Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, simplifying expressions is one of the most essential skills. The process of combining like terms allows us to rewrite long, messy expressions into concise, equivalent ones. This article will guide you through every step, from identifying like terms to handling brackets and coefficients, with clear bilingual explanations.

在代数中，化简表达式是最重要的基本功之一。通过合并同类项，我们可以把冗长复杂的表达式改写为简洁的等价形式。本文将带你一步步学习，从识别同类项到处理括号与系数，并配有清晰的中英双语讲解。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable part, with the same exponents. The coefficient (the number in front) can be different. For example, 3x and 5x are like terms because both contain x¹. But 3x and 3x² are not like terms, because the powers of x are different.

同类项是指变量部分完全相同且指数也相同的项。系数（项前面的数字）可以不同。例如，3x 和 5x 是同类项，因为它们都含有 x¹。但 3x 和 3x² 不是同类项，因为 x 的幂次不同。

2. Identifying Like Terms | 识别同类项

To recognise like terms, check the following rules:

要识别同类项，请检查以下规则：

They must have the same variable(s) raised to the same power(s). | 它们必须具有相同的变量且幂次相同。
Constants (numbers without variables) are all like terms. | 常数（不带变量的数字）彼此都是同类项。
The order of multiplication does not matter: xy and yx are like terms. | 乘法顺序无关紧要：xy 和 yx 是同类项。

For example, in the expression 4a + 3b – a + 7, the terms 4a and -a are like; 3b has no other b-term; 7 is a constant. | 例如，在表达式 4a + 3b – a + 7 中，4a 和 -a 是同类项；3b 没有其他含 b 的项；7 是常数项。

3. The Distributive Property | 分配律

The distributive property is often used when simplifying expressions that contain brackets. It states that a(b + c) = ab + ac. This property allows us to remove brackets before combining like terms.

分配律经常用于化简含有括号的表达式。它指出 a(b + c) = ab + ac。这一性质允许我们在合并同类项之前先去括号。

a(b + c) = ab + ac

For example, 3(2x + 4) = 3 × 2x + 3 × 4 = 6x + 12. | 例如，3(2x + 4) = 3 × 2x + 3 × 4 = 6x + 12。

4. Adding and Subtracting Like Terms | 合并同类项的加减

To combine like terms, simply add or subtract their coefficients while keeping the variable part unchanged. For example: 2x + 3x = 5x and 4y – 2y = 2y.

合并同类项时，只需对其系数进行加减，同时保持变量部分不变。例如：2x + 3x = 5x，4y – 2y = 2y。

2x + 3x = (2 + 3)x = 5x

Notice that we never change the variable part. We only combine the numbers in front. | 注意，我们绝不改变变量部分，只合并前面的数字。

5. Combining Multiple Terms | 多项合并

When several like terms appear in an expression, combine them in any order. For example, 2a + 3a + 4a = (2 + 3 + 4)a = 9a.

当表达式中出现多个同类项时，可以按任意顺序合并。例如，2a + 3a + 4a = (2 + 3 + 4)a = 9a。

If there are different groups of like terms, handle each group separately. For instance, 5x + 2y – 3x + y = (5x – 3x) + (2y + y) = 2x + 3y.

如果存在不同组的同类项，则分别处理每一组。例如，5x + 2y – 3x + y = (5x – 3x) + (2y + y) = 2x + 3y。

6. Combining Terms with Coefficients | 带系数的项合并

Coefficients can be positive, negative, fractions or decimals. The same rules apply. For example: 1.5x – 0.5x = 1.0x = x, and ½y + ⅓y = (½ + ⅓)y = (3/6 + 2/6)y = 5/6y.

系数可以是正数、负数、分数或小数，规则相同。例如：1.5x – 0.5x = 1.0x = x，½y + ⅓y = (½ + ⅓)y = (3/6 + 2/6)y = 5/6y。

(1.5 – 0.5)x = 1x = x

Be careful with fractions: find a common denominator before adding. | 注意分数：相加前需要找到公分母。

7. Simplifying Expressions with Brackets | 含括号的化简

When an expression contains brackets, first expand using the distributive property, then combine like terms. Example: 2(3x + 1) + x = 6x + 2 + x = 7x + 2.

当表达式含有括号时，先用分配律展开，再合并同类项。示例：2(3x + 1) + x = 6x + 2 + x = 7x + 2。

2(3x + 1) + x = 6x + 2 + x = 7x + 2

If there is a minus sign before a bracket, distribute the negative sign to every term inside. Example: 5x – (2x – 3) = 5x – 2x + 3 = 3x + 3.

如果括号前有负号，需要把负号分配给括号内的每一项。示例：5x – (2x – 3) = 5x – 2x + 3 = 3x + 3。

8. Combining Like Terms with Different Variables | 不同变量项的处理

Terms with different variables or different powers cannot be combined. For example, 2x + 3y cannot be simplified further; it must remain as 2x + 3y. Similarly, x² + x cannot be combined.

含有不同变量或不同幂次的项不能合并。例如，2x + 3y 无法再化简，只能保持为 2x + 3y。同样，x² + x 也不能合并。

This is because adding unlike terms would violate the rules of algebra: 2x + 3y does not equal 5xy. | 这是因为，合并非同类型会违反代数规则：2x + 3y 不等于 5xy。

9. Common Mistakes | 常见错误

Avoid these frequent pitfalls when combining like terms:

合并同类项时，请避免以下常见陷阱：

Combining terms with different exponents, such as x and x². | 合并不同指数的项，例如 x 和 x²。
Multiplying coefficients and variables together during addition. In addition, only coefficients are added; the variable part stays the same. | 在加法中把系数和变量相乘。加法中只加系数，变量部分保持不变。
Forgetting to distribute minus signs through brackets. | 忘记将括号前的负号分配给括号内各项。
Treating different variables as the same, e.g. 2x + 3y ≠ 5xy. | 把不同变量当作同一个变量，例如 2x + 3y ≠ 5xy。

10. Worked Examples | 例题

Let us apply the techniques to full examples.

让我们将技巧应用于完整例题。

Example 1: Simplify 7a + 2b – 3a + 5b.

例题1：化简 7a + 2b – 3a + 5b。

Step 1: Group like terms: (7a – 3a) + (2b + 5b). | 步骤1：分组同类项：(7a – 3a) + (2b + 5b)。

Step 2: Combine each group: 4a + 7b. | 步骤2：合并每组：4a + 7b。

7a + 2b – 3a + 5b = 4a + 7b

Example 2: Simplify 4(2x – 3) – 2(x + 1).

例题2：化简 4(2x – 3) – 2(x + 1)。

Step 1: Expand: 8x – 12 – 2x – 2. | 步骤1：展开：8x – 12 – 2x – 2。

Step 2: Combine like terms: (8x – 2x) = 6x, and constants: -12 – 2 = -14. Result: 6x – 14. | 步骤2：合并同类项：(8x – 2x) = 6x，常数项：-12 – 2 = -14。结果：6x – 14。

4(2x – 3) – 2(x + 1) = 6x – 14

11. Practice Questions | 练习题

Try these on your own, then check the answers below.

请先自行尝试，再查看下方答案。

1. Simplify 3x + 5x + 2x. | 化简 3x + 5x + 2x。
2. Simplify 6y – 3y + 2. | 化简 6y – 3y + 2。
3. Simplify 2(a + 3) + 4a. | 化简 2(a + 3) + 4a。
4. Simplify 9x + 2y – 4x – 3y. | 化简 9x + 2y – 4x – 3y。

Answers: 1. 10x; 2. 3y + 2; 3. 6a + 6; 4. 5x – y. | 答案：1. 10x；2. 3y + 2；3. 6a + 6；4. 5x – y。

12. Summary | 总结

Combining like terms is a foundational skill in algebra. Always identify the variable part and exponent, add only the coefficients, and never forget to distribute signs through brackets. With careful practice, simplifying expressions becomes quick and accurate.

合并同类项是代数中的基础技能。务必识别变量部分与指数，只对系数进行加减，并且不要忘记在去括号时分配符号。通过认真练习，化简表达式将变得又快又准。

Like terms: same variable, same power → combine coefficients.

同类项：相同变量、相同指数 → 合并系数。

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Collecting Like Terms | 合并同类项