📚 Cambridge KS3 Mathematics: Simplifying Algebraic Expressions | 剑桥初中数学：代数式化简与合并同类项
In Cambridge KS3 Mathematics, simplifying algebraic expressions is one of the most important skills you will build before studying equations, formulae and graphs. It allows you to rewrite a long or messy expression in a shorter, clearer form without changing its value. This article explains what terms are, how to recognise like terms, how to collect them correctly, and how to avoid common errors.
在剑桥初中数学中，代数式化简是学习方程、公式和图像之前必须掌握的重要技能之一。它可以帮助你把一个较长或较乱的代数式写成更短、更清晰的形式，同时不改变原式的值。本文将讲解什么是项、如何识别同类项、如何正确合并同类项，以及如何避免常见错误。
1. Algebraic Terms and Coefficients | 代数项与系数
An algebraic term is a number, a variable, or a number and variable multiplied together, such as 3x, −5y, 7, or a². The number part of a term is called the coefficient. In the term 4x, the coefficient is 4, and the variable is x.
代数项是指一个数、一个变量，或者一个数与变量相乘得到的结果，例如 3x、−5y、7 或 a²。一个项中的数字部分称为系数。在 4x 这个项中，系数是 4，变量是 x。
When we write terms in an expression, they are usually separated by addition or subtraction signs. For example, in 3x + 5y − 2, there are three terms: 3x, 5y and −2.
当我们写出一个代数式时，各项通常由加号或减号分隔。例如，在 3x + 5y − 2 中，共有三个项：3x、5y 和 −2。
2. Recognising Like Terms | 识别同类项
Like terms are terms that have exactly the same variable part raised to the same power. Only like terms can be added or subtracted directly. For example, 3x and 7x are like terms, while 3x and 3x² are not like terms because the powers of x are different.
同类项是指变量的部分完全相同，并且变量的指数也相同的项。只有同类项才能直接相加或相减。例如，3x 和 7x 是同类项，而 3x 和 3x² 不是同类项，因为 x 的指数不同。
Here are some examples of like and unlike terms: 2a and 5a are like; 4xy and −7xy are like; 6m² and 9m² are like; but 5p and 5q are not like, and 8n and 8n³ are not like.
下面是一些同类项和非同类项的例子：2a 和 5a 是同类项；4xy 和 −7xy 是同类项；6m² 和 9m² 是同类项；但 5p 和 5q 不是同类项，8n 和 8n³ 也不是同类项。
3. The Rule for Collecting Like Terms | 合并同类项的法则
ts while keeping the variable part exactly the same. This is because 3x + 5x means three x's plus five x's, which makes eight x's, so 3x + 5x = 8x.
合并同类项是指将系数相加或相减，而变量部分保持不变。这是因为 3x + 5x 表示三个 x 加五个 x，总共得到八个 x，所以 3x + 5x = 8x。
3x + 5x = 8x
7y − 2y = 5y
The same rule applies to subtraction. Since 7y means seven y's and 2y means two y's, their difference is five y's, giving 5y.
同样的法则也适用于减法。因为 7y 表示七个 y，2y 表示两个 y，它们的差是五个 y，因此得到 5y。
4. Working with Positive Coefficients | 正系数的运算
ll three terms are like terms, add the coefficients: 6 + 2 + 3 = 11, so the answer is 11a.
当所有系数都是正数时，合并同类项非常简单。例如，化简 6a + 2a + 3a。由于这三项都是同类项，只需将系数相加：6 + 2 + 3 = 11，所以答案是 11a。
6a + 2a + 3a = 11a
e, 2x² + 4x² = 6x², but 2x² + 4x cannot be combined because the powers of x are different.
只要指数相同，含有指数的同类项也可以合并。例如，2x² + 4x² = 6x²，但 2x² + 4x 不能合并，因为 x 的指数不同。
5. Working with Negative Coefficients | 负系数的运算
coefficient of the second term is −3. Adding them gives 9 + (−3) = 6, so the result is 6k.
当涉及负号时，可以把符号看作系数的一部分。化简 9k − 3k。第一项的系数是 9，第二项的系数是 −3。将它们相加得到 9 + (−3) = 6，所以结果是 6k。
9k − 3k = 6k
If the coefficients are both negative, add them as negative numbers. For example, simplify −4m − 5m. The coefficients are −4 and −5, so their sum is −9, giving −9m.
如果两个系数都是负数，就按负数相加。例如，化简 −4m − 5m。系数分别是 −4 和 −5，它们的和是 −9，因此得到 −9m。
6. Constants and Variable Terms | 常数项与含变量的项
A constant is a term with no variable, such as 3, −7 or 12. Constants are like terms with each other, but they cannot be combined with terms that contain variables. For example, in 4x + 3 + 2x + 5, the variable terms 4x and 2x combine to 6x, and the constants 3 and 5 combine to 8.
常数项是没有变量的项，例如 3、−7 或 12。常数项之间是同类项，但不能与含变量的项合并。例如，在 4x + 3 + 2x + 5 中，含变量的项 4x 和 2x 合并为 6x，常数项 3 和 5 合并为 8。
4x + 3 + 2x + 5 = 6x + 8
This is a common structure in Cambridge KS3 questions: group the x-terms together, group the constants together, and then write the simplified expression with the variable term first.
这是剑桥初中数学题中常见的结构：先把含 x 的项放在一起，再把常数项放在一起，然后将化简后的代数式写成变量项在前、常数项在后的形式。
7. Expressions with Several Variables | 含多个变量的代数式
When an expression contains more than one variable, collect each type of variable separately. For example, simplify 3a + 2b + 5a + 4b. The a-terms are 3a and 5a, which give 8a. The b-terms are 2b and 4b, which give 6b. The final expression is 8a + 6b.
当一个代数式中含有多个变量时，需要分别合并每一种变量。例如，化简 3a + 2b + 5a + 4b。含有 a 的项是 3a 和 5a，合并得到 8a。含有 b 的项是 2b 和 4b，合并得到 6b。最终代数式为 8a + 6b。
3a + 2b + 5a + 4b = 8a + 6b
If terms have mixed variables such as xy, they must have exactly the same combination of variables to be like terms. For example, 4xy and 9xy are like terms, but 4xy and 4x are not.
如果项中含有混合变量，例如 xy，那么它们必须具有完全相同的变量组合才是同类项。例如，4xy 和 9xy 是同类项，但 4xy 和 4x 不是同类项。
8. Expanding Before Collecting | 先展开再合并
Some expressions include brackets, such as 2(x + 3) + 4x. Before collecting like terms, you must expand the bracket. Use the distributive law: multiply the term outside the bracket by each term inside. Here, 2(x + 3) becomes 2x + 6.
有些代数式含有括号，例如 2(x + 3) + 4x。在合并同类项之前，必须先展开括号。使用分配律：用括号外的项分别乘以括号内的每一项。这里，2(x + 3) 展开为 2x + 6。
2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6
erms: 2x + 4x = 6x, and the constant remains 6. The fully simplified expression is 6x + 6.
展开后，合并含 x 的项：2x + 4x = 6x，常数项仍然是 6。完全化简后的代数式为 6x + 6。
9. Common Errors and How to Avoid Them | 常见错误及避免方法
One common mistake is adding unlike terms. For example, 3x + 4y is sometimes written incorrectly as 7xy. This is wrong because x and y are different variables, so the terms cannot be combined.
一个常见错误是把非同类项相加。例如，有人会把 3x + 4y 错误地写成 7xy。这是错误的，因为 x 和 y 是不同的变量，所以这些项不能合并。
Another mistake is changing the power of a variable. For example, 2x² + 3x² equals 5x², not 5x⁴. Only the coefficients are added; the exponent stays the same.
另一个错误是改变变量的指数。例如，2x² + 3x² 等于 5x²，而不是 5x⁴。只能把系数相加，指数保持不变。
| Common error | Correct working |
|---|---|
| 3x + 4y = 7xy | 3x + 4y cannot be simplified |
| 2x² + 3x² = 5x⁴ | 2x² + 3x² = 5x² |
| 5 − 2x = 3x | 5 − 2x cannot be simplified |
This table shows that a constant and a variable term cannot be combined, and that the exponent must not change when collecting like terms.
上表说明，常数项和含变量的项不能合并，而且合并同类项时指数不能改变。
10. Worked Example and Practice | 例题与练习
Worked example: Simplify 5x + 3y − 2x + 7y + 4. Identify the x-terms, y-terms and constants. The x-terms are 5x and −2x, giving 3x. The y-terms are 3y and 7y, giving 10y. The constant is 4. Therefore, the simplified expression is 3x + 10y + 4.
例题：化简 5x + 3y − 2x + 7y + 4。先确定含有 x 的项、含有 y 的项和常数项。含有 x 的项是 5x 和 −2x，合并得到 3x。含有 y 的项是 3y 和 7y，合并得到 10y。常数项是 4。因此，化简后的代数式为 3x + 10y + 4。
5x + 3y − 2x + 7y + 4 = 3x + 10y + 4
collect the b-terms: 4b + 2b = 6b. Finally, add the constant 6. The answer is 4a + 6b + 6.
现在尝试这道练习题：化简 7a + 4b − 3a + 2b + 6。先合并含有 a 的项：7a − 3a = 4a。再合并含有 b 的项：4b + 2b = 6b。最后加上常数项 6。答案是 4a + 6b + 6。
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