果为零时，不要写成 0n，直接在最终答案里去掉它即可。
If you are unsure, put brackets around every coefficient before adding. Writing (-4 + 9) and (3 - 8) makes the sign handling visible and is accepted in exam working.
如果不确定，就把每个系数先加上括号再相加。写成 (-4 + 9) 和 (3 - 8) 能让符号处理过程清晰可见，在考试解答中也是被接受的写法。
8. Brackets and the Distributive Law | 括号与分配律
Many IGCSE questions hide the like terms behind brackets. You must expand first, then collect. Trying to collect across brackets without expanding is the classic trap.
许多 IGCSE 题目会把同类项藏在括号后面。必须先展开，再合并。不展开就试图跨括号合并，是最经典的陷阱。
For example, expand and simplify 3(x + 4) + 2(x - 1).
例如，展开并化简 3(x + 4) + 2(x - 1)。
3(x + 4) + 2(x - 1) = 3x + 12 + 2x - 2
3x + 2x + 12 - 2 = 5x + 10
Watch the second bracket carefully. Multiplying (x - 1) by +2 gives +2x and -2, not +2x and +2.
要特别注意第二个括号。把 (x - 1) 乘以 +2，得到的是 +2x 和 -2，而不是 +2x 和 +2。
s to re-expand your answer mentally and see whether you return to the original expression.
当负数乘以括号时，括号内每一项的符号都改变：-3(x - 5) = -3x + 15。快速的检查方法是把答案在心中重新展开，看是否能回到原表达式。
9. Powers, Products and Non-Like Terms | 幂、乘积与非同类项
Powers behave differently from coefficients, and mixing the two rules is a frequent error. Adding like terms changes the coefficient; multiplying terms changes the powers.
幂的规则与系数不同，把两套规则混淆是很常见的错误。合并同类项改变的是系数；相乘则改变指数。
3a + 5a = 8a but 3a × 5a = 15a²
In the first case we are counting a's. In the second case we are multiplying, so a × a = a² and 3 × 5 = 15.
第一种情况我们是在数 a 的个数。第二种情况我们是在做乘法，所以 a × a = a²，3 × 5 = 15。
Terms with different powers never merge. The expression x² + x + 3 stays as it is, because x², x and 3 are three separate types of term. Likewise 4x²y and 4xy² are not like terms: in the first, x is squared; in the second, y is squared.
指数不同的项永远不能合并。表达式 x² + x + 3 就保持原样，因为 x²、x 和 3 是三种不同的项。同理，4x²y 与 4xy² 也不是同类项：前者 x 是平方，后者 y 是平方。
When expressions contain several letters, group by the complete letter part. In 7xy + 3x - 2xy + 5x, the xy-terms give 5xy and the x-terms give 8x, so the answer is 5xy + 8x.
当表达式含有多个字母时，要按完整的字母部分分组。在 7xy + 3x - 2xy + 5x 中，xy 项合并为 5xy，x 项合并为 8x，所以答案是 5xy + 8x。
10. Constants, Fractions and Decimals | 常数项、分数与小数
x + 3, the constants give 7 + 3 = 10, while the x-terms give 3x, so the answer is 3x + 10.
不含字母的数字称为常数项，它们自成一类。在 4x + 7 - x + 3 中，常数项给出 7 + 3 = 10，x 项给出 3x，所以答案是 3x + 10。
Fractional coefficients follow exactly the same rules, but you need a common denominator when the fractions are unlike.
分数系数的规则完全相同，但当分母不同时需要先通分。
½x + ¼x = ¾x ⅔y - ⅙y = ½y
- 0.9t = 1.5t. Keep the decimal points aligned in your working to avoid place-value slips.
小数系数与整数处理方式相同：0.5y + 1.25y = 1.75y，2.4t - 0.9t = 1.5t。书写时把小数点对齐，可以避免数位错误。
Mixed expressions such as ½x + 3 + ¼x - 1 tidy up to ¾x + 2, with the fractions collected together and the constants collected separately.
像 ½x + 3 + ¼x - 1 这样的混合表达式可以整理为 ¾x + 2，分数归分数、常数归常数。
11. Checking Your Answer by Substitution | 用代入法检验答案
Substitution turns a guess into a proof. Choose small, different numbers for each letter, evaluate both the original and the simplified expression, and confirm that the two results match.
代入法可以把猜测变成验证。给每个字母取小而不同的数字，分别计算原表达式和化简后表达式的值，确认两者结果相等。
4(2) + 8(1) = 8 + 8 = 16. The values agree, so the simplification is very likely correct.
以主要例题为例，取 x = 2、y = 1。原式给出 6(2) + 3(1) - 2(2) + 5(1) = 12 + 3 - 4 + 5 = 16。化简式给出 4(2) + 8(1) = 8 + 8 = 16。两者一致，说明化简基本正确。
Use at least one negative number in your test. If your check uses only positive values, a sign error can sometimes hide, because the same mistake appears on both sides of the comparison.
检验时至少要取一个负数。如果只用正数测试，符号错误有时会被掩盖，因为同一个错误会同时出现在对比的两边。
Substitution is a checking tool, not a method of proof, and it is not a substitute for showing the coefficient working in an exam. Use it after you have written the answer, not instead of the method marks.
代入法是检验工具，不是证明方法，也不能代替考试中展示系数运算的过程。它应当写在答案之后作为检查，而不是用来替代获得方法分的步骤。
12. Common Exam Errors and Practice | 常见考试错误与练习
Nearly every lost mark in this topic comes from a short list of repeated mistakes. Learn to recognise them and you will stop making them.
这个知识点的失分几乎都来自一份反复出现的错误清单。学会识别它们，就能停止犯这些错误。
Adding the powers: 5x + 3x = 8x² is wrong; the answer is 8x. 把指数相加：5x + 3x = 8x² 是错的，正确答案是 8x。
Merging unlike terms: x² + x cannot be written as 2x² or 2x. 合并非同类项：x² + x 不能写成 2x² 或 2x。
Losing a minus sign: -2x + 7x equals 5x, not -9x. 丢失负号：-2x + 7x 等于 5x，不是 -9x。
Forgetting to expand brackets before collecting across them. 忘记先展开括号就跨括号合并。
Writing 0x instead of removing the zero term from the answer. 把零项写成 0x，而不是从答案中去掉。
Now try these exam-style questions in the same style as a typical IGCSE paper. Simplify each expression fully.
现在用下面这些典型 IGCSE 风格的题目练一练。请把每个表达式完全化简。
(a) 9a + 4b - 5a + 2b (b) 3x² + 2x - x² + 7x (c) 5(p + 3) - 2(p - 4)
(d) 7mn - 2m + mn + 9m (e) ⅓y + ½y - ⅙y (f) 4 - 3t + 8t - 11
Answers: (a) 4a + 6b, (b) 2x² + 9x, (c) 3p + 23, (d) 8mn + 7m, (e) ⅔y, (f) 5t - 7. Check each one with a substitution before you move on.
答案：(a) 4a + 6b；(b) 2x² + 9x；(c) 3p + 23；(d) 8mn + 7m；(e) ⅔y；(f) 5t - 7。做完每一题后先用代入法检验，再继续下一题。
Collecting like terms is a small skill with a large reach. Get it right every time in Year 10 and Year 11, and every later algebra topic becomes easier.
合并同类项是一个小而影响深远的技能。在十年级和十一年级每次都做对，之后每一个代数专题都会变得更轻松。
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