Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms that can be simplified by combining those which are similar. This process is called collecting like terms. It is one of the most fundamental skills in simplifying algebraic expressions and solving equations.

在代数中，表达式往往包含多个项，可以通过合并其中相似的项来简化。这个过程叫作合并同类项。它是简化代数表达式和求解方程时最基本、最重要的技能之一。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable part, including the same power (exponent). The numerical coefficients do not need to be the same. For example, 3x and 5x are like terms because both contain the variable x to the first power.

同类项是指变量部分完全相同的项，包括相同的指数（幂）。数字系数不需要相同。例如，3x 和 5x 是同类项，因为它们都包含一次方的变量 x。

Similarly, 4y² and −2y² are like terms because they both contain y squared. However, 2x and 2x² are not like terms, because the powers of x are different.

类似地，4y² 和 −2y² 是同类项，因为它们都包含 y 的平方。然而，2x 和 2x² 不是同类项，因为 x 的指数不同。

2. Identifying Coefficients and Variables | 识别系数和变量

Every term in an algebraic expression consists of two parts: a coefficient (the number in front) and a variable part (the letter or letters). For example, in the term 7ab, the coefficient is 7 and the variable part is ab.

代数表达式中的每一项都由两部分组成：系数（前面的数字）和变量部分（字母或字母组合）。例如，在项 7ab 中，系数是 7，变量部分是 ab。

When collecting like terms, we only combine the coefficients, while keeping the variable part unchanged. For instance, 2x + 3x = (2 + 3)x = 5x.

合并同类项时，我们只合并系数，而保持变量部分不变。例如，2x + 3x = (2 + 3)x = 5x。

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

3. The Rule of Collecting Like Terms | 合并同类项的规则

The basic rule is simple: add or subtract the coefficients of like terms, and keep the variable part exactly the same. If a term has no visible coefficient, its coefficient is understood to be 1.

基本规则很简单：对同类项的系数进行加减，并保持变量部分完全不变。如果一个项没有显式系数，其系数默认为 1。

For example, x + 4x = 1x + 4x = 5x. In the same way, y − 3y = (1 − 3)y = −2y.

例如，x + 4x = 1x + 4x = 5x。同样地，y − 3y = (1 − 3)y = −2y。

Always remember that only the coefficients change; the variable part never changes when you collect like terms.

请务必记住：只有系数发生变化；合并同类项时，变量部分永远不变。

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

When adding like terms, simply add their coefficients. For example, 2a + 3a + a = (2 + 3 + 1)a = 6a.

合并同类项做加法时，直接把系数相加。例如，2a + 3a + a = (2 + 3 + 1)a = 6a。

When subtracting, be careful with negative signs. For example, 8b − 5b = 3b, and 4c − 7c = −3c.

做减法时，要小心负号。例如，8b − 5b = 3b，而 4c − 7c = −3c。

Mixed signs can appear together. For instance, 5p − 2p + 3p = (5 − 2 + 3)p = 6p.

正负号可以混合出现。例如，5p − 2p + 3p = (5 − 2 + 3)p = 6p。

5. Dealing with Different Variables | 处理不同变量

If an expression contains different variables, you must treat each group separately. Terms with x can only be combined with other x terms; terms with y can only be combined with other y terms; and so on.

如果表达式中包含不同的变量，你必须将每一组分开处理。含 x 的项只能与其它含 x 的项合并；含 y 的项只能与其它含 y 的项合并；依此类推。

For example, in 3x + 2y + 4x − y, combine the x terms: 3x + 4x = 7x; combine the y terms: 2y − y = y. The simplified expression is 7x + y.

例如，在 3x + 2y + 4x − y 中，合并 x 项：3x + 4x = 7x；合并 y 项：2y − y = y。化简后的表达式是 7x + y。

Notice that 7x and y are not like terms, so they cannot be combined further.

注意，7x 和 y 不是同类项，因此不能继续合并。

6. Combining Constants | 合并常数项

Constant terms, which have no variable part, are also like terms with each other. For example, 5 and −3 are like terms because they are both constants.

常数项没有变量部分，它们彼此之间也是同类项。例如，5 和 −3 是同类项，因为它们都是常数。

Combine constants in the same way: 4 + 7 − 2 = 9. In an expression like 2x + 5 + 3x − 2, combine the constants: 5 − 2 = 3, so the result is 5x + 3.

用同样的方法合并常数：4 + 7 − 2 = 9。在类似 2x + 5 + 3x − 2 的表达式中，合并常数：5 − 2 = 3，因此结果是 5x + 3。

Always look for constant terms separately from variable terms.

始终要把常数项与变量项分开处理。

7. Simplifying Expressions with Multiple Terms | 简化多项表达式

A typical expression may have several different types of terms. The goal is to rearrange the expression so that like terms are next to each other, then combine them.

一个典型的表达式可能包含几种不同类型的项。我们的目标是把表达式重新排列，使同类项相邻，然后再合并。

For example, simplify 3x² + 5x − 2x² + 7 − 3x + 1. First, group x² terms: 3x² − 2x² = x². Group x terms: 5x − 3x = 2x. Group constants: 7 + 1 = 8. The simplified expression is x² + 2x + 8.

例如，化简 3x² + 5x − 2x² + 7 − 3x + 1。首先，合并 x² 项：3x² − 2x² = x²。合并 x 项：5x − 3x = 2x。合并常数项：7 + 1 = 8。化简后的表达式是 x² + 2x + 8。

When writing the final answer, it is conventional to place terms in descending order of power: x², then x, then constant.

书写最终答案时，通常按幂次从高到低排列：先 x²，再 x，最后常数项。

8. Multiplication and Division: Not Collecting | 乘除：不是合并

Collecting like terms only applies to addition and subtraction. When you multiply or divide terms, different rules are used. For example, x × x = x², and 2x × 3x = 6x², not 5x.

合并同类项只适用于加减运算。当进行乘法或除法时，要使用不同的规则。例如，x × x = x²，并且 2x × 3x = 6x²，而不是 5x。

Similarly, 6x² ÷ 2x = 3x. These are not examples of collecting like terms; they are examples of simplifying products or quotients.

同样地，6x² ÷ 2x = 3x。这些不是合并同类项的例子，而是简化积或商的例子。

Be sure not to confuse the two operations. You can only collect like terms when the sign between them is plus or minus.

请一定不要混淆这两种运算。只有当项之间的符号是加号或减号时，才能合并同类项。

9. Common Mistakes | 常见错误

One common mistake is combining terms that are not alike, such as writing 3x + 2y = 5xy. This is incorrect because x and y are different variables.

一个常见错误是合并不是同类项的项，例如写成 3x + 2y = 5xy。这是不正确的，因为 x 和 y 是不同的变量。

Another mistake is changing the variable part. For example, 2x + 3x should not become 5x²; it must remain 5x.

另一个错误是改变变量部分。例如，2x + 3x 不应变成 5x²；它必须保持 5x。

A third mistake is forgetting that a letter without a number has a coefficient of 1. For example, x + 3x is 4x, not 3x.

第三个错误是忘记没有数字的字母其系数是 1。例如，x + 3x 是 4x，而不是 3x。

Finally, do not forget negative signs. For example, 4a − 6a = −2a, not 2a.

最后，不要忘记负号。例如，4a − 6a = −2a，而不是 2a。

10. Worked Examples | 例题详解

Let us work through several examples step by step.

让我们一步一步地完成几个例题。

Example 1: Simplify 7m + 2n − 3m + 5n.

例题 1：化简 7m + 2n − 3m + 5n。

Combine the m terms: 7m − 3m = 4m. Combine the n terms: 2n + 5n = 7n. Answer: 4m + 7n.

合并 m 项：7m − 3m = 4m。合并 n 项：2n + 5n = 7n。答案：4m + 7n。

Example 2: Simplify 4x² + 3x − x² + 2x + 6.

例题 2：化简 4x² + 3x − x² + 2x + 6。

Combine x² terms: 4x² − x² = 3x². Combine x terms: 3x + 2x = 5x. The constant 6 stays. Answer: 3x² + 5x + 6.

合并 x² 项：4x² − x² = 3x²。合并 x 项：3x + 2x = 5x。常数 6 保持不变。答案：3x² + 5x + 6。

Example 3: Simplify 5a − 2b − 3a + 4b + 7 − 2.

例题 3：化简 5a − 2b − 3a + 4b + 7 − 2。

Combine a terms: 5a − 3a = 2a. Combine b terms: −2b + 4b = 2b. Combine constants: 7 − 2 = 5. Answer: 2a + 2b + 5.

合并 a 项：5a − 3a = 2a。合并 b 项：−2b + 4b = 2b。合并常数项：7 − 2 = 5。答案：2a + 2b + 5。

11. Practice and Summary | 练习与总结

To master collecting like terms, practice identifying the variable part and carefully performing addition and subtraction on coefficients. Work through many examples until the process becomes automatic.

要掌握合并同类项，请练习识别变量部分，并对系数仔细地进行加减运算。大量练习例题，直到这个过程变得熟练自然。

Remember these key points: (1) only like terms can be combined; (2) keep the variable part unchanged; (3) combine constants separately; (4) watch the signs.

请记住这些要点：（1）只有同类项才能合并；（2）保持变量部分不变；（3）单独合并常数项；（4）注意符号。

Try to simplify this expression as a final check: 6x + 3y − 2x + y − 4. The answer is 4x + 4y − 4.

作为最终练习，请尝试化简这个表达式：6x + 3y − 2x + y − 4。答案是 4x + 4y − 4。

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Collecting Like Terms | 合并同类项