Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, combining like terms is one of the most essential skills you will ever learn. When an expression contains terms that share the same variable part, we can merge them into a single, simpler term. This process is called combining like terms, and it forms the backbone of simplifying algebraic expressions.

在代数学中，合并同类项是你将学到的最核心技能之一。当一个代数表达式包含具有相同变量部分的项时，我们可以把它们合并成一个更简的单项。这个过程称为合并同类项，它是化简代数表达式的基础。

1. Understanding Algebraic Terms | 理解代数项

An algebraic term is a single number, a variable, or a product of numbers and variables. For example, 5x, −3y², 7, and 4ab are all algebraic terms. Each term has two parts: a coefficient (the numerical factor) and a variable part. In the term 6x, the coefficient is 6 and the variable part is x.

代数项是单个数字、单个变量或数字与变量的乘积。例如，5x、−3y²、7 和 4ab 都是代数项。每一项都有两个部分：系数（数字因数）和变量部分。在项 6x 中，系数是 6，变量部分是 x。

When we look at an expression such as 3x + 5x − 2x, we see several terms. The expression is made up of three separate terms connected by plus and minus signs. Understanding how to identify each term individually is the first step toward combining them correctly.

当我们观察像 3x + 5x − 2x 这样的表达式时，会看到几个项。这个表达式由三个独立的项组成，它们之间用加号和减号连接。正确识别每一项是合并它们的第一步。

2. Identifying Like Terms | 识别同类项

Like terms are terms that have exactly the same variable part, including the same exponents. The coefficients may be different, but the variable part must match precisely. For instance, 4x and −7x are like terms because both have the variable part x. Similarly, 2y² and 9y² are like terms because both contain y².

同类项是指变量部分完全相同的项，包括相同的指数。系数可以不同，但变量部分必须精确匹配。例如，4x 和 −7x 是同类项，因为它们的变量部分都是 x。同样，2y² 和 9y² 是同类项，因为它们都包含 y²。

Terms that are not like terms cannot be combined. For example, 3x and 3y are not like terms because the variables are different. Likewise, 5x and 5x² are not like terms because the exponents on x differ. You can only add or subtract terms that are truly alike in their variable structure.

不是同类项的项不能合并。例如，3x 和 3y 不是同类项，因为变量不同。同理，5x 和 5x² 不是同类项，因为 x 上的指数不同。你只能对变量结构真正相同的项进行加减运算。

Like Terms: 4x and −7x ✔ | 2y² and 9y² ✔ | 3x and 3y ✘

5x and 5x² ✘ | 6xy and 2yx ✔ | 4 and 8 ✔

3. The Rule for Combining | 合并规则

The fundamental rule for combining like terms is simple: add or subtract the coefficients while keeping the variable part unchanged. If you have 2x + 3x, you add the coefficients 2 and 3 to get 5, then keep the variable x, resulting in 5x. The variable part never changes during combination.

合并同类项的基本规则很简单：对系数进行加减运算，同时保持变量部分不变。如果有 2x + 3x，将系数 2 和 3 相加得 5，然后保留变量 x，结果是 5x。变量部分在合并过程中永不改变。

This rule applies regardless of how many like terms are present. For example, 4x + 2x − 7x + x can be simplified by combining all four coefficients: 4 + 2 − 7 + 1 = 0. Since there is only one x, the result is 0x, which is simply 0. When a variable has no visible coefficient, its coefficient is understood to be 1.

这条规则适用于任意数量的同类项。例如，4x + 2x − 7x + x 可以通过合并所有四个系数来化简：4 + 2 − 7 + 1 = 0。因为变量都是 x，结果是 0x，即 0。当一个变量没有显式系数时，其系数默认为 1。

aₙxₙ + aₘxₙ = (aₙ + aₘ)xₙ

4. Combining Positive Terms | 合并正项

When all like terms have positive coefficients, the process is straightforward. Consider the expression 3m + 5m + 2m. All three terms share the variable m, so we add the coefficients: 3 + 5 + 2 = 10. The simplified expression is 10m. Notice that the variable m appears exactly once in the final answer.

当所有同类项的系数都是正数时，合并过程非常直接。考虑表达式 3m + 5m + 2m。三个项都共享变量 m，所以我们把系数相加：3 + 5 + 2 = 10。化简后的表达式是 10m。注意，变量 m 在最终答案中只出现一次。

Sometimes the terms may contain multiple variables. For instance, 2pq + 5pq + pq can be combined because every term has the same variable part pq. The coefficients 2, 5, and 1 (remember, pq has an understood coefficient of 1) add up to 8, giving us 8pq. The order of variables in a term does not matter, so pq and qp are the same.

有时项可能包含多个变量。例如，2pq + 5pq + pq 可以合并，因为每一项都有相同的变量部分 pq。系数 2、5 和 1（记住，pq 的默认系数为 1）相加等于 8，得到 8pq。项中变量的顺序无关紧要，所以 pq 和 qp 是相同的。

5. Combining Negative Terms | 合并负项

When like terms have negative coefficients, we treat the minus sign as part of the coefficient. For example, in the expression 7a − 3a − 2a, the coefficients are +7, −3, and −2. Adding them together gives 7 + (−3) + (−2) = 2, so the result is 2a. The negative signs are crucial and must not be overlooked.

当同类项的系数为负数时，我们把减号视为系数的一部分。例如，在表达式 7a − 3a − 2a 中，系数分别是 +7、−3 和 −2。将它们相加得到 7 + (−3) + (−2) = 2，所以结果是 2a。负号至关重要，绝不能忽略。

Consider a more complex example: 4x − 9x + 6x − x. The coefficients are 4, −9, +6, and −1 (since x has an implied coefficient of −1 when preceded by a minus sign). Adding: 4 − 9 + 6 − 1 = 0. Therefore, the entire expression simplifies to 0x = 0. This shows that combining like terms can sometimes eliminate the variable completely.

考虑一个更复杂的例子：4x − 9x + 6x − x。系数分别是 4、−9、+6 和 −1（因为 x 前面有减号时，隐含系数为 −1）。相加：4 − 9 + 6 − 1 = 0。因此，整个表达式化简为 0x = 0。这表明合并同类项有时会完全消除变量。

6. Combining Constant Terms | 合并常数项

Constant terms are numbers without any variables. All constant terms are like terms because they share the same “variable part,” which is none at all. Thus, 5, −3, 12, and 7 can all be combined together. For instance, 8 + 5 − 3 + 2 simplifies to 12, because 8 + 5 = 13, then 13 − 3 = 10, and finally 10 + 2 = 12.

常数项是不含任何变量的数字。所有常数项都是同类项，因为它们共享相同的”变量部分”——即什么都没有。因此，5、−3、12 和 7 都可以合并在一起。例如，8 + 5 − 3 + 2 化简为 12，因为 8 + 5 = 13，然后 13 − 3 = 10，最后 10 + 2 = 12。

In an expression that mixes variables and constants, such as 3x + 7 + 2x − 4, we must separate the like terms. Combine the x terms: 3x + 2x = 5x. Combine the constants: 7 − 4 = 3. The simplified expression is 5x + 3. Constant terms and variable terms cannot be combined with each other.

在混合了变量和常数的表达式中，如 3x + 7 + 2x − 4，我们必须分开合并同类项。合并 x 项：3x + 2x = 5x。合并常数项：7 − 4 = 3。化简后的表达式是 5x + 3。常数项和变量项之间不能相互合并。

7. Combining Terms with Exponents | 合并含指数的项

When variables have exponents, the exponents must match exactly for terms to be considered like terms. The expression 2x² + 5x² + x² contains three like terms, all with the variable part x². Adding the coefficients 2 + 5 + 1 = 8 gives us 8x². The exponent 2 remains unchanged throughout.

当变量带有指数时，指数必须完全一致，项才能被视为同类项。表达式 2x² + 5x² + x² 包含三个同类项，它们的变量部分都是 x²。将系数 2 + 5 + 1 = 8 相加得到 8x²。指数 2 在整个过程中保持不变。

However, x and x² are not like terms and cannot be combined. Consider the expression 4x² + 3x − 2x² + x. The x² terms are 4x² and −2x², which combine to give 2x². The x terms are 3x and x, which combine to give 4x. The final simplified expression is 2x² + 4x. We cannot go any further because x² and x are different types of terms.

然而，x 和 x² 不是同类项，不能合并。考虑表达式 4x² + 3x − 2x² + x。x² 项是 4x² 和 −2x²，合并后得到 2x²。x 项是 3x 和 x，合并后得到 4x。最终化简的表达式为 2x² + 4x。我们不能再继续化简，因为 x² 和 x 是不同类型的项。

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

Terms that contain more than one variable, such as xy or ab², follow the same rules. Like terms must have the exact same variables with the exact same exponents. The term 3xy and 5xy are like terms because both contain xy. They combine to give 8xy. The term 3xy and 5x²y are not like terms because the exponent on x differs.

包含多个变量的项，如 xy 或 ab²，遵循同样的规则。同类项必须具有完全相同的变量和完全相同的指数。项 3xy 和 5xy 是同类项，因为两者都包含 xy。它们合并得到 8xy。项 3xy 和 5x²y 不是同类项，因为 x 的指数不同。

Let us simplify a longer expression: 2ab + 3a − 4ab + b + 6ab. First, identify the like terms. The ab terms are 2ab, −4ab, and 6ab; combining them gives 4ab. The term 3a has no other a terms, and b has no other b terms. The simplified expression is 4ab + 3a + b. When terms cannot be combined, they simply remain as they are.

让我们化简一个更长的表达式：2ab + 3a − 4ab + b + 6ab。首先，识别同类项。ab 项有 2ab、−4ab 和 6ab；合并它们得到 4ab。项 3a 没有其他 a 项，b 也没有其他 b 项。化简后的表达式是 4ab + 3a + b。当无法合并时，项就原样保留。

9. Simplifying Expressions Step by Step | 逐步化简表达式

To simplify any expression reliably, follow a systematic procedure. First, rewrite the expression, removing any unnecessary brackets. Second, identify groups of like terms and mark them with different symbols. Third, combine the coefficients within each group. Finally, write the simplified expression, ordering terms conventionally from highest power to lowest.

要可靠地化简任何表达式，请遵循系统化的步骤。第一步，重写表达式，去掉任何不必要的括号。第二步，识别同类项组，用不同符号标记它们。第三步，合并每组内的系数。最后，写出化简后的表达式，按指数从高到低的惯例排列各项。

Let us apply this to a full example. Simplify: 5x − 3y + 2x + 7y − x + 4.

让我们将此应用于一个完整示例。化简：5x − 3y + 2x + 7y − x + 4。

Step 1 — Group x terms: 5x + 2x − x = 6x

Step 2 — Group y terms: −3y + 7y = 4y

Step 3 — Constant: 4

Final Answer: 6x + 4y + 4

10. Application in Equations | 在方程中的应用

Combining like terms is not only useful for simplifying expressions but also essential for solving equations. When solving an equation like 3x + 5 + 2x − 8 = 17, we first combine like terms on each side. The left side becomes 5x − 3. The equation simplifies to 5x − 3 = 17, which we can solve in two more steps: add 3 to both sides to get 5x = 20, then divide by 5 to get x = 4.

合并同类项不仅用于化简表达式，而且对方程求解至关重要。在解方程 3x + 5 + 2x − 8 = 17 时，我们首先在等式两边分别合并同类项。左边变为 5x − 3。方程化简为 5x − 3 = 17，这可以通过两步求解：两边同时加 3 得到 5x = 20，然后除以 5 得到 x = 4。

In equations with variables on both sides, combining like terms is even more important. Take 4x − 7 = 2x + 9. Subtract 2x from both sides to get 2x − 7 = 9. Then add 7 to both sides to get 2x = 16, and divide by 2 to obtain x = 8. Without combining like terms, such equations would be impossible to simplify to a solvable form.

在两边都有变量的方程中，合并同类项更为重要。以 4x − 7 = 2x + 9 为例。两边同时减去 2x 得到 2x − 7 = 9。然后两边加 7 得到 2x = 16，再除以 2 得到 x = 8。如果不合并同类项，这样的方程就无法化简为可解的形式。

11. Common Mistakes | 常见错误

One of the most common mistakes students make is combining terms that are not like terms. Writing 3x + 5y = 8xy is incorrect because x and y are different variables. Another frequent error is forgetting the hidden coefficient of 1: in expressions like x + 3x, many students only add the visible coefficients and write 3x instead of 4x.

学生最常犯的错误之一是合并并非同类项的项。写 3x + 5y = 8xy 是不正确的，因为 x 和 y 是不同的变量。另一个常见错误是忘记隐藏的系数 1：在 x + 3x 这类表达式中，许多学生只加可见的系数，写出 3x 而不是 4x。

A third common mistake is mishandling subtraction signs. In the expression 5a − 2a + 3a, some students mistakenly combine 5 + 2 + 3 = 10a, forgetting that the middle term is negative. The correct calculation is 5 − 2 + 3 = 6a. Always pay careful attention to the sign that precedes each term when collecting coefficients.

第三个常见错误是处理减号不当。在表达式 5a − 2a + 3a 中，有些学生错误地合并 5 + 2 + 3 = 10a，忘记了中间项是负的。正确的计算是 5 − 2 + 3 = 6a。在收集系数时，始终注意每个项前面的符号。

Incorrect \| 错误	Correct \| 正确	Reason \| 原因
3x + 5y = 8xy	3x + 5y (cannot combine)	Different variables
x + 3x = 3x	x + 3x = 4x	Coefficient of x is 1
5a − 2a + 3a = 10a	5a − 2a + 3a = 6a	Middle term is negative
4x² + 2x = 6x	4x² + 2x (cannot combine)	Exponents differ

12. Practice Problems | 练习题目

To master combining like terms, consistent practice is essential. Below are several problems of increasing difficulty. Try each one on your own before checking the answers. Remember the golden rule: keep the variable part unchanged and add or subtract only the coefficients.

要掌握合并同类项，持续练习必不可少。下面是几个难度递增的题目。先独立尝试每道题，再核对答案。记住黄金法则：保持变量部分不变，只对系数进行加减运算。

Problem 1: Simplify 7x + 3x − 2x

题目 1：化简 7x + 3x − 2x

Answer: 8x
Problem 2: Simplify 4a + 9b − 2a + b

题目 2：化简 4a + 9b − 2a + b

Answer: 2a + 10b
Problem 3: Simplify 5x² + 3x − 2x² + 7 − 4

题目 3：化简 5x² + 3x − 2x² + 7 − 4

Answer: 3x² + 3x + 3
Problem 4: Simplify 6mn − 3m + 2mn + 8m − mn

题目 4：化简 6mn − 3m + 2mn + 8m − mn

Answer: 7mn + 5m
Problem 5: Solve 2x + 5 + 3x − 5 = 20

题目 5：解方程 2x + 5 + 3x − 5 = 20

Answer: x = 4

Combining like terms is a fundamental algebraic skill that appears throughout mathematics, from basic arithmetic to advanced calculus. Master it well, and every future topic becomes more manageable.

合并同类项是一项基础的代数技能，贯穿于从基础算术到高等微积分的整个数学体系。熟练掌握它，未来的一切数学主题都将变得更加轻松。

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Combining Like Terms | 合并同类项