📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, collecting like terms is one of the first essential simplification skills. When an expression contains several terms that share exactly the same variable part, you can add or subtract their coefficients to make the expression shorter and easier to use.
在 IGCSE 代数中，合并同类项是最基础的化简技能之一。当表达式中的若干项含有完全相同的变量部分时，你可以将它们的系数相加或相减，使表达式更短、更易使用。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable letters and exactly the same powers. For example, 3x and 5x are like terms because both contain the variable x raised to the power 1. Constants such as 4 and −7 are also like terms because they have no variable part.
同类项是指含有完全相同的字母以及完全相同幂次的项。例如，3x 和 5x 是同类项，因为两者都含有变量 x 的一次幂。像 4 和 −7 这样的常数项也是同类项，因为它们没有变量部分。
The coefficients can be different; that does not affect whether terms are like terms. What matters is the variable part only.
系数可以不同，这不会影响它们是否为同类项。关键在于变量部分是否相同。
2. Identifying Like Terms | 识别同类项
To identify like terms, ignore the number in front and look only at the letters and their powers. For instance, 7y, −2y and 0.5y are like terms, while 7y and 7y² are not like terms because the powers differ.
要识别同类项，可以忽略前面的数字，只看字母及其幂。例如，7y、−2y 和 0.5y 是同类项，而 7y 和 7y² 不是同类项，因为幂次不同。
| Expression | Like terms? |
| 4x and 9x | Yes |
| 3a and 3b | No |
| 6x² and 2x² | Yes |
| 5xy and 5x | No |
This table shows that both the letter and the power must match for terms to be collected.
上表说明，只有字母和幂次都相同，项才能被合并。
3. Why We Collect Like Terms | 为什么要合并同类项
Collecting like terms simplifies an expression so that it contains fewer terms. A shorter expression is easier to substitute into, easier to solve with, and easier to compare with other expressions.
合并同类项可以化简表达式，使其包含更少的项。更短的表达式在代入数值、求解以及与其他表达式比较时都更方便。
For example, x + x + x + x looks long, but it can be written as 4x. This is a basic form of algebraic simplification.
例如，x + x + x + x 看起来很长，但它可以写成 4x。这是代数化简的基本形式之一。
4. Adding and Subtracting Like Terms | 合并时的加减运算
rt unchanged. For example, 3p + 5p means (3 + 5)p = 8p, and 10q − 4q means (10 − 4)q = 6q.
合并同类项时，只需将系数相加或相减，变量部分保持不变。例如，3p + 5p 等于 (3 + 5)p = 8p，而 10q − 4q 等于 (10 − 4)q = 6q。
3p + 5p = 8p
10q − 4q = 6q
Remember that 1x is usually written as x, and −1x is written as −x.
请记住，1x 通常写成 x，而 −1x 写作 −x。
5. Combining More Than Two Terms | 合并两个以上的项
An expression may contain several groups of like terms. Work group by group, and then write the simplified groups together. For example, 2x + 3y + 5x − y can be rearranged so that x terms and y terms are handled separately.
一个表达式可能包含多组同类项。可以逐组处理，然后把化简后的各组写在一起。例如，2x + 3y + 5x − y 可以重新排列，使 x 项和 y 项分别处理。
2x + 5x + 3y − y = 7x + 2y
Rearranging is allowed because addition is commutative; changing the order of the terms does not change the value.
重新排列是允许的，因为加法具有交换性；改变项的顺序不会改变原式的值。
6. Constants and Single Variables | 常数项与单个变量
Constant terms are numbers with no variable. Collect all constants together in the same way as variable terms. For example, 3a + 8 − 2 + 5a becomes 8a + 6 after collecting 3a with 5a and 8 with −2.
常数项是不含变量的数字。将常数项像变量项一样合并。例如，3a + 8 − 2 + 5a 在合并 3a 与 5a、8 与 −2 后得到 8a + 6。
3a + 8 − 2 + 5a = 8a + 6
It is useful to underline or circle like terms first, so no term is missed.
先用下划线或圆圈标出同类项很有帮助，这样就不会漏掉任何一项。
7. Terms with Powers | 含幂的项
Only terms with exactly the same power can be collected. For instance, 2x² and 5x² are like terms, so they give 7x². However, 2x² and 3x are not like terms because one has x² and the other has x.
只有幂次完全相同的项才能合并。例如，2x² 和 5x² 是同类项，合并后得到 7x²。但 2x² 和 3x 不是同类项，因为一个是 x²，另一个是 x。
2x² + 5x² = 7x²
2x² + 3x cannot be simplified further as a single term
In questions, make sure you do not combine x, x² and x³; they are separate groups.
解题时务必注意，不要把 x、x² 和 x³ 混在一起；它们是不同的组。
8. Negative Coefficients | 负系数
p the minus sign with the coefficient. For example, 7x − 3x − 2x equals (7 − 3 − 2)x = 2x.
负系数的处理规则与正系数相同。做减法时，减号要跟着系数一起走。例如，7x − 3x − 2x 等于 (7 − 3 − 2)x = 2x。
t zero, the term disappears. For example, 4y − 4y = 0, and the answer is simply 0, not 0y.
如果结果为系数零，该项就消失了。例如 4y − 4y = 0，答案就是 0，而不是 0y。
9. Expressions with Brackets | 含有括号的表达式
Before collecting like terms, you often need to remove brackets by using the distributive law or by simplifying signs. For example, 3 + 2(x + 1) becomes 3 + 2x + 2, which then simplifies to 2x + 5.
在合并同类项之前，通常需要先使用分配律或化简符号来去掉括号。例如，3 + 2(x + 1) 先变为 3 + 2x + 2，然后化简为 2x + 5。
3 + 2(x + 1) = 3 + 2x + 2 = 2x + 5
When a minus sign sits before a bracket, change the signs of every term inside the bracket before collecting.
当括号前是减号时，先去括号并改变括号内每一项的符号，然后再合并同类项。
10. Common Mistakes | 常见错误
s is incorrect because x and y are different variables and cannot be combined into one term.
一个常见错误是将不同类项相加，例如把 3x + 2y 写成 5xy。这是不正确的，因为 x 和 y 是不同的变量，不能合成一项。
ple writing 2x + 3x = 5x². The correct answer is 5x, since the variable x does not change.
另一个常见错误是在合并时改变了变量部分，例如将 2x + 3x 写成 5x²。正确答案是 5x，因为变量 x 不变。
Also remember that x² and x are not the same thing; 2x² + 3x cannot be combined.
还要记住 x² 和 x 是不同的；2x² + 3x 不能合并。
11. Worked Examples | 例题精讲
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