Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, combining like terms is a fundamental skill that allows us to simplify expressions, solve equations, and understand mathematical relationships more clearly. Like terms are terms that have exactly the same variable parts, including the same exponents.

在代数中，合并同类项是一项基础技能，它帮助我们化简表达式、解方程，并更清晰地理解数学关系。同类项是指变量部分完全相同（包括相同的指数）的项。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that contain the same variable(s) raised to the same power. For example, 3x and 5x are like terms because both contain the variable x to the first power. However, 3x and 3x² are not like terms because the exponents are different.

同类项是含有相同变量且变量指数相同的项。例如，3x 和 5x 是同类项，因为它们都含有一次的变量 x。然而，3x 和 3x² 不是同类项，因为指数不同。

Examples of like terms: 2a, 7a, −4a

同类项的例子：2a、7a、−4a
Examples of unlike terms: 2a, 2b, 2a²

非同类的例子：2a、2b、2a²
Constant terms (numbers without variables) are all like terms: 3, −8, 15

常数项（不含变量的数字）都是同类项：3、−8、15

2. Why Do We Combine Like Terms? | 为什么合并同类项？

Combining like terms makes expressions shorter and easier to work with. It is the first step in solving equations, simplifying formulas, and comparing algebraic quantities. Without this skill, expressions become long and difficult to interpret.

合并同类项可以让表达式更简洁、更容易处理。它是解方程、化简公式和比较代数量的第一步。没有这项技能，表达式会变得冗长且难以理解。

For example, the expression 4x + 3 + 2x + 5 can be simplified to 6x + 8. This simplified form is much easier to evaluate or substitute into.

例如，表达式 4x + 3 + 2x + 5 可以简化为 6x + 8。这种简化形式更容易求值或代入。

3. The Golden Rule: Same Variable, Same Power | 黄金法则：变量相同，指数相同

Only terms with exactly the same variable and exactly the same exponent can be combined. This means the variable letter must match, and the power (index) must match. The coefficients (numbers in front) are added or subtracted, while the variable part stays unchanged.

只有变量完全相同且指数完全相同的项才能合并。这意味着变量字母必须匹配，幂（指数）也必须匹配。系数（前面的数字）相加或相减，变量部分保持不变。

5x + 3x = (5 + 3)x = 8x

Here, the coefficient 5 and 3 are added, and x remains as the variable part.

在这里，系数 5 和 3 相加，x 保留为变量部分。

4. Combining Constant Terms | 合并常数项

Constants are numbers without variables. All constants are like terms with each other. To combine them, simply perform the arithmetic operation indicated.

常数是不含变量的数字。所有常数之间互为同类项。要合并它们，只需执行所指示的算术运算。

Consider the expression: 7 + 2x + 3 − 5. The constants 7, 3, and −5 combine to give 5. The expression simplifies to 2x + 5.

考虑表达式：7 + 2x + 3 − 5。常数 7、3 和 −5 合并得到 5。表达式简化为 2x + 5。

Example: 12 + 4a − 7 + 3a = (4a + 3a) + (12 − 7) = 7a + 5

例：12 + 4a − 7 + 3a = (4a + 3a) + (12 − 7) = 7a + 5

5. Visualising with Shapes | 用图形形象化理解

Some students find it helpful to think of like terms as similar objects. If x represents one apple, then 3x + 5x means 3 apples plus 5 apples, giving 8 apples. But x² cannot be added to x because it represents a different object, such as one box of apples.

有些学生发现把同类项看作相似的物体很有帮助。如果 x 代表一个苹果，那么 3x + 5x 表示 3 个苹果加 5 个苹果，得到 8 个苹果。但 x² 不能与 x 相加，因为它代表不同的物体，比如一箱苹果。

This analogy helps reinforce why 2x + 3y cannot be simplified: we cannot add apples and oranges together unless we count them as separate groups.

这个类比帮助理解为什么 2x + 3y 不能简化：除非把它们作为不同的组分别计数，否则我们不能把苹果和橘子加在一起。

6. Combining Terms with Different Powers | 合并不同指数的项

Terms with different exponents must be treated separately. For example, in the expression x² + 4x + 3x² + 2x, we combine x² with 3x² to get 4x², and 4x with 2x to get 6x. The final simplified form is 4x² + 6x.

指数不同的项必须分别处理。例如，在表达式 x² + 4x + 3x² + 2x 中，我们将 x² 与 3x² 合并得到 4x²，将 4x 与 2x 合并得到 6x。最终简化形式为 4x² + 6x。

x² + 4x + 3x² + 2x = (1 + 3)x² + (4 + 2)x = 4x² + 6x

Notice that the exponents do not change when we combine; only the coefficients are added.

请注意，合并时指数不会改变；只有系数相加。

7. Combining Terms with Multiple Variables | 合并多变量项

When terms contain more than one variable, the same rule applies. For instance, 3xy and 5xy are like terms, but 3xy and 3x are not. The variables must be identical in both type and exponent.

当项包含多个变量时，同样的规则适用。例如，3xy 和 5xy 是同类项，但 3xy 和 3x 不是。变量必须在类型和指数上完全一致。

Simplify: 2ab + 7ab − 3ab = (2 + 7 − 3)ab = 6ab

化简：2ab + 7ab − 3ab = (2 + 7 − 3)ab = 6ab
Simplify: 4x²y + 2xy² + x²y − xy² = 5x²y + xy²

化简：4x²y + 2xy² + x²y − xy² = 5x²y + xy²

In the second example, x²y and xy² are different because the exponents are attached to different variables, so they are not combined.

在第二个例子中，x²y 和 xy² 是不同的，因为指数附着在不同的变量上，所以它们不能合并。

8. Using the Distributive Property | 运用分配律

Sometimes, before combining like terms, we must first expand brackets using the distributive property. The distributive property states that a(b + c) = ab + ac.

有时，在合并同类项之前，我们必须先用分配律展开括号。分配律指出 a(b + c) = ab + ac。

Consider: 3(2x + 4) + 5x. First expand: 3 × 2x = 6x and 3 × 4 = 12. The expression becomes 6x + 12 + 5x. Now combine 6x and 5x to get 11x + 12.

考虑：3(2x + 4) + 5x。先展开：3 × 2x = 6x，3 × 4 = 12。表达式变为 6x + 12 + 5x。现在合并 6x 和 5x 得到 11x + 12。

3(2x + 4) + 5x = 6x + 12 + 5x = 11x + 12

9. Step-by-Step Method | 分步解法

A reliable method for combining like terms involves four clear steps. These steps keep the process organised and prevent mistakes.

一个可靠的合并同类项方法包含四个清晰的步骤。这些步骤使过程有条理并防止错误。

Step	Example with 4a + 2b − a + 3b
1. Identify like terms	4a and −a are like terms; 2b and 3b are like terms.
2. Group them together	(4a − a) + (2b + 3b)
3. Add or subtract coefficients	(3a) + (5b)
4. Write the final expression	3a + 5b

This method works for any expression, no matter how many terms it contains.

这个方法适用于任何表达式，无论它包含多少项。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make mistakes when variables have coefficients of 1 or −1, or when negative signs are involved. Pay special attention to the following points.

当变量系数为 1 或 −1 时，或涉及负号时，学生常犯错误。请特别注意以下几点。

Mistake: Combining 2x + 3x² as 5x³. Correct: the exponents do not change, so 2x + 3x² cannot be combined.

错误：将 2x + 3x² 合并为 5x³。正确：指数不变，因此 2x + 3x² 不能合并。
Mistake: Writing x as 0x. Correct: x means 1x, so x + x = 2x.

错误：把 x 写成 0x。正确：x 表示 1x，所以 x + x = 2x。
Tip: Always combine constants first, then variable terms.

技巧：总是先合并常数项，再合并变量项。
Tip: Use a highlighter to mark matching variable parts before combining.

技巧：合并前用荧光笔标出匹配的变量部分。

11. Worded Problems Involving Like Terms | 涉及同类项的应用题

Like terms appear frequently in real-world problems. For example, if a rectangle has length 2x + 1 and width x + 3, its perimeter can be written as 2(2x + 1) + 2(x + 3). Expanding gives 4x + 2 + 2x + 6, which simplifies to 6x + 8.

同类项经常出现在实际应用题中。例如，如果一个矩形的长为 2x + 1，宽为 x + 3，它的周长可以写成 2(2x + 1) + 2(x + 3)。展开得到 4x + 2 + 2x + 6，化简为 6x + 8。

Another example: three friends have x − 2, 2x + 5, and 3x − 1 stickers respectively. The total number of stickers is (x + 2x + 3x) + (−2 + 5 − 1) = 6x + 2.

另一个例子：三个朋友分别有 x − 2、2x + 5 和 3x − 1 张贴纸。贴纸总数为 (x + 2x + 3x) + (−2 + 5 − 1) = 6x + 2。

12. Practice Exercises | 巩固练习

The best way to master combining like terms is through consistent practice. Try the following exercises and check your answers below.

掌握合并同类项的最佳方法就是持续练习。尝试以下练习并核对下方答案。

Simplify 5m + 3m − 2m.

化简 5m + 3m − 2m。
Simplify 7p + 3 − 2p + 8.

化简 7p + 3 − 2p + 8。
Simplify 4x² + 3x − x² + 2x.

化简 4x² + 3x − x² + 2x。
Simplify 2(a + 4) + 3a.

化简 2(a + 4) + 3a。

Answers: 1. 6m 2. 5p + 11 3. 3x² + 5x 4. 5a + 8

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Combining Like Terms | 合并同类项