📚 Collecting Like Terms | 合并同类项
In algebra, expressions often contain several terms that can be simplified by combining those that are alike. This process is called collecting like terms. It is one of the most fundamental skills you need for solving equations, rearranging formulas, and simplifying algebraic expressions. Mastering this topic will save time and prevent errors in every subsequent area of mathematics.
在代数中，表达式常常包含多个可以合并的同类项。这个过程称为合并同类项。这是解方程、重排公式和化简代数表达式所需的最基本技能之一。掌握这一主题将在后续各个数学领域中节省时间并避免错误。
1. What Are Terms? | 什么是“项”？
A term is a single mathematical expression made up of numbers, variables, or the product of numbers and variables. For example, in the expression 3x + 2y - 5, the terms are 3x, 2y, and -5. Terms are usually separated by plus or minus signs.
项是由数字、变量或数字与变量的乘积组成的单个数学表达式。例如，在表达式 3x + 2y - 5 中，项为 3x、2y 和 -5。项通常用加号或减号分隔。
- 3x is a term: 3 is the coefficient, x is the variable.
- 3x 是一个项：3 是系数，x 是变量。
- 2y is a term: 2 is the coefficient, y is the variable.
- 2y 是一个项：2 是系数，y 是变量。
- -5 is a constant term because it has no variable.
- -5 是常数项，因为它不含变量。
2. What Are 'Like Terms'? | 什么是“同类项”？
Like terms are terms that have exactly the same variable(s) raised to the same power(s). The coefficient can be different, but the variable part must match. For example, 4x and 7x are like terms because both contain x. However, 4x and 4x² are not like terms because the powers of x differ.
同类项是指具有完全相同变量和相同指数的项。系数可以不同，但变量部分必须匹配。例如，4x 和 7x 是同类项，因为它们都包含 x。但是，4x 和 4x² 不是同类项，因为 x 的指数不同。
Only like terms can be added or subtracted. This is the key idea behind collecting like terms.
只有同类项才能进行加减运算。这是合并同类项的核心思想。
3. Identifying Like Terms | 识别同类项
To identify like terms, look at both the variable letters and their exponents. The order of multiplication does not matter, so ab and ba are like terms because multiplication is commutative.
要识别同类项，需要同时观察变量字母及其指数。乘法顺序无关紧要，因此 ab 和 ba 是同类项，因为乘法满足交换律。
- 5a and -2a are like terms.
- 5a 和 -2a 是同类项。
- 3x² and 5x are not like terms (different exponents).
- 3x² 和 5x 不是同类项（指数不同）。
- 2xy and 7yx are like terms (same variables, same powers).
- 2xy 和 7yx 是同类项（变量相同，指数相同）。
Remember: a variable without an explicit exponent has an exponent of 1. For instance, x is the same as x¹.
记住：没有明确写出指数的变量，其指数为 1。例如，x 等同于 x¹。
4. The Rule for Collecting Like Terms | 合并同类项的法则
When collecting like terms, add or subtract the coefficients and keep the variable part unchanged. For example:
合并同类项时，将系数相加或相减，变量部分保持不变。例如：
3x + 5x = (3 + 5)x = 8x
9y - 4y = (9 - 4)y = 5y
The variable part (x or y) is not altered during the operation. Only the numerical coefficients are combined.
变量部分（x 或 y）在运算过程中不改变，只合并数值系数。
5. Adding and Subtracting Like Terms | 同类项的加减
For addition and subtraction, always work with signed numbers. For instance, when combining 7a, -2a, and 3a, treat the coefficients as +7, -2, and +3.
进行加减运算时，始终使用带符号的数。例如，合并 7a、-2a 和 3a 时，将系数视为 +7、-2 和 +3。
7a - 2a + 3a = (7 - 2 + 3)a = 8a
If there are multiple like terms with different signs, combine them carefully.
如果有多个符号不同的同类项，要小心合并。
6. Collecting Terms with Different Variables | 合并含有不同变量的项
Terms with different variables are not like terms and cannot be combined. For example, 2x + 3y cannot be simplified to 5xy. Each type of variable term must be collected separately.
不同变量的项不是同类项，不能合并。例如，2x + 3y 不能化简为 5xy。每种变量项必须分别合并。
4x + 2y - x + 3y = (4x - x) + (2y + 3y) = 3x + 5y
Notice that x-terms are grouped together, and y-terms are grouped together.
注意，x 项分为一组，y 项分为另一组。
7. Combining Constant Terms | 合并常数项
Constant terms are like terms because they have no variable. All constants can be combined into a single term.
常数项因为没有变量，所以彼此是同类项。所有常数可以合并为一个项。
5 + 3x - 2 + 7x = (5 - 2) + (3x + 7x) = 3 + 10x
Always write the constant term last in a simplified expression.
在化简后的表达式中，通常把常数项写在最后。
8. Using the Distributive Law | 使用分配律
nd brackets before collecting like terms. The distributive law states: a(b + c) = ab + ac.
有时需要先去掉括号，再合并同类项。分配律指出：a(b + c) = ab + ac。
2(x + 3) = 2·x + 2·3 = 2x + 6
After expanding, you can then collect like terms across the whole expression.
展开后，就可以在整个表达式中合并同类项了。
9. Simplifying Expressions with Brackets | 化简含括号的表达式
When an expression contains more than one bracket, expand each bracket first, then simplify. For example:
当表达式包含多个括号时，先展开每个括号，再化简。例如：
3(x + 2) + 2(2x - 1) = 3x + 6 + 4x - 2
= (3x + 4x) + (6 - 2) = 7x + 4
Be careful with negative signs outside brackets: -2(x - 3) = -2x + 6.
注意括号外的负号：-2(x - 3) = -2x + 6。
10. Common Mistakes to Avoid | 避免常见错误
Many students make errors when collecting like terms. Here are the most common pitfalls:
许多学生在合并同类项时会出错。以下是最常见的陷阱：
- Combining non-like terms, such as 2x + 3y = 5xy (wrong).
- 合并非同类项，例如 2x + 3y = 5xy（错误）。
- Forgetting to keep the variable part unchanged: 3x + 4x = 7x² (wrong).
- 忘记保持变量部分不变：3x + 4x = 7x²（错误）。
- Dropping negative signs: 2x - 5x = 3x (wrong, should be -3x).
- 漏掉负号：2x - 5x = 3x（错误，应为 -3x）。
Always double-check that the variables and exponents match before adding.
在相加之前，务必再次确认变量和指数是否匹配。
11. Worked Examples | 例题解析
Let us apply the rules to some typical exam questions.
让我们将规则应用到一些典型考题中。
Example 1 / 例1: Simplify 5x + 3y - 2x + y.
5x + 3y - 2x + y = (5x - 2x) + (3y + y) = 3x + 4y
Example 2 / 例2: Simplify 4a² + 3a - a² + 2a.
4a² + 3a - a² + 2a = (4a² - a²) + (3a + 2a) = 3a² + 5a
Example 3 / 例3: Expand and simplify 2(3x - 1) + 4(x + 2).
2(3x - 1) + 4(x + 2) = 6x - 2 + 4x + 8 = 10x + 6
12. Practice Questions | 练习题
Try these questions to test your understanding. Solutions are shown below each question.
尝试以下问题来测试你的理解。每个问题下方显示答案。
| Question / 问题 | Answer / 答案 |
| 1. 3x + 8x | 11x |
| 2. 7y - 3y + 2y | 6y |
| 3. 2a + 5b - a + 3b | a + 8b |
| 4. 5 + 2x - 3 + 6x | 2 + 8x |
| 5. 4(x - 2) + 3(2x + 1) | 10x - 5 |
13. Summary | 总结
Collecting like terms is the process of simplifying algebraic expressions by adding or subtracting terms that have the same variable and exponent. Remember to combine only the coefficients, keep the variable part unchanged, and deal with different variable groups separately. Expand any brackets first, and always be mindful of negative signs.
合并同类项是通过将具有相同变量和指数的项相加或相减来化简代数表达式的方法。记住只合并系数，保持变量部分不变，并分别处理不同变量组。先展开所有括号，并始终注意负号。
The ability to collect like terms quickly and accurately builds a strong foundation for algebra. Keep practising, and it will become second nature.
快速准确地合并同类项的能力为代数打下坚实基础。继续练习，这会成为本能反应。
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