📚 Collecting Like Terms | 合并同类项
In algebra, expressions often contain several terms that can be simplified. Collecting like terms is the process of adding or subtracting terms that have exactly the same variable part. This skill is essential for solving equations, expanding brackets, and rearranging formulae in IGCSE Mathematics.
在代数中，表达式常包含多个可以化简的项。合并同类项就是把变量部分完全相同的项相加或相减。这项技能对解方程、去括号和变形公式都很关键。
1. What Are Terms? | 什么是项
In algebra, an expression is built from terms separated by plus or minus signs. For example, 5x + 3y - 2 has three terms: 5x, 3y, and -2. A term can be a single number, a single variable, or a product of numbers and variables.
在代数中，表达式由用加号或减号分隔的项组成。例如，5x + 3y - 2 有三个项：5x、3y 和 -2。项可以是一个数、一个变量，或数与变量的乘积。
When we simplify an expression, we often collect terms that are alike. This makes the expression shorter and easier to use in later calculations.
化简表达式时，我们通常合并相同类型的项，使表达式更短、更容易用于后续计算。
2. Like Terms vs Unlike Terms | 同类项与不同类项
Like terms have exactly the same variable part, including the same powers. For example, 4x and -7x are like terms because both contain x to the power 1. However, 4x and 4x² are not like terms because their exponents are different. Constants are also like terms: 5 and -3 can be combined to give 2.
同类项必须具有完全相同的变量部分，包括相同的幂。例如，4x 和 -7x 是同类项，因为两者都含有 1 次方的 x。但 4x 和 4x² 不是同类项，因为它们的指数不同。常数也是同类项：5 和 -3 可以合并为 2。
Unlike terms cannot be combined by addition or subtraction. For instance, 3a and 3b remain separate, and 2x and 2y cannot be written as one single term.
不同类项不能通过加法或减法合并。例如，3a 和 3b 必须保持分离，2x 和 2y 也不能写成一个项。
3. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and check the letters and their powers. The coefficient is the number in front of the variable. The table below shows some examples.
识别同类项时，不需要看系数，只要检查字母及其幂。系数是变量前面的数。下表给出了一些例子。
| First term | Second term | Are they like? |
|---|---|---|
| 5x | -3x | Yes |
| 2a | 2b | No |
| x² | x | No |
| 7xy | -xy | Yes |
The table shows that coefficients do not matter: 5x and -3x are like terms. The variables and their powers decide whether terms can be combined.
上表说明系数并不重要：5x 和 -3x 是同类项。能否合并取决于变量及其幂。
4. The Basic Rule: Add the Coefficients | 基本规则：系数相加
To collect like terms, keep the variable part unchanged and add or subtract the coefficients only. In the expression 3x + 5x, the coefficients 3 and 5 add to give 8, so:
合并同类项时，保持变量部分不变，只把系数相加或相减。在 3x + 5x 中，系数 3 和 5 相加得 8，因此：
3x + 5x = 8x
= 5y. The variable y stays exactly the same; only the coefficients 9 and 4 are subtracted.
另一个例子是 9y - 4y = 5y。变量 y 保持不变，只有系数 9 和 4 相减。
5. Working with Negative Coefficients | 处理负系数
Signs are very important when collecting terms. For example, 7x - 2x gives 5x, because the coefficient of the second term is -2.
合并时符号非常重要。例如，7x - 2x 等于 5x，因为第二项的系数是 -2。
7x - 2x = 5x
rm has a larger coefficient than the positive term, the result is negative: 3p - 8p = -5p.
如果负项的系数大于正项的系数，结果就是负的：3p - 8p = -5p。
ive terms are combined, add their coefficients and keep the negative sign: -4y - 2y = -6y.
当两个负项合并时，应把系数相加并保留负号：-4y - 2y = -6y。
6. Collecting Terms with Different Variables | 含不同变量的合并
In an expression such as 2x + 3y + 5x - y, collect the x terms and the y terms separately. The x terms are 2x and 5x, giving 7x. The y terms are 3y and -y, giving 2y.
在 2x + 3y + 5x - y 这样的表达式中，应分别合并 x 项和 y 项。x 项为 2x 和 5x，得到 7x；y 项为 3y 和 -y，得到 2y。
2x + 3y + 5x - y = 7x + 2y
Do not combine x and y terms because they represent different unknown quantities.
不要把 x 项和 y 项合并，因为它们代表不同的未知量。
7. Terms with Powers | 含幂的项
Power terms can only be combined when both the variable and its exponent are the same. For example, x² and x are different, so x² + 3x cannot be simplified further.
只有变量和指数都相同，幂项才能合并。例如，x² 和 x 不同，所以 x² + 3x 不能继续化简。
x² + 3x² = 4x²
+ 3x² + 4x, collect x² terms and x terms separately: (5x² + 3x²) + (-2x + 4x) = 8x² + 2x.
在 5x² - 2x + 3x² + 4x 中，应分别合并 x² 项和 x 项：(5x² + 3x²) + (-2x + 4x) = 8x² + 2x。
8. Expanding Brackets Before Collecting | 先展开括号再合并
When brackets appear, expand them first using the distributive law, then collect like terms. For example:
当出现括号时，应先使用分配律展开括号，然后再合并同类项。例如：
x - 5). First, 3(x + 2) = 3x + 6 and 2(x - 5) = 2x - 10. Then collect x terms and numbers:
展开 3(x + 2) + 2(x - 5)。首先，3(x + 2) = 3x + 6，2(x - 5) = 2x - 10。然后合并 x 项和数字项：
3(x + 2) + 2(x - 5) = 3x + 6 + 2x - 10 = 5x - 4
Always expand fully before collecting. If you collect terms inside brackets incorrectly, the final answer will be wrong.
合并前一定要充分展开括号。如果括号内的项合并不正确，最终答案就会出错。
9. Fractions and Decimals in Coefficients | 分数与小数系数
Fractions and decimals follow the same rule: add or subtract the coefficients. For example:
分数和小数遵循同样的规则：只对系数进行加减。例如：
1/2 x + 3/2 x = 4/2 x = 2x
example is 0.3y + 1.7y = 2.0y. Since 2.0y is the same as 2y, the simplified answer is 2y.
另一个例子是 0.3y + 1.7y = 2.0y。因为 2.0y 与 2y 相同，所以化简结果为 2y。
to a common denominator before adding. For example, 1/2 x + 1/3 x = 3/6 x + 2/6 x = 5/6 x.
当分母不同时，应先通分再相加。例如，1/2 x + 1/3 x = 3/6 x + 2/6 x = 5/6 x。
10. Common Mistakes | 常见错误Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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