📚 Combining Like Terms: Complete IGCSE Algebra Guide | 合并同类项：IGCSE 代数完整指南
In IGCSE algebra, simplifying an expression often begins with collecting, or combining, like terms. A term is a single number, a variable, or a product of numbers and variables. Like terms have exactly the same variable part, so they can be added or subtracted. This skill is essential for solving equations, rearranging formulae, and working with brackets, and it appears in almost every algebra question you will meet at IGCSE, O Level and A Level.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。项是一个单独的数、一个变量，或数与变量的乘积。同类项具有完全相同的变量部分，因此可以相加或相减。这项技能对于解方程、变换公式和处理括号都至关重要，它几乎出现在 IGCSE、O Level 和 A Level 的每一道代数题中。
1. What Is an Algebraic Term? | 什么是代数项
An algebraic expression is built from terms separated by + or − signs. Each term has a coefficient (the numerical part) and possibly a variable part. For example, in 4x + 3y − 7, the terms are 4x, 3y, and −7.
代数表达式由用 + 或 − 分隔的项组成。每个项都有一个系数（数字部分），可能还有变量部分。例如，在 4x + 3y − 7 中，项是 4x、3y 和 −7。
In the term 4x, the coefficient is 4 and the variable part is x. In 3y, the coefficient is 3 and the variable part is y. The term −7 has no variable at all, so it is called a constant term.
在项 4x 中，系数是 4，变量部分是 x。在 3y 中，系数是 3，变量部分是 y。项 −7 完全没有变量，因此称为常数项。
A term can also contain powers or several variables multiplied together. For example, 5x² has coefficient 5 and variable part x², while −2ab has coefficient −2 and variable part ab.
一个项也可以包含幂，或几个变量相乘。例如，5x² 的系数是 5，变量部分是 x²；而 −2ab 的系数是 −2，变量部分是 ab。
Every expression can be read as a sum of its terms. Reading it this way is the safest habit, because it stops you from losing minus signs later on.
每个表达式都可以读作它各项的和。这样读是最安全的习惯，因为它能防止你在后面丢掉负号。
2. Like Terms vs Unlike Terms | 同类项与不同类项
Terms are 'like' if they have exactly the same variable part, including the same exponents. The coefficients can be different. For example, 6x and −2x are like terms, but 6x and 6x² are not like terms.
如果各项具有完全相同的变量部分，包括相同的指数，它们就是同类项。系数可以不同。例如，6x 和 −2x 是同类项，但 6x 和 6x² 不是同类项。
Because multiplication is commutative, the order of variables inside a term does not matter. So 3ab, −5ba and 7ab are all like terms, and xy is the same as yx.
由于乘法满足交换律，项内部变量的书写顺序并不重要。因此 3ab、−5ba 和 7ab 都是同类项，xy 与 yx 也完全相同。
| Pair of terms | 项的组合 | Like or unlike? | 是否同类项 | Reason | 原因 |
|---|---|---|
| 6x and −2x | Like | 同类 | Same variable part x | 变量部分同为 x |
| 6x and 6x² | Unlike | 不同类 | Different exponents | 指数不同 |
| 3ab and −5ba | Like | 同类 | ab = ba | 乘法可交换 |
| x²y and xy² | Unlike | 不同类 | Exponents attached to different letters | 指数属于不同字母 |
| 7 and −4 | Like | 同类 | Both constant terms | 都是常数项 |
A quick test: cover the numbers and compare only the letters. If the letters and their powers match exactly, the terms can be combined.
一个快速的检验方法：遮住数字，只比较字母。如果字母及其指数完全一致，这两项就可以合并。
3. The Sign Belongs to the Term | 符号属于该项
The sign in front of a term belongs to that term. So in 3a − 2b + 5, the terms are 3a, −2b, and +5. When you rearrange terms, you must carry the sign with each term.
项前面的符号属于该项。因此在 3a − 2b + 5 中，项是 3a、−2b 和 +5。当你重新排列各项时，必须把符号随项一起移动。
A useful trick is to rewrite every subtraction as adding a negative. So 7 − 4m − m becomes 7 + (−4m) + (−m). Now nothing can be lost when you reorder.
一个实用技巧是把每个减法都改写成加上一个负数。于是 7 − 4m − m 变成 7 + (−4m) + (−m)。这样在重新排序时就不会丢掉任何符号。
a minus sign in front means coefficient −1. So −m means −1m, and x + x means 1x + 1x = 2x.
记住，单独写出的字母系数为 1，前面有负号则表示系数为 −1。因此 −m 就是 −1m，而 x + x 就是 1x + 1x = 2x。
4. How to Combine Like Terms: Step by Step | 合并同类项的步骤
Use the same four steps every time, and you will rarely make a mistake under exam pressure.
每次都使用同样的四步方法，你在考试压力下就很少会出错。
Step 1: Identify each term together with its sign. | 第一步：连同符号一起识别每一个项。
Step 2: Group terms that have the same variable part. | 第二步：把变量部分相同的项归为一组。
Step 3: Add or subtract the coefficients only. | 第三步：只对系数进行加减。
Step 4: Keep the variable part unchanged and write the simplified answer. | 第四步：保持变量部分不变，写出化简后的答案。
. The x terms give 5x − 2x = 3x, and the y terms give 3y + 4y = 7y. The answer is 3x + 7y.
例 1：化简 5x + 3y − 2x + 4y。把 x 项和 y 项分别分组。x 项得到 5x − 2x = 3x，y 项得到 3y + 4y = 7y。答案是 3x + 7y。
9. The a terms give 8a − 3a = 5a, and the constants give 5 − 9 = −4. The answer is 5a − 4.
例 2：化简 8a + 5 − 3a − 9。a 项得到 8a − 3a = 5a，常数项得到 5 − 9 = −4。答案是 5a − 4。
e x² terms give 4x² − x² = 3x², and the x terms give 3x + 6x = 9x. The answer is 3x² + 9x.
例 3：化简 4x² + 3x − x² + 6x。x² 项得到 4x² − x² = 3x²，x 项得到 3x + 6x = 9x。答案是 3x² + 9x。
5. Combining Like Terms with Brackets | 含括号的合并同类项
When an expression contains brackets, expand first, then collect. This order is not optional: you cannot combine a term inside a bracket with a term outside it until the bracket is gone.
当表达式含有括号时，先展开，再合并。这个顺序不是可选的：在括号消失之前，你不能把括号内的项与括号外的项合并。
Expanding uses the distributive law:
展开使用分配律：
a(b + c) = ab + ac
So 3(2x − 1) = 6x − 3, and 2(x + 4) = 2x + 8. Adding these gives 6x − 3 + 2x + 8 = 8x + 5.
因此 3(2x − 1) = 6x − 3，2(x + 4) = 2x + 8。把它们相加得到 6x − 3 + 2x + 8 = 8x + 5。
nd −(2a − 5) = −2a + 5. This single rule causes more lost marks than any other in algebra.
括号前的负号会翻转括号内每一项的符号。因此 −(x + 3) = −x − 3，−(2a − 5) = −2a + 5。这一条规则造成的失分比代数中任何其他规则都多。
. Collecting gives 3x + 11. Notice that −3 × (−1) = +3, which is where most students slip.
现在试试 2(3x + 4) − 3(x − 1)。展开得到 6x + 8 − 3x + 3。合并后得到 3x + 11。注意 −3 × (−1) = +3，这正是大多数学生出错的地方。
6. Powers and Multiple Variables | 幂与多变量
o exponents must match exactly. 4x² + 3x² = 7x², but 4x² + 3x cannot be simplified at all.
只有完全相同的变量部分才能合并，因此指数必须完全一致。4x² + 3x² = 7x²，但 4x² + 3x 根本无法化简。
When several letters appear, compare the whole variable part. 5a²b and 2ab² look similar but are unlike terms, because in the first the a is squared and in the second the b is squared.
当出现多个字母时，要比较整个变量部分。5a²b 与 2ab² 看起来很相似，但它们是不同类项，因为前者的 a 是平方，后者的 b 是平方。
Here is the key rule to keep straight: when you multiply powers of the same letter you add the indices, but when you combine like terms you add the coefficients and leave the index alone.
需要牢记的关键规则是：同底数幂相乘时指数相加，而合并同类项时是系数相加、指数保持不变。
x² × x³ = x⁵ | x² + x³ cannot be combined
4x², while 2x² × 3x² = 6x⁴. The two lines look alike but mean completely different things.
因此 2x² + 3x² − x² = 4x²，而 2x² × 3x² = 6x⁴。这两行看起来相似，含义却完全不同。
7. Fractions, Decimals and Negative Coefficients | 分数、小数与负系数
of integers. To combine ½x + ¼x, write both with denominator 4 to get 2⁄4x + 1⁄4x = 3⁄4x.
分数系数的处理方式完全相同，只是把整数相加换成分数相加。要合并 ½x + ¼x，把两者都化成以 4 为分母，得到 2⁄4x + 1⁄4x = 3⁄4x。
It is usually safer to keep fractions throughout rather than converting to decimals, because decimals such as 0.333… are rounded and can lose accuracy in later steps.
通常保留分数比化成小数更安全，因为像 0.333… 这样的小数是四舍五入的结果，会在后续步骤中损失精度。
rules. So −3x + x = −2x, and 5x − 8x = −3x. When the result is −1x, write it simply as −x.
负系数遵循普通的有理数运算规则。因此 −3x + x = −2x，5x − 8x = −3x。当结果为 −1x 时，直接写成 −x。
A common exam request is 'write down the coefficient of x'. For 6 − 4x + 2y the coefficient of x is −4, not 4, since the sign travels with the term.
考试中常见的要求是"写出 x 的系数"。对于 6 − 4x + 2y，x 的系数是 −4 而不是 4，因为符号随项一起移动。
8. Checking Your Answer by Substitution | 用代入法检验答案
Substitution is the fastest way to prove your simplification is correct. Choose simple numbers for the variables, evaluate the original expression, then evaluate your answer. The two values must be equal.
代入法是证明化简结果正确的最快方式。为变量选取简单的数值，先计算原表达式的值，再计算你答案的值。两个值必须相等。
12 = 27, and the simplified form gives 6 + 21 = 27. They match, so the answer is correct.
以 5x + 3y − 2x + 4y 为例，我们把它化简为 3x + 7y。令 x = 2、y = 3。原式给出 10 + 9 − 4 + 12 = 27，化简式给出 6 + 21 = 27。两者一致，说明答案正确。
If the two values differ, you have either misread a sign or combined two unlike terms. Go back to the grouping step and check the expAansion of any brackets.
如果两个值不相等，说明你要么看错了符号，要么把两个不同类项合并了。回到分组那一步，并检查括号的展开过程。
9. Common Mistakes and Exam Traps | 常见错误与考试陷阱
Most lost marks in this topic come from a small number of predictable errors. Learning to recognise them is worth more than doing ten extra questions.
这个知识点上的失分大多来自少数几个可预测的错误。学会识别它们，比多做十道题更有价值。
| Mistake | 错误 | Correct | 正确 |
|---|---|
| 3x + 2y = 5xy | 3x + 2y stays as it is | 3x + 2y 保持不变 |
| x + x = x² | x + x = 2x |
| −(x + 3) = −x + 3 | −(x + 3) = −x − 3 |
| 4x² + 3x = 7x³ | Cannot be combined | 无法合并 |
| 2(3x + 4) = 6x + 4 | 2(3x + 4) = 6x + 8 |
Another trap is forgetting that a term written without a number still has a coefficient of 1. Also watch out for question wording such as 'simplify' (collect terms) versus 'expand' (remove brackets) - you often need to do both.
另一个陷阱是忘记：没有写数字的项系数仍然是 1。同时要留意题目用词，"化简"（合并同类项）与"展开"（去掉括号）含义不同——很多时候你需要两者都做。
10. Using It in Equations and Formulae | 在方程与公式中的应用
Solving a linear equation is really just combining like terms on each side, then moving terms across the equals sign. Collect on both sides first, then rearrange.
解一元一次方程实际上就是先在等号两边各自合并同类项，再移项。先两边各自合并，然后再重新排列。
to get 4x − 4 = 12, then add 4 to get 4x = 16, so x = 4. Always substitute back to check.
来看 7x − 4 = 3x + 12。两边同时减去 3x 得 4x − 4 = 12，再加 4 得 4x = 16，所以 x = 4。务必代回原方程检验。
ject of y = 3x + 2z, subtract 2z from both sides and divide by 3, giving x = (y − 2z) ÷ 3.
同样的技能也是变换公式的基础。要把 y = 3x + 2z 变成以 x 为主元，两边同时减去 2z 再除以 3，得到 x = (y − 2z) ÷ 3。
des 3x + 1 and x + 4 is P = 2(3x + 1) + 2(x + 4) = 8x + 10 after expanding and collecting.
几何题中它无处不在。一个边长为 3x + 1 和 x + 4 的矩形，其周长为 P = 2(3x + 1) + 2(x + 4) = 8x + 10，展开并合并后得到。
11. Exam-Style Practice Questions | 考试风格练习题
Work through these without a calculator, then check your answers below. They cover every skill in this article.
请在不使用计算器的情况下完成以下题目，然后对照下面的答案。它们涵盖了本文的每一项技能。
Q1. Simplify 9x + 4 − 5x + 7. | 化简 9x + 4 − 5x + 7。
Q2. Simplify 3(2a − 5) − 2(a − 4). | 化简 3(2a − 5) − 2(a − 4)。
Q3. Simplify 6m² + 2m − 4m² + 7m. | 化简 6m² + 2m − 4m² + 7m。
Q4. Simplify 5xy − 3x + 2yx − 7x. | 化简 5xy − 3x + 2yx − 7x。
Q5. Simplify ½p + ⅔p − ¼p. | 化简 ½p + ⅔p − ¼p。
Q6. Solve 8x − 3 = 5x + 9. | 解方程 8x − 3 = 5x + 9。
x. A5 uses denominator 12, giving 6⁄12 + 8⁄12 − 3⁄12 = 11⁄12p. A6 gives 3x = 12, so x = 4.
答案：A1 = 4x + 11。A2 = 6a − 15 − 2a + 8 = 4a − 7。A3 = 2m² + 9m。A4 = 7xy − 10x。A5 用以 12 为分母通分，得到 6⁄12 + 8⁄12 − 3⁄12 = 11⁄12p。A6 得到 3x = 12，所以 x = 4。
12. Quick Revision Summary | 速记总结
If you remember nothing else, remember these five points. They account for virtually every mark available on this topic.
如果你什么都记不住，至少记住下面五点。它们几乎涵盖了本知识点的全部得分点。
A term includes the sign in front of it. | 一个项包含它前面的符号。
Like terms must have identical variable parts and identical indices. | 同类项必须变量部分相同、指数也相同。
Combine coefficients only; never change the variable part. | 只合并系数，绝不改变变量部分。
Expand brackets before collecting, and flip signs after a minus. | 先展开括号再合并，负号后要变号。
Check by substituting simple numbers. | 用简单数值代入检验。
The whole method compresses into a single line:
整个方法可以压缩成一行：
same letters + same powers → add the coefficients, keep the letters
Practise ten minutes a day on mixed expressions with brackets, powers and fractions, and combining like terms will become automatic - which frees your attention for the real challenge of the question.
每天花十分钟练习含括号、幂和分数的混合表达式，合并同类项就会变得自动化——这样你就能把注意力留给题目真正的难点。
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