📚 Collecting Like Terms | 合并同类项
In algebra, expressions often contain several terms that can be simplified. Collecting like terms is one of the most fundamental skills for IGCSE Mathematics: it makes expressions shorter, neater, and ready for solving equations.
在代数中，表达式常常包含多个可以化简的项。合并同类项是 IGCSE 数学最基础的技能之一，它能让表达式更短、更整洁，也为解方程做好准备。
1. What is a term? | 什么是项
An algebraic expression is made of terms. A term can be a number, a variable, or a product of numbers and variables. For example, in 3x + 2y - 5, the terms are 3x, 2y, and -5.
代数表达式由项组成。项可以是一个数、一个变量，或者数与变量的乘积。例如，在 3x + 2y - 5 中，项分别是 3x、2y 和 -5。
The number part of a term is called the coefficient. In -7a, the coefficient is -7. A term with no variable, such as 5 or -3, is called a constant.
项中的数字部分叫作系数。在 -7a 中，系数是 -7。没有变量的项（如 5 或 -3）叫作常数项。
2. What are like terms? | 什么是同类项
Like terms have exactly the same variable part, including the same powers. The coefficients can be different. For example, 3x and -7x are like terms because both contain x to the power 1.
同类项具有完全相同的变量部分，包括相同的幂。系数可以不同。例如，3x 和 -7x 是同类项，因为它们都含有 x 的一次幂。
Similarly, 2x² and 5x² are like terms; 4xy and -xy are like terms. However, 3x and 3x² are not like terms because the powers of x are different.
同样，2x² 和 5x² 是同类项；4xy 和 -xy 是同类项。但 3x 和 3x² 不是同类项，因为 x 的幂不同。
| Expression 表达式 | Like terms? 是否同类项 | Reason 原因 |
|---|---|---|
| 3x and 5x | Yes | Same variable x |
| 2x² and -4x² | Yes | Same power x² |
| 4xy and 2yx | Yes | Multiplication order does not matter |
| 3x and 3y | No | Different variables |
| 5a and 5a² | No | Different powers |
3. Identifying like terms | 识别同类项
To identify like terms, ignore the coefficient and focus on the variable part. Circle or underline terms with the same variable letters and powers.
识别同类项时，先忽略系数，重点看变量部分。把具有相同字母和相同幂的项圈出或划线标记。
- 2x, -5x, 8x are like terms. 2x、-5x、8x 是同类项。
- 2x and 2x² are not like terms. 2x 和 2x² 不是同类项。
- 3ab and -ab are like terms. 3ab 和 -ab 是同类项。
A useful method is to rewrite the expression so that like terms appear next to each other before simplifying.
一个实用的方法是先重新排列表达式，使同类项靠在一起，再进行化简。
4. The golden rule: add coefficients only | 黄金法则：只合并系数
You can only add or subtract like terms. To combine them, add or subtract the coefficients only and keep the variable part exactly the same.
只有同类项才能相加或相减。合并时，只把系数相加或相减，变量部分保持不变。
3x + 5x = (3 + 5)x = 8x
of x plus 5 lots of x give 8 lots of x. The same idea works for subtraction: 9a - 4a = 5a.
这是因为 3 个 x 加上 5 个 x 等于 8 个 x。减法同理：9a - 4a = 5a。
7y - 2y = 5y
5. Positive and negative like terms | 正数与负数同类项
When terms have negative coefficients, pay close attention to signs. For example, 2x - 7x means 2x + (-7x), so the coefficients add to -5.
当项有负系数时，要特别注意符号。例如，2x - 7x 可以看作 2x + (-7x)，所以系数相加为 -5。
2x - 7x = -5x
Another example: -4a + 9a - 2a = (-4 + 9 - 2)a = 3a.
另一个例子：-4a + 9a - 2a = (-4 + 9 - 2)a = 3a。
Always remember that the sign in front of a term belongs to that term. Moving terms around must keep their signs attached.
请始终记住：项前面的符号属于该项。重新排列项时，符号必须跟着项一起移动。
6. Collecting like terms with powers | 含幂的同类项合并
Only terms with the same variable and the same power can be combined. For example, x² and x² can be combined, but x² and x cannot.
只有变量相同且幂相同的项才能合并。例如，x² 和 x² 可以合并，但 x² 和 x 不能。
4x² - x² + 3x + 2x = 3x² + 5x
Notice that the x² terms and the x terms are collected separately. Do not change the power when adding or subtracting like terms.
注意，x² 项和 x 项要分别合并。合并同类项时不要改变幂。
x² + x² = 2x², not x⁴
7. Terms with more than one variable | 含多个变量的项
Terms with more than one variable are like terms only if every variable and every power match. For example, 3xy and -5xy are like terms; 3xy and 3x²y are not.
含有多个变量的项，只有当每个变量和每个幂都完全相同时才是同类项。例如，3xy 和 -5xy 是同类项；3xy 和 3x²y 不是。
2xy + 5xy - xy = 6xy
Also, xy and yx are like terms because multiplication is commutative: xy = yx.
此外，xy 和 yx 是同类项，因为乘法满足交换律：xy = yx。
3a²b + 4a²b - 2a²b = 5a²b
8. Brackets before collecting | 先去括号再合并
If an expression contains brackets, expand them first before collecting like terms. Multiply each term inside the bracket by the factor outside.
如果表达式含有括号，先展开括号再合并同类项。把括号外的因数乘以括号内的每一项。
3(x + 2y) + 2(x - y) = 3x + 6y + 2x - 2y = 5x + 4y
Be careful with a minus sign before a bracket: it changes the sign of every term inside the bracket.
括号前有负号时要小心：它会改变括号内每一项的符号。
2(3a - b) - (a + 2b) = 6a - 2b - a - 2b = 5a - 4b
9. Common mistakes and how to avoid them | 常见错误与避免方法
Mistake 1: writing x² + x² = x⁴. The power stays the same; only the coefficient changes.
错误 1：把 x² + x² 写成 x⁴。幂保持不变，只有系数相加。
Mistake 2: confusing addition with multiplication. x + x = 2x, but x × x = x².
错误 2：混淆加法与乘法。x + x = 2x，但 x × x = x²。
Mistake 3: forgetting the sign before a term. 3x - 2x = x, not 5x.
错误 3：忘记项前面的符号。3x - 2x = x，而不是 5x。
Mistake 4: trying to combine unlike terms. 3x + 4y cannot be simplified further.
错误 4：试图合并非同类项。3x + 4y 不能进一步化简。
10. Worked examples | 例题精讲
Example 1: Simplify 6m + 3n - 2m + 5n.
例 1：化简 6m + 3n - 2m + 5n。
(6m - 2m) + (3n + 5n) = 4m + 8n
Example 2: Simplify 4p² + 3p - 2p² + 7p.
例 2：化简 4p² + 3p - 2p² + 7p。
(4p² - 2p²) + (3p + 7p) = 2p² + 10p
Example 3: Simplify 5xy - 2x + 3xy + x.
例 3：化简 5xy - 2x + 3xy + x。
(5xy + 3xy) + (-2x + x) = 8xy - x
Example 4: Simplify 2(3x - y) + 3(x + 2y).
例 4：化简 2(3x - y) + 3(x + 2y)。
6x - 2y + 3x + 6y = 9x + 4y
11. Practice questions | 练习题
Simplify each expression, then check your answer in the table.
化简下列各式，然后在表格中核对答案。
| Question 题目 | Answer 答案 |
|---|---|
| 7x + 3x - 2x | 8x |
| 5a - 9a + 4b + 2b | -4a + 6b |
| 3x² + 2x - x² +
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